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58beb316c7ffe6a68746e2ebd3689ecffd94dcf9 | subsection | 21 | 77 | Polynomials of degree | Then, by Lemma REF , we haved(k(m)+1,m) > 1 + c_4(k(m)+1,m).If we let s=q_1q_2\cdots q_{k(m)} and t=q_{k(m)+1}\cdots q_m
we have
2\phi (t)/t=d(k(m)+1,m), andc_4(k(m)+1,m)&=&8\cdot \sqrt{\frac{q_1q_2\cdots q_{k(m)}}{q_{k(m)+1}q_{k(m)+2}\cdots q_m}}\\
&=&\frac{8s}{\sqrt{q_1q_2\cdots q_m}}\ge \frac{8s}{\sqrt{p-1}}Since s ... | {
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"arxiv_id": "",
"doi": "10.2140/pjm.1982.98.123",
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"raw": "D. J. Madden and W. Y. Vélez, Polynomials that represent quadratic residues at primitive roots, Pacific J. Math. 98 (1982), 123–137.",
"s... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
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291e1b3ee39d2285f15bd27e96f8e89c111ea687 | subsection | 22 | 77 | Polynomials of degree | Moreover,
amongst these exceptions only for
(p,k) \in \lbrace (13,1),(37,28),(61,18)\rbrace
there exists \bar{\xi }\in S^*\cap S^*+1
such that k=- 2(1-2\bar{\xi }). In particular,
\bar{\xi }=4 for (p,k)=(13,1),
\bar{\xi }=26 for (p,k)=(37,28),
and \bar{\xi }=5 for (p,k)=(61,18). | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
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44cfbc50c8a14707c6f157f5b45f95b4453043e5 | subsection | 23 | 77 | Vertex-transitive graphs of order | The goal of this paper is to prove that the Petersen
graph is the only connected vertex-transitive graph of order a product of two primes without a Hamilton cycle.
Recall that vertex-transitive graphs of prime-squared order are necessarily Cayley graphs of abelian groups.
The existence of Hamilton cycles in such graphs... | {
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"raw": "D. Marušič, Hamilonian circuits in Cayley graphs, Discrete Math. 46 (1983), 49–54.",
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"arxiv_i... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
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"Dragan Marusic"
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d0aa9f21ba075d10f2b218ef587d5062b893bfbd | subsection | 24 | 77 | Vertex-transitive graphs of order | If \gcd (c,q)=1, where c=a/\gcd (a,q)
and a is the order of \alpha \in \mathbb {Z}_p^*,
then it follows by that X
is a Cayley graph of
the group \langle \rho ,\sigma ^c\rangle =\langle \rho \rangle \rtimes \langle \sigma ^c\rangle .
Thus in this case the result about existence
of Hamilton cycles in Cayley graphs of sem... | {
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"arxiv_id": "",
"doi": "10.4153/cjm-1982-020-8",
"end": 233,
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"source_ref_id": "2b23bb6... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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beb29c56fc38620b2800324cf1237c64b7ee22d7 | subsection | 25 | 77 | Vertex-transitive graphs of order | Since the above actions are primitive
and hence the corresponding basic orbital graphs are connected,
it suffices to prove the existence of Hamilton cycles solely in
basic orbital graphs of these actions.
This is done in the next three sections.Graphs arising from the actions in the first four rows and the last row of
... | {
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"arxiv_id": "",
"doi": "10.1016/0012-365x(89)90157-x",
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"raw": "B. Alspach, Lifting Hamilton cycles of quotient graphs, Discrete Math. 78 (1989), 25–36.",
"source_ref_id": "7c0067fe920d8045ec54c42... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
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44e2a9396f2383f02bfb8922848764128c6caaa5 | subsection | 26 | 77 | Vertex-transitive graphs of order | Since the quotient graph X_\rho has a Hamilton cycle
containing a double edge and since 13 is a prime,
this cycle lifts to a Hamilton
cycle in the original graph X (see Figure REF ).
[Figure: A vertex-transitive graph arising from theaction of \textrm {PSL}(2,13) on cosets of D_{12}given in Frucht's notation with resp... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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4239f2892d7c6b522cbf126193b8305fc6d7f753 | subsection | 27 | 77 | Vertex-transitive graphs of order | Let \mathcal {P}= \lbrace S_1, S_2, \ldots , S_m\rbrace
be the set of orbits of \rho , and let \pi : X \rightarrow X_\mathcal {P}
be the corresponding projection of X to its quotient X_\mathcal {P}.
For a (possibly closed) path
W = S_{i_1}S_{i_2}\ldots S_{i_k} in X_\mathcal {P} we let
the lift of W be
the set of all p... | {
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"source... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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99ee6007497ef90983d5ad3e1f698c5888a17f94 | subsection | 28 | 77 | Graphs arising from certain small rank/degree group actions | We deal here with the first four rows and the last row
of Table REF of which the first three rows
correspond to
groups of rank 3 and 4.In the first proposition we show the existence of Hamilton cycles in
the graphs arising from the first three rows of Table REF .
With the exception of one of the graphs arising from the... | {
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"start... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
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"math.CO"
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eeb8de59c61222f1a795380a253345ca5cafec77 | subsection | 29 | 77 | Graphs arising from certain small rank/degree group actions | More precisely, one suborbit is of length 6,
one of length 10, two of length 12, four of length 20,
five of length 30 and 27 of length 60. Of the latter
8 are non-self-paired.
In Figure REF
the graph of valency 6 is shown together with a Hamilton
cycle in the quotient graph with respect to a semiregular
automorphism o... | {
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"arxiv_id": "",
"doi": "10.1006/jsco.1996.0125",
"end": 579,
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"source_ref... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
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c8c6dc2ca8a9af2aa24a2ff79fc98df9f9feddee | subsection | 30 | 77 | Actions of | In this section the existence of Hamilton cycles
in basic orbital graphs arising
from the group action \hbox{\rm PSL}(2,q^2)
on the cosets of \hbox{\rm PGL}(2,q) given in Row 5 of Table REF
is considered. The following group-theoretic result due to
Manning will
be needed in this respect.Proposition 6.1
Let G be a tr... | {
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} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
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560772a7b6ddd7f6247693117ace5100c10110e1 | subsection | 31 | 77 | Actions of | (Note that there is only one conjugacy class of subgroups in G
isomorphic to N.)
A subgroup of N is isomorphic to one of the following groups
\mathbb {Z}_q^2\rtimes \mathbb {Z}_{q-1}, \mathbb {Z}_q, \mathbb {Z}_q\rtimes \mathbb {Z}_{q-1},
\mathbb {Z}_q\rtimes \mathbb {Z}_l, where 2\le l<q-1, and
\mathbb {Z}_l, where l ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
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8a6c1d3663c62363f3b12eed532338d0d6e6d769 | subsection | 32 | 77 | Actions of | Since |\mathcal {H}|=q(q^2+1)/2 and since
the length of an orbit of N
on \mathcal {H} with coset stabilizer isomorphic to \mathbb {Z}_2 or to \mathbb {Z}_1
equals, respectively, \frac{q^2(q-1)}{2} and q^2(q-1), we have\frac{q(q^2+1)}{2} = q\frac{q+1}{2} + a\frac{q^2(q-1)}{2} + bq^2(q-1),where a is the number of orbits ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
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30f867f9079f28eb6a3deed179be664f5e9020c2 | subsection | 33 | 77 | Actions of | Formal proofs are given in Propositions REF and REF .It follows from the previous paragraph that we only need to prove
that the subgraph of X_\mathcal {P}
induced on the large orbit of N contains a Hamilton cycle with
at least one double edge in the corresponding multigraph
or two components each of which contains a Ha... | {
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primes | [
"Shaofei Du",
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"Dragan Marusic"
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99bacdfbc15ae1f4cf048c21994151f7a439965b | subsection | 34 | 77 | Actions of | One can now see that
\cdot \colon W\otimes \overline{W}\times \hbox{\rm SL}(2,q^2) \rightarrow W\otimes \overline{W}
defined by the rule(\mathbf {w}\otimes \mathbf {w}^{\prime })\cdot g=
\mathbf {w}g\otimes \mathbf {w}^{\prime } * g=
\mathbf {w}g\otimes \mathbf {w}^{\prime }g^\phiis an action of \hbox{\rm SL}(2,q^2) on... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
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3ceccf63eade17982cd53ca360ff77f53f5577fb | subsection | 35 | 77 | Actions of | It follows that
G fixes a non-degenerate symmetric bilinear
form of W\otimes \overline{W} defined by the rule(\mathbf {w}_1^{\prime }\otimes \mathbf {w}_2^{\prime },
\mathbf {w}_1^{\prime \prime }\otimes \mathbf {w}_2^{\prime \prime })=
f(\mathbf {w}_1^{\prime }, \mathbf {w}_1^{\prime \prime })f(\mathbf {w}_2^{\prime }... | {
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} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
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"Dragan Marusic"
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0a147356f81bd706a7a3f0da8cac2c87275bd840 | subsection | 36 | 77 | Actions of | The following result characterizing suborbits of
the action of G on the cosets of \hbox{\rm PGL}(2,q)
in the context of the action of
P\Omega ^{-}(4, q) on \Omega was proved in .Proposition 6.3
For any \langle \mathbf {v}\rangle \in \Omega , the nontrivial suborbits of the action of G on \Omega
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primes | [
"Shaofei Du",
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4d1000f341ad171cd2bc8dc1ceb6343d5ce28c34 | subsection | 37 | 77 | Actions of | For k\in F_q we have\mathbf {v}_1\rho ^k&=&\mathbf {v}_1,\\
\mathbf {v}_2\rho ^k&=&k^2\mathbf {v}_1+\mathbf {v}_2+k\mathbf {v}_3,\\
\mathbf {v}_3\rho ^k&=&2k\mathbf {v}_1+\mathbf {v}_3,\\
\mathbf {v}_4\rho ^k&=&\mathbf {v}_4,and so \rho ^k maps the vector x=\sum _{i=1}^4x_i\mathbf {v}_i\in V to\mathbf {x}\rho ^k=(x_1+k... | {
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} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.019775526598095894,
0.023361368104815483,
-0.04180937260389328,
0.0007576994830742478,
0.0006938028382137418,
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0.... | |
fa7072b908ed077e255baa857d87e4c12ee66db8 | subsection | 38 | 77 | Actions of | By \mathcal {I}_P and \mathcal {L}_P, we denote, respectively, the set of blocks in \mathcal {I} and \mathcal {L}; that is,
\mathcal {P}=\mathcal {I}_P\cup \mathcal {L}_P.Remark 6.4 Note thatN=N_G(P)=\langle \left[
\begin{array}{cc}
a&b\\0&a^{-1}
\end{array}\right]\bigm |a\in \langle \alpha \rangle , b\in F_{q^2}\rangl... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.057599987834692,
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0.03476756066083908,
0.0... | |
61dd1a9237fb18537b33e287d908442cd3c4d941 | subsection | 39 | 77 | Actions of | Then y_2\ne 0 and x_3\ne 0, and \langle \mathbf {x}\rangle \sim \langle \mathbf {y}\rho ^k\rangle if and only if(\mathbf {x},\mathbf {y}\rho ^k)=((0,0,x_3,x_4),
(y_1+k^2y_2,y_2,ky_2, y_4))=\pm 2\lambda ,that is, if and only if-2x_3ky_2+2\theta x_4y_4=\pm 2\lambda .From (REF )
we get that k=\frac{\theta x_4y_4\mp \lambd... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1152,
"openalex_id": "",
"raw": "T. Tchuda, Combinatorial-geometric characterization of some primitive representations of the groups \\hbox{\\rm PSL}_n(q), n=2,3, PhD Thesis, Kiev, 1986 (in Russian).",
"source_ref_id": "86ba... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.029068170115351677,
0.014373866841197014,
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-... | |
b625a3ebea4f01c363ed3e212a93431335b3415b | subsection | 40 | 77 | Actions of | We may choose H to consist of all the matrices of the form\left[ \begin{array}{cc}
x & 0 \\
0 & x^{-1}
\end{array} \right]
\ (x\in F^*) \ \textrm { and } \
\left[ \begin{array}{cc}
0 & -x \\
x^{-1} & 0
\end{array} \right]
\ (x\in F^*).Note that, since p\ge 13, H is a dihedral subgroup D_{p-1}.
Further, letg=
\left[ ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1333,
"openalex_id": "",
"raw": "D. Marušič and R. Scapellato, A class of non-Cayley vertex-transitive graphs associated with \\hbox{\\rm PSL}(2,p), Discrete Math. 109 (1992), 161–170.",
"source_ref_id": "dc31cf3db22a1b23ec4... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
0.0013263262808322906,
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0.01236506085842847... | |
2db2de35f69e72f11e3f1649448b99e6ce0b5b37 | subsection | 41 | 77 | Actions of | If ab\ne 0 and g^{\prime }\in gH then either
\chi (g^{\prime })=(\xi ,y\eta ) or \chi (g^{\prime })=(1-\xi ,-y\eta ^{-1}) for some y\in S^*.Let \approx be the equivalence relation on F \times F^* defined by(\xi ,\eta ) \approx (1-\xi ,\frac{\xi \eta }{\xi - 1}) \textrm { for } \xi \ne 0,1.There is then a natural identi... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.02640535868704319,
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0.009488234296441078,
0.02... | |
c75f50c710c9bacc6b4f641d2d69423c7655994d | subsection | 42 | 77 | Actions of | Finally, all remaining cosets in \mathcal {H} contain matrices with no entry equal to zero,
where we have to bring in the equivalence relation \approx on characters
defined by (REF ).For each \xi \in F define the
following subsets of \mathcal {H} (in fact subsets of
(F\times F^*)/_{\approx } \cup \lbrace \infty \rbrace... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1327,
"openalex_id": "",
"raw": "D. Marušič and R. Scapellato, A class of non-Cayley vertex-transitive graphs associated with \\hbox{\\rm PSL}(2,p), Discrete Math. 109 (1992), 161–170.",
"source_ref_id": "dc31cf3db22a1b23ec4... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.015659872442483902,
... | |
2f6d6d54047f7d423d796282807e6e0bb8308204 | subsection | 43 | 77 | Actions of | 2 suborbits of length \frac{p-1}{4}, namely
{\cal S}_{\frac{1}{2}}^+ and {\cal S}_{\frac{1}{2}}^- which
are self-paired if and only if p\equiv {1\pmod {8}}.Example 7.3
The smallest admissible pair of primes p = 13 and q = 7
gives rise to the action of G=\hbox{\rm PSL}(2,13) on cosets of H=D_{12}
with the following sub... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1393,
"openalex_id": "",
"raw": "C. E. Praeger and M. Y. Xu, Vertex primitive transitive graphs of order a product of two distinct primes, J. Combin. Theory Ser. B 59 (1993), 245–266.",
"source_ref_id": "5dfe58ee8c60a5542d53... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.02534826099872589,
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0.025806086137890816,
-0.005... | |
cf8314aad7cf0728fa0c1d66dd4fa01b1cb039b0 | subsection | 44 | 77 | Actions of | Also, the basic orbital graphs arising from Rows 8 and 9 of Table REF
are isomorphic, that is, X(G,H,\mathcal {S}_\xi ^+) \cong X(G,H,\mathcal {S}_\xi ^-),
where \xi \in S^*\cap S^*+1 and \xi \ne \frac{1}{2},1.With the explicit description of the suborbits
of G on {\cal H} the construction of the
corresponding general... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.020017879083752632,
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0.028577353805303574,
... | |
838c4c6f2b3e69c871017c36df9924e77c2ac9f5 | subsection | 45 | 77 | Actions of | These areV_\infty &=&\lbrace H
\left[ \begin{array}{cc}
1 & a \\
0 & 1
\end{array} \right]
\colon a\in F\rbrace \ \textrm { and }\\
V_x &=&\lbrace H
\left[ \begin{array}{cc}
1 & x \\
-x^{-1} & 0
\end{array} \right]
\left[ \begin{array}{cc}
1 & a \\
0 & 1
\end{array} \right]
\colon a\in F\rbrace
= V_{-x}, \ x\in F^*... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.030672967433929443,
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0.015992671251296997,
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-0.029833657667040825,
0.05020600184798241,
0.0... | |
b9303a08bbaffc0991de208cfca742a938e29f8d | subsection | 46 | 77 | Actions of | Since, by Theorem REF ,
\mathcal {S}_0 ^+ \cup \mathcal {S}_1 ^+ and \mathcal {S}_0 ^- \cup \mathcal {S}_1 ^-
are the only two suborbits with nontrivial intersection with \mathcal {S}_1
we obtain the value of d(V_\infty ) as given in Table REF .To determine the values for d(V_\infty , V_x) let us consider
the neighbors... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.4134/jkms.2013.50.6.1199",
"end": 1962,
"openalex_id": "https://openalex.org/W2068452045",
"raw": "K. Kutnar, A. Malnič, L. Martínez and D. Marušič, Quasi m-Cayley strongly regular graphs, J. Korean Math. Soc. 50 (2013), 1199–1211.",
... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.02713887020945549,
0.... | |
aa9434a0d44ebcf7a4ded708d8cf9b2341346fd4 | subsection | 47 | 77 | Actions of | Equivalently, g fixes a vertex and the only power g^i fixing another vertex is the identity mapping.
If a group G is quasi-semiregular on the vertex set of the graph with m+1 orbits, then the graph is called a quasi m-Cayley graph on G.
If G is cyclic and quasi-semiregular with two nontrivial orbits
then the graph is s... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.045147836208343506,
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0.02739701420068741,
0.0... | |
cc8ca3147dc79e3120203ffb64104bf2b7be507a | subsection | 48 | 77 | Actions of | Applying the action of on any of these elements gives
\begin{eqnarray*}
H
\left[ \begin{array}{cc}
1 & x \\
-x^{-1} & 0
\end{array} \right]
\left[ \begin{array}{cc}
1 & a \\
0 & 1
\end{array} \right]
\sigma &=&
H
\left[ \begin{array}{cc}
1 & x \\
-x^{-1} & 0
\end{array} \right]
\sigma
\left[ \begin{array}{cc}
1 & az^... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.002580819418653846,
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0.01022408902645111,
-0... | |
b3927189c674cbccf6ed33efa2f9d42ce15760cf | subsection | 49 | 77 | Actions of | If this neighbor is inside an orbit Vx, xF*, then there exists jF
such that
\left[ \begin{array}{cc}
1-\eta y^{-1} & y \\
\frac{\xi -1}{\eta }-\xi y^{-1} & \frac{y(\xi -1)}{\eta }
\end{array} \right]
\equiv
\left[ \begin{array}{cc}
1 & x \\
-x^{-1} & 0
\end{array} \right]
\left[ \begin{array}{cc}
1 & j \\
0 & 1
\en... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.004954067058861256,
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... | |
cfb398b611c3606dc0a8469f0a08bde251c132b0 | subsection | 50 | 77 | Actions of | Then for y \in F^* we haved(V_y)=\left\lbrace
\begin{array}{cl}
0, & \xi \in N^*\cap N^*+1 \\
2, & \xi \in S^*\cap N^*+1 \textrm { or } \xi \in N^*\cap S^*+1\\
4, & \xi \in S^*\cap S^*+1 \\
0, & \xi =1/2 \textrm { and } p\equiv {5\pmod {8}}\\
2, & \xi =1/2 \textrm { and } p\equiv {1\pmod {8}}\\
\end{array}\right.,wher... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.019027164205908775,
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0.02708357200026512,
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-0.030867643654346466,... | |
3c392ca4457bd792d30f04ad895b990682f6cb79 | subsection | 51 | 77 | Case | Let\varepsilon =\left\lbrace \begin{array}{ll}
2, & q\equiv {1,3\pmod {8}}\\
0, & q\equiv {5,7\pmod {8}}
\end{array}\right..The following lemma gives us
the number of edges inside a block
and between two blocks from
\mathcal {L}_P for the orbital graph X(G,H,\mathcal {S}_0).Lemma 6.6
Let X=X(G,H,\mathcal {S}_0). Then ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.019682444632053375,
0.02493109554052353,
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0.009925139136612415,
... | |
928e00a19fabee17d293423fe9ff5a5dea8b060e | subsection | 52 | 77 | Case | This implies that
d(\langle \mathbf {x}\rangle P,\langle \mathbf {y}\rangle P)=1 for \frac{q+1}{2} blocks
\langle \mathbf {y}\rangle P\in \mathcal {L}_P,
proving part (ii).To prove part (i), take \mathbf {y}=\pm \mathbf {x}=\pm (1,1,0,0). Then, by (REF ), there are edges inside the block \langle \mathbf {x}\rangle P if... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.033787235617637634,
0.011582886800169945,
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0.01802290976047516,
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0.014581617899239063,
... | |
d768339aa5294a7f6152df449cadf0bb1c1f4eb6 | subsection | 53 | 77 | Case | By Lemma REF , X\langle \mathcal {L} \rangle _\mathcal {P} is vertex-transitive,
and consequently
one can see that also X\langle \mathcal {L}\rangle ^{\prime }_\mathcal {P} is vertex-transitive.If q\equiv {1\pmod {4}} then
Proposition REF and Lemma REF (iii) combined together
imply that X\langle \mathcal {L}\rangle ^{\... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.030481282621622086,
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0.008589070290327072,
... | |
571f28a2790df6c0fdd5b0cbf4ca02ef5c02a0c8 | subsection | 54 | 77 | Case | Take a respective Hamilton path for each component, say \mathcal {U}=U_1U_2\cdots U_l, and
\mathcal {U}^{\prime }=U^{\prime }_1 U^{\prime }_2\cdots , U^{\prime }_l, where l=\frac{q(q-1)}{4}.
Choose any two isolated vertices W_1 and W_2
and construct the cycle
\mathcal {D}=W_1\mathcal {U}W_2\mathcal {U}^{\prime }W_1. Ch... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.030860047787427902,
0.027809128165245056,
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0.0009000211139209569,
0.0... | |
d1f8dc4132352c9a09e50dc9343c2aa5008d183f | subsection | 55 | 77 | Case | We can therefore conclude that X\langle \mathcal {L}\rangle _\mathcal {P},
which is vertex-transitive by Lemma REF ,
has at most two connected components.
The rest of the argument follows word by word from the argument given in the proof of Proposition REF , since, by Lemma REF , d(\langle \mathbf {x}\rangle P, \langle... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.004822519142180681,
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0.00843177828937769,
0.012735418975353241,
0.0... | |
3101250c1cdcb9c3cbbb53a6837b816ca80ba3c9 | subsection | 56 | 77 | Case | Consequently,
vertices inside each of these two sets
are of the same valency.)Since we are in the case \mathcal {S}_\xi =\mathcal {S}_\xi ^+\cup \mathcal {S}_\xi ^-
the solutions of () and ()
for x=y depends solely on \xi . Consequently,
d(V_x)=d(V_y) for all x,y\in F^*.
Suppose first that d(V_x)=0, x\in F^*.
Note that... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.002679936122149229,
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-0.0013743017334491014,
0.02217194065451622,
0.0017... | |
6c1122a8d17e04f8a87df13646fcc5a0dacf158f | subsection | 57 | 77 | Case | By Proposition REF we have
d(V_\infty )=0 and
d(V_\infty ,V_y)=1 for every y\in F^*.We now need to compute the valency val_{X_\mathcal {P}-\lbrace V_\infty \rbrace }(V_y),
y\in F^*. Let x\in F^*.
The number of edges d(V_y,V_x) between V_y and V_x in X_\rho
is obtained from () and () by
letting \xi =1/2. We obtainj_{1,... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.03234866261482239,
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0.016479507088661194,
-0.00048160599544644356,
-0.... | |
ff0edf9ca16c392acf32e22f3218b5c94446be40 | subsection | 58 | 77 | Cases | Proposition 7.9
Let X=X(G,H,\mathcal {W}), where \mathcal {W}\in \lbrace \mathcal {S}_0^+\cup \mathcal {S}_1^+, \mathcal {S}_0^-\cup \mathcal {S}_1^-),
be one of the graphs arising from Rows 3 or 4 of Table REF .
Then X is hamiltonian.Proof.
Suppose that \mathcal {W}=\mathcal {S}_0^+\cup \mathcal {S}_1^+.
By Propositi... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.021604357287287712,
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0.011549786664545536,
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0.0... | |
6e4f1d407803317a5915f9e630485871887d25a8 | subsection | 59 | 77 | Cases | Let \eta _i be the possible solution for \eta given in the i-th row of
Table REF .
Then \eta _1\eta _4=\eta _2\eta _3=y^2. This implies that
either (\eta _1,\eta _4)\in S^*\times S^* or
(\eta _1,\eta _4)\in N^*\times N^*, and that
either (\eta _2,\eta _3)\in S^*\times S^* or
(\eta _2,\eta _3)\in N^*\times N^*.
[Table: ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.003869131440296769,
0.0026061... | |
39ed5a3dc62ebd297c98a58bce89c58a041a8c09 | subsection | 60 | 77 | Cases | (Namely, the number of possible
number-theoretic solutions
in Propositions REF and REF
needs to be divided by 2 because of the fact that the
equivalence relation (REF ) implies V_y=V_{-y}.)
Moreover, (p-5)/8 is not an integer if p\equiv {1\pmod {8}}
and so there are at least (p-1)/8 vertices in \mathcal {O}(S^*)
adjac... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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-0.015169952996075153,
0.009820789098739624,
-0... | |
bcc022cd2dd539f9fd71ceed26aac6fffbdc99a5 | subsection | 61 | 77 | Cases | Thus, since \pm 1\in N^*-1\cap S^*
for p\equiv {5\pmod {8}},
Propositions REF and REF
combined together imply thatval(V_x)_{X_\mathcal {P}\langle \mathcal {O}(N^*)\rangle }=
\left\lbrace
\begin{array}{cr}
(p+7)/8, & p\equiv {1\pmod {8}}\\
(p-5)/8, & p\equiv {5\pmod {8}}
\end{array}\right..If follows that, with the ex... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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-0.0030859694816172123,
-0.011077447794377804,
-... | |
ec4c2efd3e4ae985cbf602f340157495f8bd5339 | subsection | 62 | 77 | Cases | We obtainj_{1,2}=j_{3,4}=\frac{1}{2}(y-x\pm \sqrt{x^2+y^2})
\textrm { and }
\eta _{1,2}=\eta _{3,4}=\frac{y}{x}(\pm \sqrt{x^2+y^2}-y).By Proposition REF ,
2\in S^*, and so d(V_x)=2 for every x\in F^* for which
\eta _{1,2}=\eta _{3,4}= \pm x(\sqrt{2}-1)\in S^*.
Further, if x^2+y^2=0 then y^2=-x^2, and so
y=\pm \sqrt{-1}... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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... | |
ea9297ad81f2037e149e984c9d65cd9c4f763ee0 | subsection | 63 | 77 | Cases | The corresponding values \eta _1 and \eta _2 for the pairs
V_1,V_s and V_g, V_{gs} are, respectively,\eta _{1,2}(1,s)=s(\pm \sqrt{1+s^2} - s) \textrm { and }
\eta _{1,2}(g,gs)=gs(\pm \sqrt{1+s^2} - s).Therefore\frac{\eta _{i}(g,gs)}{\eta _{i}(1,s)}=g\in N^*, i\in \lbrace 1,2\rbrace .And consequently, the bicirculant X_... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.02044353075325489,
0.0... | |
f7157553106742beca7171d0d496d9d4b9b75780 | subsection | 64 | 77 | Cases | Of course, this cycle contains an edge of the form
V_xV_y, x,y\in S^*.
Replacing this edge with a path V_xV_\infty V_y gives
a Hamilton cycle in X_\rho .
Obviously this Hamilton cycle contains double edges in X_\rho ,
and so, by Lemma REF ,
lifts to a Hamilton cycle in X.
\rule {2.5mm}{3mm}In Proposition REF graphs ari... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0... | |
74cc2ddfe6b702f8cb7e80f53595f7e5357bb744 | subsection | 65 | 77 | Cases | (For example, this can be checked using Magma .)
The below matrix gives the symbol of this graph with respect to
the orbits S_i=\lbrace v_i^j\colon j\in \mathbb {Z}_{13}\rbrace , i\in \mathbb {Z}_7,
of a (7,13)-semiregular automorphism:\left[
\begin{array}{ccccccc}
\emptyset & \lbrace 0,2\rbrace & \lbrace 6,12\rbrace &... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1006/jsco.1996.0125",
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"raw": "W. Bosma, J. Cannon, and C. Playoust, The Magma Algebra System I: The User Language, J. Symbolic Comput. 24 (1997), 235–265.",
"source_ref_... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.... | |
5c6441082865eb73a8b57809fb57b94df5a04b28 | subsection | 66 | 77 | Cases | Because of the isomorphism given in Proposition REF
we may assume that X arises from Row 8 of Table REF ,
that is, it is associated with a suborbit
\mathcal {S}_\xi ^+,
where \xi \in S^*\cap S^*+1, \xi \ne \frac{1}{2} and \xi \ne 1.There are 8 self-paired suborbits of length (p-1)/2=18,
giving 4 non-isomorphic vertex-... | {
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"arxiv_id": "",
"doi": "10.1006/jsco.1996.0125",
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"source_ref... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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-0.030248817056417465,
0.022... | |
a2a10abd5bde900d0a4874f9584f7cdab52812ac | subsection | 67 | 77 | Cases | It follows by () and ()
that vertices V_1 and V_x, x\in F_{37}^*, are adjacent
in X_\mathcal {P} if and only ifx^2+2(1-2\xi )x+1 \in S^*\cup \lbrace 0\rbrace \textrm { and }
\eta _{1,2}(1,x)=1 - \frac{2}{1+x\pm \sqrt{x^2+2(1-2\xi )x+1}} \in S^*orx^2-2(1-2\xi )x+1 \in S^*\cup \lbrace 0\rbrace \textrm { and }
\eta _{3,4}... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
-0.02449876442551613,
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0.0... | |
90aff9c414fefe958bbebbee4f6382a59f2e775d | subsection | 68 | 77 | Cases | Therefore, connectedness of X implies that
the bipartite graph induced by the edges with one endvertex
in
\mathcal {O}(S^*) and the other in \mathcal {O}(N^*)
contains a matching preserved by the \sigma .
Since, by (), we have\frac{\eta _{i}(\tau ,\tau ^3)}{\eta _{i}(1,\tau ^2)}=\tau \in N^*, i\in \lbrace 1,2,3,4\rbrac... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0... | |
0eb9213edb0eaf68d59d2f170d8c3602b4a60aa3 | subsection | 69 | 77 | Cases | As before this cycle/path can be extended to a Hamilton
cycle in X_\rho which, by Lemma REF ,
lifts to a Hamilton cycle in X.
[Table: Information about existence of edges inX(\hbox{\rm PSL}(2,37),D_{36},\mathcal {S}_\xi ^+) where \xi \in S^*\cap S^*+1, \xi \ne \frac{1}{2} and \xi \ne 1.]Example 7.14
Let X be a basic ... | {
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"source_ref... | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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... | |
a9782329dff3346ad9758ae23a252801f0aec897 | subsection | 70 | 77 | Cases | Next, by (), we have\frac{\eta _{i}(\tau ,\tau ^3)}{\eta _{i}(1,\tau ^2)}=\tau \in N^*, i\in \lbrace 1,2,3,4\rbrace ,and so Table REF implies that
vertices in X\langle \mathcal {O}(N^*)\rangle are of
valency at least 2. Namely, for every \xi
there exists at least one \tau \in \mathcal {T}^+\cup \mathcal {T}^-
such tha... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.01... | |
b397ce572c525dbe61a934d2ae63f0a6c01e316e | subsection | 71 | 77 | Cases | The number of edges d(V_y,V_x), x,y\in F^*,
between V_y and V_x in X_\rho
is obtained from () and ():j_{1,2}&=&\frac{1}{2}(y-x\pm \sqrt{x^2+2(1-2\xi )xy+y^2}), \\
j_{3,4}&=&\frac{1}{2}(y-x\pm \sqrt{x^2-2(1-2\xi )xy+y^2}), \\
\eta _{1,2}&=&\frac{y}{2\xi x}((2\xi -1)x-y \pm \sqrt{x^2+2(1-2\xi )xy+y^2}),\\
\eta _{3,4}... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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... | |
9519f8df2c6b14fc7e9992d844130e20c388e5f9 | subsection | 72 | 77 | Cases | Consequently, Theorem REF and Proposition REF combined together imply
that for the two polynomialsf(z)=1+2(1-2\xi )z^2+z^4 \textrm { and } h(z)=1-2(1-2\xi )z^2+z^4there exist g,g^{\prime }\in F^* such that
F^*=\langle g\rangle =\langle g^{\prime }\rangle and f(g),h(g^{\prime })\in S^*\cup \lbrace 0\rbrace .
It follows ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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c2da4c9772177b762fa3b5b8fced5cf9efca8b76 | subsection | 73 | 77 | Cases | Hence,\frac{\eta _{i}(\bar{g},\bar{g}\bar{s})}{\eta _{i}(1,\bar{s})}=\bar{g}\in N^*, i\in \lbrace 1,2,3,4\rbrace .Consequently, for each i exactly one of
\eta _{i}(\bar{g},\bar{g}\bar{s}) and
\eta _{i}(1,\bar{s}) belongs to S^*, implying
that in the bicirculant X_\rho -\lbrace V_\infty \rbrace
we have a full induced c... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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5197d940a34e7b084d8b5315aabd357434714fee | subsection | 74 | 77 | Cases | This implies thatval(X)=\frac{p-1}{2}&\ge & 4 + d(V_x) + |\sum d(V_x,V_y) \colon y\in S^*|.On the other hand, since, by assumption
the valency of the graph induced on \mathcal {O}(S^*) is either 0 or 1,
we have by calculating valency val(X) at V_x, x\in S^*:val(X)=\frac{p-1}{2}&=&d(V_\infty ,V_1) + \epsilon + d(V_1) + ... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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52bfdb047565ac18f72680cd8fb3c62e836d46bb | subsection | 75 | 77 | Cases | Replacing this edge with a path V_xV_\infty V_y gives
a Hamilton cycle in X_\rho .
Obviously this Hamilton cycle contains double edges,
and so, by Lemma REF , it lifts to a Hamilton cycle in X.
We may therefore assume that (p-1)/4\equiv {5\pmod {6}}
and that the generalized Petersen graph in X_\mathcal {P}-\lbrace V_\i... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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fe1fd748c339b1a835470129c8016a401e959c20 | subsection | 76 | 77 | Proof of Theorem | Proof of Theorem REF .
Let X be a connected vertex-transitive graph of order pq,
where p and q are primes and p\ge q, other than the
Petersen graph.
If q\in \lbrace 2,p\rbrace then X admits a Hamilton cycle
by Proposition REF .
We may therefore assume that q\notin \lbrace 2,p\rbrace .
Then X is a generalized orbital gr... | {
"cite_spans": []
} | 1808.08553 | Hamilton cycles in vertex-transitive graphs of order a product of two
primes | [
"Shaofei Du",
"Klavdija Kutnar",
"Dragan Marusic"
] | [
"math.CO"
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0440db2af16208ef9627ca40512cebb3b9451539 | abstract | 0 | 53 | Abstract | In this paper, we present new stochastic methods for solving two important
classes of nonconvex optimization problems. We first introduce a randomized
accelerated proximal gradient (RapGrad) method for solving a class of nonconvex
optimization problems consisting of the sum of $m$ component functions, and
show that it ... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
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7cc1a0ad9f945c2b03ac583f73b81c6e4ed922af | subsection | 1 | 53 | Introduction | Nonconvex optimization plays a fundamental role in modern statistics and machine learning, e.g.,
for empirical risk minimization with either nonconvex loss () or regularization
(, , ), as well as the training of deep neural networks ().
In this paper, we consider two classes of nonconvex optimization problems that are ... | {
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08f46c39d989084b71443542865a2924a037af85 | subsection | 2 | 53 | Introduction | It can be shown that
the condition number for these problems is usually larger than m (see Section for more details).In addition to (REF ), we consider an important class of nonconvex multi-block optimization problems with linearly coupled constraints, i.e.,\min _{x_i\in X_i} &~ \textstyle {\sum }_{i=1}^mf_i(x_i)\\
\t... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
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b5b702c3552ebfa42c894423619337330f7432e4 | subsection | 3 | 53 | Introduction | \Vert \nabla f(\bar{x})\Vert ^2 \le \epsilon .
Since each GD iteration
requires a full gradient computation, i.e., m gradient computations for f_i's, totally this algorithm needs {\cal O}(m L / \epsilon ) gradient computations
for all the component functions f_i's.
Ghadimi and Lan (see also ) show that by using the s... | {
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"raw": "Saeed Ghadimi and Guanghui Lan, Stochastic first-and zeroth-order methods for nonconvex stochastic programming, SIAM Journal on Optimization, 23 (2013), pp. 2341–2368.",
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1804f64f3d87a9d0ace4dd6c1e03bb8599ec692c | subsection | 4 | 53 | Introduction | For example,
In , Hong et al. established the complexity for a variant of ADMM for nonconvex multi-block problems,
see also and for some previous work on the asymptotic analysis of ADMM for nonconvex optimization.
In , Melo and Monteiro presented a linearized proximal multiblock ADMM
with complexity \mathcal {O}\left(1... | {
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fc87f2078d822986cfd07faa96155dacee944691 | subsection | 5 | 53 | Introduction | In particular, RapGrad does not require the computation of full gradients throughout its
entire procedure by properly initializing a few intertwined search points and gradients using information obtained from the previous subproblems.
This comes with the price of requiring additional storage (memory) for maintaining \m... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
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07e8520a83636c5ffb92a3af51a581fb0f4074ee | subsection | 6 | 53 | Introduction | Moreover, we demonstrate that the total number primal block updates that RapGrad requires
can be much smaller, up to a factor of {\cal O}(\sqrt{m}), than its batch counterpart.
To the best of our knowledge, this is the first time that the complexity of randomized methods for solving this special class of nonconvex mul... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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2531fc370a1053fb8b17e9abd43d7ffea4730459 | subsection | 7 | 53 | Notation and terminology | Let \mathbb {R} denote the set of real numbers.
All vectors are viewed as column vectors, and
for a vector x \in \mathbb {R}^d, we use x^{\top } to denote its transpose.
For any n \ge 1, the set of integers \lbrace 1,\ldots ,n\rbrace is denoted by [n].
We use \mathbb {E}_s[X] to denote the expectation of a random varia... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
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1d0e5490dbbc765d6f047b83d7902297f556f73a | subsection | 8 | 53 | Nonconvex finite-sum optimization | In this section, we develop a randomized accelerated proximal gradient (RapGrad) method
for solving the nonconvex finite-sum optimization problem in (REF )
and demonstrate that it can significantly improve the existing rates of convergence
for solving these problems, especially when their objective functions have large... | {
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75707fafb35355d580512fc0b5a9198a7908aea4 | subsection | 9 | 53 | Nonconvex finite-sum optimization | Using p_{\lambda ,\gamma ,\epsilon }, our problem of interest is given by\min _{x\in \mathbb {R}^n} \tfrac{1}{2m}\Vert Ax-b\Vert ^2 + \tfrac{\rho }{2}\textstyle {\sum }_{i=1}^np_{\lambda ,\gamma ,\epsilon }(x_i),which can be viewed as a special case of problem (REF ) with f_i(x) = \tfrac{1}{2}(a_i^{\top }x-b_i)^2 + \tf... | {
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Multi-block Optimization | [
"Guanghui Lan",
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d0ebccb2294d22f445de209ad49b53aad050a23e | subsection | 10 | 53 | Nonconvex finite-sum optimization | Figure REF shows our Algorithm REF not only reduces the function value as well as gradient norm faster than both SVRG and AG.
[Figure: Comparison on function value f and square of gradient norm \Vert \nabla f\Vert ^2 for Algorithm , SVRG and AG. Left Figure: Comparison on function value f. Right Figure: Comparison on s... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
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"Guanghui Lan",
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0cf4fd392de98321da5465240c939ab73995e278 | subsection | 11 | 53 | The Algorithm | The basic idea of RapGrad is to solve problem (REF ) iteratively by using the proximal-point type method.
More specifically, given a current search point \bar{x}^{\ell -1} at the l-th iteration, we will employ a randomized accelerated gradient (RaGrad) obtained by properly modifying the
randomized primal-dual gradient ... | {
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Multi-block Optimization | [
"Guanghui Lan",
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4e8175a5cc90a371f30c7adb2840014a87eaab5f | subsection | 12 | 53 | The Algorithm | By using the strong convexity of the objective functions,
we will be able to show that all the search points \underline{x}^i_s, i = 1, \ldots , m, will converge,
similarly to the search point x^s, to the optimal solution of the subproblem in (REF ) (see Lemma REF below).
Therefore, we can use y^i_s to approximate \nabl... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
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"math.OC"
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b066f2ec75837ba81a7a4f57b255182d0cca4359 | subsection | 13 | 53 | The Algorithm | Assume nonnegative parameters \lbrace \alpha _t\rbrace , \lbrace \tau _t\rbrace , \lbrace \eta _t\rbrace are given.
t = 1, \ldots , s
1. Generate a random variable i_t uniformly distributed over [m].
2. Update x^t and y^t according to\tilde{x}^t =&~ \alpha _t (x^{t-1}-x^{t-2}) + x^{t-1}.\\
\underline{x}_i^t = &~
\left\... | {
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} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
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eacb88009bbe22e42157d8901b734504324f565a | subsection | 14 | 53 | The Algorithm | A point x \in X is called an approximate stationary point if it
sits within a small neighborhood of a point \hat{x} \in X which approximately satisfies the first-order optimality condition.A point x \in X is called an (\epsilon ,\delta )-solution of (REF ) if there exists some \hat{x}\in X such that[d\left( \nabla f(\h... | {
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Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
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"math.OC"
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11eaef66071cb684f8a3df4208c7211f145d5de6 | subsection | 15 | 53 | The Algorithm | \Vert \nabla f(\hat{x})\Vert ^2\le \epsilon and \Vert x-\hat{x}\Vert ^2\le \epsilon /L^2, which implies that\Vert \nabla f(x)\Vert ^2&=\Vert \nabla f(x)-\nabla f(\hat{x})+\nabla f(\hat{x})\Vert ^2
\le 2 \Vert \nabla f(x)-\nabla f(\hat{x})\Vert ^2+2\Vert \nabla f(\hat{x})\Vert ^2\\
&\le 2 L^2\Vert x-\hat{x}\Vert ^2+2\Ve... | {
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"end": 1203,
"openalex_id": "https://openalex.org/W2962970637",
"raw": "Damek Davis and Benjamin Grimmer, Proximally guided stochastic subgradient method for nonsmooth, nonconvex problems, arXiv preprint arXiv: 1707.035... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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... | |
1295d9599e96ffb31e06fe038198d9d6699cc589 | subsection | 16 | 53 | The Algorithm | Suppose that in Algorithm REF , the number of iterations s=\lceil -\log \widetilde{M} / \log \alpha \rceil with\widetilde{M}:=\textstyle {6\left(5+\tfrac{2L}{\mu }\right)}\max \left\lbrace \tfrac{6}{5},\tfrac{L^2}{\mu ^2}\right\rbrace , \quad \alpha =1-\tfrac{2}{m\left(\sqrt{1+16c/m}+1\right)}, \quad c = 2+\tfrac{L}{\m... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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bd09d6a27e78725c6e93e45497588886336f09e9 | subsection | 17 | 53 | The Algorithm | Indeed, observe that the full gradient is computed only once in the first outer loop,
and that for each subproblem (REF ),
we only need to compute s gradients with\textstyle {s= \left\lceil -\tfrac{\log \widetilde{M}}{\log {\alpha }}\right\rceil \sim \mathcal {O}\left(\left(m+\sqrt{m\tfrac{ L}{\mu }}\right)\log \left(\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1703.10993",
"end": 1481,
"openalex_id": "https://openalex.org/W2604676813",
"raw": "Courtney Paquette, Hongzhou Lin, Dmitriy Drusvyatskiy, Julien Mairal, and Zaid Harchaoui, Catalyst acceleration for gradient-based non-conve... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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... | |
84691da186d6a723f69b06874cc3f982123ac727 | subsection | 18 | 53 | The Algorithm | In fact, our complexity bound minorizes
those for variance-reduced stochastic algorithms as long as L/\mu \log (L/\mu ) > m^{\frac{1}{3}}.Theorem REF only shows the convergence of RapGrad in expectation.
Similarly to the nonconvex SGD methods in , , we can
establish and then further improve the convergence of RapGrad w... | {
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{
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"doi": "",
"end": 695,
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"raw": "Saeed Ghadimi and Guanghui Lan, Stochastic first-and zeroth-order methods for nonconvex stochastic programming, SIAM Journal on Optimization, 23 (2013), pp. 2341–2368.",
"source_ref_id": "9520... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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7d6104d2009d78472c7a1c6cab9ca029477f9912 | subsection | 19 | 53 | The Algorithm | Indeed, by the method of multipliers and Fenchel conjugate duality, we have&\min _{{\textbf {x}}\in X,x_m\in \mathbb {R}^{n}} \lbrace \psi ({\textbf {x}})+\psi _m(x_{m}) + \max _{y\in \mathbb {R}^{n}} \left\langle \textstyle {\sum }_{i=1}^{m}\mathbf {A}_ix_i -\mathbf {b}, y\right\rangle \rbrace
\\
=&\min _{{\textbf {x... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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e3d91a64bfc7460d061da63a8fbf66e5a518dcf6 | subsection | 20 | 53 | The Algorithm | Specifically,y^t &=\textstyle \operatornamewithlimits{arg\,min}\limits _{y \in \mathbb {R}^n}~ h(y)+\langle -\mathbf {A}\tilde{{\textbf {x}}}^t+\mathbf {b},y\rangle + \tau _t V_{ h}(y,y^{t-1})\\
&=\textstyle \operatornamewithlimits{arg\,max}\limits _{y \in \mathbb {R}^n}~ \langle (\mathbf {A}\tilde{{\textbf {x}}}^t-\ma... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.050... | |
4f1abd7aee3b6b634ee08009f255e39f616accb3 | subsection | 21 | 53 | The Algorithm | We will derive
the convergence result for Algorithm REF in terms of primal variables and construct relations between successive search points ({\textbf {x}}^{\ell },x_m^l),
which will be used to prove the final convergence of RapDual.
[H]
RapDual for nonconvex multi-block optimization
Compute A_m^{-1} and reformulate... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.0... | |
627cfc1c6ac5b80f979700f64896c4ebaaf9b515 | subsection | 22 | 53 | The Algorithm | Update x^t and y^t according to\tilde{{\textbf {x}}}^t =&~ \alpha _t ({\textbf {x}}^{t-1}-{\textbf {x}}^{t-2}) + {\textbf {x}}^{t-1},\\
g^t=&~(\tau _t g^{t-1}+\mathbf {A}\tilde{{\textbf {x}}}^t-\mathbf {b})/(1+\tau _t), \\
y^t =&~\textstyle \operatornamewithlimits{arg\,min}\limits _{y \in \mathbb {R}^n}~ h(y)+\langle -... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.0... | |
e1b44e532e182ce0a1e981843c3eae6a89dadb9c | subsection | 23 | 53 | The Algorithm | A point (\emph {\textbf {x}},x_m) \in X\times \mathbb {R}^{n} is called an (\epsilon ,\delta ,\sigma )-solution of (REF ) if there exists some \hat{\emph {\textbf {x}}}\in X, and \lambda \in \mathbb {R}^n such that\left[d(\nabla f(\hat{\emph {\textbf {x}}})+ \mathbf {A}^{\top }\lambda , -N_{X} (\hat{\emph {\textbf {x}}... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.... | |
b71cbd07102f54d85880abf4940c6c2863f1ab8a | subsection | 24 | 53 | The Algorithm | Besides, the definition of a (\epsilon ,\delta ,\sigma )-solution guarantees \Vert \nabla f_m(x_m)+\lambda \Vert ^2\le \epsilon and \Vert \mathbf {A}{\textbf {x}}+x_m-\mathbf {b}\Vert ^2\le \sigma , which altogether justify that ({\textbf {x}},x_m) is a reasonably good solution.
Let the iterates (\emph {\textbf {x}}^... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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... | |
0f04bd24c969510848a4320a71a2f6c424aa104e | subsection | 25 | 53 | The Algorithm | Suppose in Algorithm REF , number of iterations s=\left\lceil -\log \widehat{\mathcal {M}}/\log \alpha \right\rceil with\widehat{\mathcal {M}}=(2+\tfrac{L}{\mu })\cdot \max \left\lbrace 2,\tfrac{L^2}{\mu ^2}\right\rbrace , \quad \alpha =1-\tfrac{2}{(m-1)(\sqrt{1+8c}+1)},\quad c = \tfrac{\bar{A}^2}{\mu \bar{\mu }}=\tfra... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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... | |
8052ed4df3c4166f35bec1e5604586937b0b1a89 | subsection | 26 | 53 | The Algorithm | According to Theorem REF ,
we can bound the complexity of RapDual to compute a
stochastic (\epsilon ,\delta ,\sigma )-solution of (REF ) in terms of block updates in ().
Note that for each subproblem (REF ), we only need to update s primal blocks with\textstyle {s= \left\lceil -\tfrac{\log \widehat{\mathcal {M}}}{\log ... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0... | |
366270a187e108640862e190b716f31e62f1c72c | subsection | 27 | 53 | The Algorithm | \mathbf {A}_1 = \mathbf {A}_2=\ldots =\mathbf {A}_{m-1}, we immediately have \Vert \mathbf {A}\Vert = \sqrt{m-1}\bar{A}, which means
that RapDual can
potentially save the number of primal block updates by a factor of {\cal O}(\sqrt{m}) than its batch counterpart.It is also interesting to compare RapDual with the noncon... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/130936361",
"end": 372,
"openalex_id": "https://openalex.org/W1975768153",
"raw": "Cong D. Dang and Guanghui Lan, Stochastic block mirror descent methods for nonsmooth and stochastic optimization, SIAM Journal on Optimization, 25 (2... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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23c5595e7224ecf2e7157e0bfc9612eda651882c | subsection | 28 | 53 | The Algorithm | To do so, we can iteratively solve the following saddle-point subproblems in place of the ones in (REF ):&\min _{{\textbf {x}}\in X,x_m\in \mathbb {R}^{n}} \lbrace \psi ({\textbf {x}})+\psi _m(x_{m}) + \max _{y\in \mathbb {R}^{n}} \left\langle \textstyle {\sum }_{i=1}^{m} A_ix_i - b , y\right\rangle \rbrace
\\
=&\min ... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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... | |
7955f641a12d4bb08d8866315d2bc31dee56ca8f | subsection | 29 | 53 | Convergence analysis for RapGrad | In this section, we will first develop the convergence results for Algorithm REF applied to the convex finite-sum subproblem (REF ),
and then using them to establish the convergence of RapGrad.
Observe that the component functions \psi _i and \varphi in (REF ) satisfy:\tfrac{\mu }{2}\Vert x-y\Vert ^2\le \psi _i(x)-\ps... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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-0.00868862122297287,... | |
8d307086c228973acbe4b674b89c9935adfb5696 | subsection | 30 | 53 | Convergence analysis for RapGrad | Let \hat{\underline{x}}_i^t = (1+\tau _t)^{-1}(\tilde{x}^t+\tau _t\underline{x}_i^{t-1}), for i = 1,\ldots , m,~ t=1,\ldots , s.\mathbb {E}_{i_t} [\psi (\hat{\underline{x}}_i^t)] &= m \psi (\underline{x}_i^t)-(m-1) \psi (\underline{x}_i^{t-1}) ,\\
\mathbb {E}_{i_t} [\nabla \psi (\hat{\underline{x}}_i^t)] &= m\nabla \ps... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s10107-017-1173-0",
"end": 1162,
"openalex_id": "https://openalex.org/W2964037929",
"raw": "Guanghui Lan and Yi Zhou, An optimal randomized incremental gradient method, Mathematical programming, (2017), pp. 1–49.",
"source_ref... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.0... | |
102d377e1578ca4e7ac04f229b52234ed0663142 | subsection | 31 | 53 | Convergence analysis for RapGrad | If the parameters in Algorithm REF satisfy for all t = 1, \ldots , s-1,\alpha _{t+1}\gamma _{t+1}= &~ \gamma _{t}, \\
\gamma _{t+1}[m(1+\tau _{t+1})-1] \le &~ m\gamma _{t}(1+\tau _{t}),\\
\gamma _{t+1}\eta _{t+1}\le &~ \gamma _{t}(1+\eta _{t}),\\
\tfrac{\eta _s \mu }{4}\ge &~ \tfrac{(m-1)^2\hat{L}}{m^2\tau _s},\\
\tfra... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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0.... | |
dd3c9fd58a404afc349fa6ee90935604b9dd6e0a | subsection | 32 | 53 | Convergence analysis for RapGrad | If \varphi (x)=\tfrac{\mu }{2}\Vert x-z\Vert ^2, for some z\in X, then, for any s \ge 1, we have\mathbb {E}_s\left[\Vert x^*-x^s\Vert ^2\right]&~\le \textstyle {\alpha ^s(1+2\tfrac{\hat{L}}{\mu })~\mathbb {E}_s\left[\Vert x^*-x^0\Vert ^2+\textstyle {\tfrac{1}{m}\sum _{i=1}^m}\Vert {\underline{x}}^{0}_i-x^0\Vert ^2\righ... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
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bb14e6b9b2d6574dc354f917df457b46a7dfb7d3 | subsection | 33 | 53 | Convergence analysis for RapGrad | Plugging into (REF ), we obtain the following two relations:\mathbb {E}_s\left[\Vert x^*-x^s\Vert ^2\right]&~\le \alpha ^s\mathbb {E}_s\left[\Vert x^*-x^0\Vert ^2+\textstyle {\sum }_{i=1}^m\tfrac{\hat{L}}{mr}\Vert {\underline{x}}^{0}_i-x^*\Vert ^2\right]\\
&~\le \alpha ^s\mathbb {E}_s\left[\Vert x^*-x^0\Vert ^2+\textst... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04939063638448715,
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0.0... | |
b3e41b71413f575e99c065aa08f4762df83c3801 | subsection | 34 | 53 | Convergence analysis for RapGrad | Actually, as shown below we do not need to solve the subproblem too accurately, and a constant number of iteration of Algorithm REF
for each subproblem is enough to guarantee the convergence of Algorithm REF .Let the number of inner iterations s\ge \left\lceil -\log (7M/6)/\log \alpha \right\rceil with M:=\textstyle {... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.017039164900779724,
0.0003096166474279016,
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0.012607760727405548,
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0.008397546596825123,
-0.015185754746198654... | |
92834c578d20c434f25999447d1f7c05a18d21be | subsection | 35 | 53 | Convergence analysis for RapGrad | According to Theorem REF (with \hat{L} = 2\mu + L), we have, for \ell \ge 1,\mathbb {E}\Vert x_*^{\ell }-\bar{x}^{\ell }\Vert ^2 &\le \textstyle {\alpha ^s(5+\tfrac{2L}{\mu })\mathbb {E}\left[\Vert x^{\ell }_*-\bar{x}^{\ell -1}\Vert ^2+\textstyle {\sum }_{i=1}^m\tfrac{1}{m}\Vert \bar{\underline{x}}^{\ell -1}_i-\bar{x}^... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.04973955079913139,
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0.005538483615964651,
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0.041530996561050415,
-0.044490959495306015,
... | |
de971b2dba7283b622932ca1353c403d4ab017a5 | subsection | 36 | 53 | Convergence analysis for RapGrad | By the optimality condition of the \hat{\ell }-th subproblem (REF ),\nabla \psi ^{\hat{\ell }}(x_*^{\hat{\ell }})+\nabla \varphi ^{\hat{\ell }}(x_*^{\hat{\ell }})\in -N_X(x_*^{\hat{\ell }}).From the definition of \psi ^{\hat{\ell }} and \varphi ^{\hat{\ell }}, we have\nabla f(x_*^{\hat{\ell }})+3\mu (x_*^{\hat{\ell }}-... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.019368596374988556,
0.018391773104667664,
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0.030861547216773033,
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0.023260632529854774,
-0.0014013233594596386,
... | |
61006ed21b739e37d058cf98a990733bfa1b66f6 | subsection | 37 | 53 | Nonconvex multi-block optimization with linear constraints | In this section, we present a randomized accelerated proximal dual (RapDual) algorithm for solving the nonconvex multi-block optimization problem
in (REF ) and show the potential advantages in terms of the total number of block updates.As mentioned in Section 1, we assume the inverse of the last block of the constraint... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/130936361",
"end": 1900,
"openalex_id": "https://openalex.org/W1975768153",
"raw": "Cong D. Dang and Guanghui Lan, Stochastic block mirror descent methods for nonsmooth and stochastic optimization, SIAM Journal on Optimization, 25 (... | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.010576962493360043,
-0.0012419918784871697,
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-0.026602663099765778,
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0.03733225166797638,
-0.06434700638055801,
-0.0025583887472748756,
-0.02706054039299488,
0... | |
1aea0a59acf5aa3f1ebcb443e8aa096906f87aa0 | subsection | 38 | 53 | Convergence analysis for RapDual | In this section, we first show the convergence of Algorithm REF for solving the convex multi-block subproblem (REF ) with\psi _i(x)-\psi _i(y)-\langle \nabla \psi _i(y),x-y\rangle \ge \tfrac{\mu }{2}\Vert x-y\Vert ^2, \ \ \forall x, y\in X_i, \quad i=1,\ldots ,m-1,
\tfrac{\mu }{2}\Vert x-y\Vert ^2\le \psi _m(x)-\psi _... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
0.011220754124224186,
-0.01322691235691309,
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0.022090621292591095,
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0.013501519337296486,
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0.005679028574377298,
-0.018291888758540154,
0.... | |
d0967fce72ebeeeccee3cc59775de070cfe8c3a4 | subsection | 39 | 53 | Convergence analysis for RapDual | Let the iterates \emph {\textbf {x}}^t and y^t for t = 1, \ldots , s be generated by Algorithm REF
and (\emph {\textbf {x}}^*, y^*) be a saddle point of (REF ).
Assume that the parameters in Algorithm REF satisfy for all t = 1, \ldots , s-1\alpha _{t+1} = &~ (m-1) \tilde{\alpha }_{t+1},\\
\gamma _t = &~\gamma _{t+1}\t... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.004168548155575991,
-0.010167748667299747,
-0.03206693008542061,
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0.01731492206454277,
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0.019023530185222626,
-0.00512964092195034,
0... | |
7e9403e863d88ec1e3186a1fe46e4a4441318e33 | subsection | 40 | 53 | Convergence analysis for RapDual | Then, for any s \ge 1, we have\mathbb {E}_s\left\lbrace \Vert \emph {\textbf {x}}^s-\emph {\textbf {x}}^*\Vert ^2+\Vert x_m^s-x_m^*\Vert ^2\right\rbrace \le \alpha ^s\mathcal {M}(\Vert \emph {\textbf {x}}^0-\emph {\textbf {x}}^*\Vert ^2 +\Vert x_m^0-x_m^*\Vert ^2),where x_m^*=\operatornamewithlimits{arg\,min}_{x_m\in {... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.008131741546094418,
0.003207689616829157,
-0.006987500004470348,
-0.014676800929009914,
-0.02506651170551777,
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0.041986025869846344,
0.016248226165771484,
-0.015020073391497135,
0.01585155539214611,
0.005747905466705561,
... | |
f3e2414c844a69d401a58b319b9b785dd6b99239 | subsection | 41 | 53 | Convergence analysis for RapDual | Then we have\textstyle {\mathbb {E}_s\left\lbrace \tfrac{(m-1)\gamma _s(\eta _s+\mu )}{2}\Vert {\textbf {x}}^s-{\textbf {x}}^*\Vert ^2+\tfrac{\gamma _s(\tau _s+1)\bar{\mu }}{2}V_{ h}(y^s,y^*)\right\rbrace
\le \tfrac{\gamma _1((m-1)\eta _1+(m-2)\mu )}{2}\Vert {\textbf {x}}^0-{\textbf {x}}^*\Vert ^2 +\gamma _1\tau _1V_{... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.039691463112831116,
-0.0007627106970176101,
-0.008458461612462997,
-0.019357597455382347,
-0.005811855662614107,
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0.030538935214281082,
0.02550504542887211,
0.02431521750986576,
-0.03426096588373184,
0.023308439180254936,
-0.009732188656926155,
... | |
f0f4e46e63f70bcf8ab1207f1773fde9bc97e7c2 | subsection | 42 | 53 | Convergence analysis for RapDual | Also the iterates (\emph {\textbf {x}}^{\ell },x_m^{\ell }) for \ell = 1, \ldots , k be generated by Algorithm REF and \hat{\ell } be randomly selected from [k]. Then\mathbb {E}\left(\Vert {\textbf {x}}^{\ell }_*-\bar{\textbf {x}}^{\ell -1}\Vert ^2+\Vert x_{m^*}^{\ell }-\bar{x}_m^{\ell -1}\Vert ^2\right)
& \le \tfrac{1... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.0174453966319561,
-0.0017361451173201203,
-0.01846800372004509,
-0.033578190952539444,
-0.012011834420263767,
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0.02187161147594452,
0.028343044221401215,
0.009310316294431686,
0.04700946807861328,
-0.024466291069984436,
0.0035772218834608793,
-0.024664707481861115,
... | |
27e09bf0c84a3080c4dac63bb944d2174aed464f | subsection | 43 | 53 | Convergence analysis for RapDual | Since ({\textbf {x}}^{\ell }_*,x_{m^*}^{\ell }) is optimal and ({\textbf {x}}_*^{\ell -1}, x_{m^*}^{\ell -1}) is feasible to the \ell -th subproblem, we have\psi ^{\ell }({\textbf {x}}_*^{\ell })+\psi _m^{\ell }(x_{m^*}^{\ell })\le \psi ^{\ell }({\textbf {x}}_*^{\ell -1})+\psi _m^{\ell }(x_{m^*}^{\ell -1}).Plugging in ... | {
"cite_spans": []
} | 1805.05411 | Accelerated Stochastic Algorithms for Nonconvex Finite-sum and
Multi-block Optimization | [
"Guanghui Lan",
"Yu Yang"
] | [
"math.OC"
] | 2,018 | en | Mathematics | [
-0.027972964569926262,
0.012384103611111641,
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-0.038060326129198074,
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0.05155083164572716,
-0.019228551536798477,
0.033726271241903305,
-0.005451905075460672,
0... |
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