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[EQUATION] This means that the maximum attainable domain wall velocity scales proportional to the anisotropy constant [MATH] if we keep the ratio [MATH] constant. There is no theoretical limit to the velocity, and this is set only by the availability of materials with high anisotropy and DMI parameters. Furthermore, th... |
Numerical calculation of propagating domain walls We solve Eq. ( 12 ) numerically by applying the relaxation algorithm [EQUATION] |
The velocity [MATH] is determined self-consistently in terms of an arbitrary input parameter [MATH] and the linear momentum for the one-dimensional system, defined as |
[EQUATION] If we note that [MATH] it becomes evident that Eq. ( 62 ) is a minimization algorithm for the functional [MATH] When the algorithm converges to a minimum of the functional, where [MATH] , the magnetization configuration satisfies Eq. ( 12 ) and represents a solitary wave with velocity [MATH] |
The form ( 63 ) for the definition of the linear momentum is not unique. Among the possible definitions, we have chosen the one which is well-defined (i.e., the one that contains a non-divergent integrand) and takes the value [MATH] for the static Bloch wall ( ). One could add a total derivative in the integrand of Eq.... |
For a definite numerical calculation we choose the anisotropy and DM parameters [EQUATION] so that [MATH] and thus the ground state is the (uniform) ferromagnetic. We insert the Bloch wall ( ) (choosing the plus signs) as an initial configuration in the numerical algorithm ( 62 ). We vary the input parameter [MATH] in ... |
For [MATH] we have [MATH] and, as the velocity increases, the component [MATH] increases, i.e. , the magnetization tilts to the direction of motion. In Fig. we show propagating domain wall profiles for an intermediate velocity [MATH] and for a high velocity [MATH] In the limit [MATH] we have [MATH] at the domain wall c... |
The complex structure of the domain wall profiles is more apparent for the faster moving walls in Fig. We have verified that our numerically calculated domain wall profiles verify the results stated in Theorem Fig. shows the angle [MATH] for the domain wall shown in Fig. with velocity [MATH] The line follows the asympt... |
We can obtain further traveling domain walls by giving as initial condition to our numerical algorithm the Bloch wall profile ( ) where we choose the plus sign for [MATH] and the minus sign for [MATH] This is a local energy maximum due to the contribution of the DMI and it it thus an unstable solution. The algorithm co... |
The disparity is due to the chirality of the DMI. On the other hand, the general features of all domain wall profiles are similar, in particular, they satisfy the asymptotic behavior given in Theorem |
For a parametrization of the domain walls we consider the tilting angle [MATH] of the magnetization at the center of the wall. In Fig. a we plot the velocity for the family of propagating domain walls versus the titling angle [MATH] The maximum velocity is obtained for [MATH] (Nèel wall). The function [MATH] is periodi... |
Acknowledgement SK and CM gratefully acknowledge financial support by the DFG under the grant no. ME 2273/3-1 including a three-month Mercator fellowship during which this work was initiated. SV gratefully acknowledges financial support by the NSF through contract DMS-1211638. The authors thank the University of Crete ... |
Appendix A Walker wall Let us assume the model with symmetric exchange and easy-axis anisotropy where we add the magnetostatic interaction. The latter is reduced to a relatively simple term, equivalent to modeling a hard axis in the film plane, if we assume that the film thickness is infinite. The effective field for t... |
is [EQUATION] where the third term on the right hand side is an anisotropy of the easy-plane type which is modeling the magnetostatic field. Distances are measured in exchange length units. |
The LL equation ( ) with effective field ( 65 ) has propagating domain wall solutions [EQUATION] under the following conditions [EQUATION] |
These solutions follow a geodesic (meridian) on the magnetization sphere connecting the poles [MATH] The angle [MATH] is typically referred to as the wall tilting The Bloch wall of Eq. ( ) is obtained for [MATH] , and the Nèel wall is obtained for [MATH] They are both static [MATH] ) solutions, but the Bloch wall is an... |
In Fig. b we plot the velocity of the wall given in Eq. ( 67 ) as a function of the titling angle [MATH] The velocity is [MATH] -periodic due to the equivalence of the walls under the reflexion [MATH] There is a maximum, that we shall call [MATH] , obtained for a value of the wall tilting [MATH] The tilting angle [MATH... |
Appendix B Non-chiral magnet The Landau-Lifshitz equation ( ) where the effective field [MATH] contains only exchange and anisotropy terms has been shown to have a Lax pair and is completely integrable producing single and multi-soliton solutions as well as periodic solutions (see |
and references therein). Single soliton solutions without the use of integrability have been derived in Here, we are using the formulation in the main text of the present study and derive single soliton and domain wall solutions without the use of integrability. In the absence of DMI ( [MATH] ) the third equation in ( ... |
[EQUATION] Cross-multiplying the two equations and further multiplying the result by [MATH] gives a perfect derivative [EQUATION] |
We have the integral [EQUATION] Inserting in Eq. ( 70 [MATH] we obtain [EQUATION] Multiplying both sides of the second equation by [MATH] and using the second of Eqs. ( 68 ) gives [MATH] on the left side, which equals [MATH] . Thus, |
[EQUATION] where we have defined [EQUATION] For domain walls we have [MATH] and [MATH] at spatial infinity. If we assume that [MATH] at spatial infinity then Eq. ( 70 ) gives [MATH] , while for [MATH] we have [MATH] An immediate conclusion is that, for a domain wall where [MATH] takes both values [MATH] and [MATH] at s... |
We continue our investigation anticipating soliton solutions with the asymptotic value [MATH] Inserting [MATH] in Eq. ( 73 ) we obtain |
[EQUATION] Soliton solutions with far-field [MATH] are obtained since [MATH] has a double root at [MATH] and a neighboring single root. For [MATH] the single root is at [MATH] , which allows for a soliton velocity in the range [MATH] For [MATH] the single root is at [MATH] , which allows for a soliton velocity in the r... |
In the general case, the polynomial [MATH] factors to [EQUATION] For values of the constant [MATH] the polynomial [MATH] has exactly one root in the interval [MATH] and another root in [MATH] In order to see this one calculates that [MATH] and [MATH] Necessarily there are two roots in [MATH] and two roots outside it. S... |
# Source: arxiv 1806.02098 # Title: k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front # Sections: all # Downloaded: 2026-03-03T01:45:51.976955+00:00 |
k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front |
Abstract This paper examines a common extension of k-medoids and k-median clustering in the case of a two-dimensional Pareto front, as A characterization of optimal clusters is provided, which allows to solve the optimization problems to optimality in polynomial time using a common dynamic programming algorithm. More p... |
Keywords: Bi-objective optimization clustering algorithms k-medoids p-median Euclidean sum-of-squares clustering Pareto Front Dynamic programming Complexity |
Introduction This paper is motivated by real-life applications of Multi-Objective Optimization (MOO). Some optimization problems can be driven by more than one objective function, with some conflicts among objectives. For example, one may minimize financial costs, while maximizing the robustness to uncertainties dupin2... |
MOO approaches may generate large PF, for a trade-off evaluation by a decision maker. The problem is here to select [MATH] good compromise solutions from [MATH] non dominated solutions while maximizing the representativity of these [MATH] solutions. This problem can be seen as an application of clustering algorithms, p... |
k-means clustering is one of the most famous unsupervised learning problem, and is widely studied in the literature since the seminal algorithm provided by Lloyd in lloyd1982least The k-medoids problem, the discrete variant of the k-means problem, fits better with our application to maximize the dissimilarity around a ... |
In section 2, we define formally the problem and unify the notation. In section 3, intermediate results and a characterization of optimal clusters are presented. In section 4, it is described how to compute efficiently the costs of the previous clusters. In section 5, a first DP algorithm is presented with a proven pol... |
Problem statement and notation We consider a set [MATH] of [MATH] elements of [MATH] , such that for all [MATH] [MATH] defining the binary relations [MATH] for all [MATH] with: |
[EQUATION] These hypotheses on [MATH] characterizes 2-d discrete PF minimizing two objectives, as illustrated in Figure . We note that the convention leading to the definitions of [MATH] considered the minimization of two objectives. This is not a loss of generality, any bi-objective optimization problem can be transfo... |
and also meta-heuristics talbi2009metaheuristics We consider in this paper the Euclidian distance : [EQUATION] Let [MATH] a strictly positive integer. [MATH] denotes the set of the possible partitions of [MATH] in [MATH] subsets: |
[EQUATION] Defining a cost function [MATH] for each subset of [MATH] to measure the dissimilarity, we investigate [MATH] -clustering problems which can be written as following combinatorial optimization problems, minimizing the sum of the measure [MATH] for all the [MATH] clusters partitioning [MATH] |
[EQUATION] K-medoids and K-median problems are in the shape of ( ). On one hand, discrete K-median cost function [MATH] considers the minimal sum of the distances from one chosen point of [MATH] , denoted as the median , to the other points of [MATH] On the other hand, |
K-medoids cost function [MATH] considers the minimal sum of the squared distances from one chosen point of [MATH] , denoted as the medoid , to the other points of [MATH] |
[EQUATION] [EQUATION] We unify notations with [MATH] , considering the generic dissimilarity function [MATH] [EQUATION] Definition 1 [MATH] -medoids) |
For all [MATH] the [MATH] -medoid of a subset [MATH] denotes the point [MATH] such that [MATH] Definition 2 [MATH] [MATH] -Med2dPF ) |
For all [MATH] the [MATH] -clustering problem ( ) using generic function [MATH] is denoted [MATH] [MATH] -Med2dPF. With [MATH] , the [MATH] -medoids are the medians and [MATH] [MATH] -Med2dPF corresponds to the discrete [MATH] -median clustering problem in a 2d PF. With [MATH] , the [MATH] -medoids are the canonical me... |
Definition 3 (Local minimums of [MATH] [MATH] -Med2dPF) For all [MATH] and [MATH] local minimums of [MATH] [MATH] -Med2dPF are characterized by the encoding of partitioning subsets [MATH] and their respective [MATH] -medoids [MATH] with the property: |
[EQUATION] State-of-the-art This section describes related works to appreciate our contributions, in the state of the art of the k-median and k-medoid problems. |
3.1 The general p-median problem The p-median problem was originally a logistic problem, having a set of customers and defining the places of depots in order to minimize the total distance for customers to reach the closest depot. We give here the general form of the p-median problem. Let [MATH] be the number of client... |
[MATH] to [MATH] . The p-median problem consists of opening [MATH] facilities and assigning each client to its closest open facility, in order to minimize the total distance. We note that in some version of the general p-median problem, the graph of the possible assignments is not complete. Considering complete grahs o... |
The p-median problem is naturally formulated within the Integer Linear Programming (ILP) framework. A first ILP formulation defines binary variables [MATH] and [MATH] |
[MATH] if and only if the customer [MATH] is assigned to the depot [MATH] [MATH] if and only if the point [MATH] is chosen as a depot. Following ILP formulation expresses the p-median problem: |
[EQUATION] The p-median problem was proven NP-hard in the general case kariv1979algorithmic The p-median problem in [MATH] with an Euclidian distance is also NP-hard megiddo1984complexity In a tree structure, the p-median problem is solvable in polynomial time, with a DP algorithm running in [MATH] time tamir1998polyno... |
hassin1991improved The p-median problem can be solved to optimality using ILP techniques. The formulation ( 11 ) is tightened in elloumi2010tighter for a more efficient resolution with a Branch & Bound (B& B) solver. For larger sizes of instances, Lagrangian relaxations were investigated in |
beltran2006solving santos2009solving or with column-and-row generation as in avella2007computational Heuristic algorithms are also widely studied mladenovic2007p |
3.2 From p-median to [MATH] [MATH] -Med2dPF clustering problems In our application, the graph is complete, the points [MATH] are exactly [MATH] |
and [MATH] is the Euclidian distance in [MATH] The k-medoid problem is the case [MATH] , whereas the k-median problem is the case [MATH] Both cases are well known clustering problems. |
[MATH] [MATH] -Med2dPF is not only an extension for k-median and k-medoids, the value [MATH] is a parameter of interest, varying [MATH] leads to different clustering solutions, as in fomin2019parameterized |
To the best of our knowledge, no specific studies concerned p-median or k-medoids problems in a PF before the preliminary work dupin2019medoids A complexity in [MATH] time and [MATH] memory space for k-medoids problems was proven in dupin2019medoids We note that an affine 2d PF is a line in [MATH] , such case is equiva... |
General planar cases of p-median and k-medoid problems can also be seen as specific cases of three-dimensional (3d) PF: affine 3d PF. Having a NP-hard complexity proven for the planar cases of p-median problems, it implies that the corresponding p-median problems are also NP-hard for 3d PF thanks to megiddo1984complexi... |
3.3 Clustering/selecting points in Pareto frontiers We summarize here results related to the selection or the clustering of points in PF, with applications to MOO algorithms. Maximizing the quality of discrete representations of Pareto sets was studied with the hypervolume measure in the Hypervolume Subset Selection (H... |
Selecting points in a 2d PF, maximizing the diversity, can be formulated also using p-dispersion problems. Max-Min and Max-Sum p-dispersion problems are NP-hard problems erkut1990discrete hansen1995dispersing Max-Min and Max-Sum p-dispersion problems are still NP-hard problems when distances fulfill the triangle inequa... |
dupin2020polynomial Other variants of p-dispersion problems are also proven to be solvable in polynomial time, using also DP algorithms dupin2020polynomial |
Some similar results exist also for k-means clustering. K-means is NP-hard for 2d cases, and thus for 3d PF mahajan2012planar The 1d case of k-means is also solvable by a DP algorithm, with a complexity in [MATH] using memory space in [MATH] |
gronlund2017fast The restriction to 2d PF would be also solvable in [MATH] time with a DP algorithm if a conjecture is proven dupin2018dynamic |
Lastly, p-center problems present also similar results. The discrete and continuous p-center problems are NP-hard in general. The discrete p-center problem in [MATH] with a Euclidian distance is also NP-hard megiddo1984complexity it implies also that the specific case of 3f PF are also NP hard using discrete p-center c... |
dupin2019planar The discrete p-center problem in a 2d PF is solvable [MATH] time and [MATH] space dupin2019planar Intermediate results |
4.1 Local optimality and interval clustering In this section, local minimums for [MATH] [MATH] -Med2dPF clustering problems are proven to fulfill an interval clustering property. These results extend the previous ones, concerning only the global optimum of k-medoids problems in a 2d PF dupin2019medoids |
Lemma 1 [MATH] is an order relation, and [MATH] is a transitive relation: [EQUATION] Proof : This is a trivial consequence of the transitivity of [MATH] and [MATH] in [MATH] |
Lemma 2 (Total order) Points [MATH] can be indexed such that: [EQUATION] This property is stronger than the property that [MATH] induces a total order in [MATH] Furthermore, the complexity of the sorting re-indexation is in [MATH] |
Proof We reindex [MATH] such that the first coordinate is increasing: [EQUATION] This sorting procedure has a complexity in [MATH] Let [MATH] , with [MATH] . We have thus [MATH] Having [MATH] implies [MATH] |
[MATH] and [MATH] is by definition [MATH] [MATH] Lemma 3 We suppose that points [MATH] are sorted following Proposition [EQUATION] |
Proof We note firstly that the equality cases are trivial, so that we can suppose [MATH] in the following proof. We prove here ( 15 ), the proof of ( 16 ) is analogous. |
Let [MATH] . We note [MATH] [MATH] and [MATH] Proposition ordering ensures [MATH] and [MATH] [MATH] With [MATH] [MATH] With [MATH] [MATH] |
Thus [MATH] [MATH] Proposition 1 [Interval clustering and local minimums of [MATH] [MATH] -Med2dPF]We suppose that points [MATH] are sorted following Lemma For all [MATH] local minimums of [MATH] [MATH] -Med2dPF encoded with [MATH] are necessarily on the shape |
[MATH] [MATH] , far a given [MATH] Hence, there is at most [MATH] local minimums for [MATH] [MATH] -Med2dPF problems. Proof : Let [MATH] encoding a local minimum of [MATH] [MATH] -Med2dPF, indexed such that [MATH] (thanks the total order of Lemma ). Let [MATH] Having [MATH] , Lemma implies that [MATH] |
and ( 10 ) implies that [MATH] and [MATH] By definition of [MATH] [MATH] and [MATH] is the minimal index of points of [MATH] , and we have [MATH] In terms of distances with equation ( 10 ), |
[MATH] and [MATH] We prove now that for all [MATH] [MATH] We prove by contradiction that [MATH] , supposing [MATH] . We would have thus |
[MATH] and lemma implies [MATH] which is in contradiction with ( 10 ). Let [MATH] Having [MATH] , the total order implies [MATH] |
or [MATH] In the first case, [MATH] is implied by lemma in the second case, [MATH] and [MATH] using lemma Hence, using [MATH] , we have thus [MATH] |
i.e. [MATH] This proves that [MATH] , and thus [MATH] [MATH] Proposition 2 (Optimal interval clustering) We suppose that points [MATH] are sorted following Lemma Each local minimum of the [MATH] [MATH] -Med2dPF problems is only composed of clusters [MATH] As a consequence, there is at most [MATH] local optima for [MATH... |
Proof We prove the result by contradiction on [MATH] We suppose having a local minimum for [MATH] [MATH] -Med2dPF in [MATH] encoded [MATH] and their respective [MATH] -medoids [MATH] which does not fulfill Proposition There exist a cluster [MATH] such that [MATH] is not on the shape [MATH] Denoting [MATH] the minimal a... |
[MATH] being nested, this is in contradiction with Proposition [MATH] Remark : a similar property was proven for the global optima of continuous and discrete K-center clustering problems in a 2d PF dupin2019planar , which is the key ingredient to derive a polynomial DP algorithm. However, a main difference exist in the... |
4.2 Costs of interval clusters With Proposition , the computation of cluster costs [MATH] and the [MATH] -medoids of interval clusters [MATH] |
are of special interest. We analyze here properties of these clusters to compute the costs efficiently thereafter. Lemma 4 Let [MATH] .We have [MATH] |
Especially, for all [MATH] [MATH] and [MATH] Proof : Let [MATH] , and let [MATH] (resp [MATH] ) index defining the [MATH] -medoids of [MATH] (resp [MATH] Using [MATH] and the positivity of distances, we have: |
[EQUATION] For all [MATH] , applying the previous result with [MATH] and [MATH] , we have [MATH] , and with [MATH] and [MATH] [MATH] [MATH] |
We define [MATH] as the cost of cluster [MATH] for the [MATH] [MATH] -Med2dPF clustering. This section aims to compute efficiently the costs [MATH] for all [MATH] By definition: |
[EQUATION] We define for all [MATH] , elements [MATH] giving the [MATH] -Med2dPF of cluster [MATH] with [MATH] chosen as the [MATH] -medoid: |
[EQUATION] Lemma 5 Let [MATH] , such that [MATH] . We have [MATH] and [MATH] It implies: [EQUATION] Proof : By definition, [MATH] and |
[MATH] Hence, [MATH] With Lemma , for all [MATH] [MATH] , which implies [MATH] The proof of [MATH] is analogous, which implies also ( 20 ). |
[MATH] Lemma 6 Let [MATH] , such that [MATH] . Let [MATH] an index of an [MATH] -medoid of [MATH] , ie [MATH] If [MATH] , there exist [MATH] an [MATH] -medoid of [MATH] , such that [MATH] , i.e. [MATH] |
If [MATH] , there exist [MATH] an [MATH] -medoid of [MATH] , such that [MATH] , i.e. [MATH] Proof : We prove the first assertion, the last one is proven similarly. Let [MATH] , such that [MATH] . Let [MATH] an index of an [MATH] -medoid of [MATH] |
Let [MATH] . The optimality of [MATH] impose [MATH] Then, Proposition assures that [MATH] Hence, [MATH] . In other words, any [MATH] is not optimal in the minimization defining [MATH] and [MATH] [MATH] |
4.3 The case [MATH] [MATH] -Med2dPF For all [MATH] , the straightforward computation of [MATH] has a time complexity in [MATH] , and thus in [MATH] . In this section, the case of convex or concave 2d PF are proven to be solvable in [MATH] time when [MATH] |
Proposition 3 We suppose that points [MATH] are sorted following Lemma , and that the 2d PF [MATH] is convex (or concave): it exist a [MATH] , decreasing and convex (or concave) function [MATH] such that |
[MATH] Then, computing [MATH] [MATH] -Med2dPF in [MATH] has a complexity in [MATH] time and [MATH] additional space, using Algorithm 1. |
Proof : Firstly, we notice that concave cases are implied by the convex cases, using the axial symmetry around the line [MATH] . Hence, we suppose that [MATH] is convex for the rest of the proof. |
Let [MATH] . h is strictly convex and for all [MATH] [MATH] . Furthermore, [MATH] and [MATH] using Lemma Hence, [MATH] is strictly decreasing, reach a minimum and then is strictly increasing. This monotony hold for the restrictions [MATH] . Algorithm 1 proceeds by a dichotomic, a decreasing (resp increasing) phase dete... |
Remark : An open question is to generalize hypothesis where the Algorithm 1 is valid to compute optimally [MATH] [MATH] -Med2dPF. A counter example, with [MATH] and a PF of size [MATH] is given as following: |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] We have [MATH] [MATH] [MATH] and [MATH] Computing efficiently the costs of interval clustering |
Computing [MATH] independently for all [MATH] using straightforward computations induces a complexity in [MATH] time. In the case of convex (or concave) 2d PF and [MATH] the time complexity is in [MATH] To improve the complexity, we notice that [MATH] fulfills following relations: |
[EQUATION] In dupin2019medoids , relations ( 21 ) are used to compute the whole matrix of cluster costs in [MATH] time and [MATH] space. Actually, the following development will avoid to store the whole DP matrix, which induces a space complexity in [MATH] For the improved DP algorithm, we need two types of computation... |
Proposition 4 Algorithm 2 computes the cluster costs [MATH] for all [MATH] with a complexity in [MATH] time and in [MATH] memory space. |
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