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26 | We evaluated ACSSM’s performance on the extrapolation task following the experimental setup of Schirmer et al. (2022). Each model infer values for all time stamps t∈𝒯′𝑡superscript𝒯′t\in\mathcal{T}^{\prime}, where 𝒯′superscript𝒯′\mathcal{T}^{\prime} denotes the union of observed time stamps 𝒯={ti}i∈[1:k]𝒯subscr... | 553 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
4 Experiment
4.2 Sequence Interpolation &\& Extrapolation
Extrapolation
We evaluated ACSSM’s performance on the extrapolation task following the experimental setup of Schirmer et al. (2022). Each model infer values for all time stam... | 594 |
27 | To evaluate the training costs in comparison to dynamics-based models that depend on numerical simulations, we re-ran the CRU model on the same hardware used for training our model, indicated by ∗ in Table 3. Specifically, we utilized a single NVIDIA RTX A6000 GPU. As illustrated in Table 3, ACSSM significantly lowers ... | 182 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
4 Experiment
4.2 Sequence Interpolation &\& Extrapolation
Computational Efficiency
To evaluate the training costs in comparison to dynamics-based models that depend on numerical simulations, we re-ran the CRU model on the same hardw... | 222 |
28 | In this work, we proposed the method for modeling time series with irregular and discrete observations, which we called ACSSM. By using a multi-marginal Doob’s hℎh-transform and a variational inference algorithm by exploiting the theory of SOC, ACSSM efficiently simulates conditioned dynamics. It leverages amortized in... | 259 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
5 Conclusion and limitation
In this work, we proposed the method for modeling time series with irregular and discrete observations, which we called ACSSM. By using a multi-marginal Doob’s hℎh-transform and a variational inference al... | 284 |
29 | On the theoretical part, all proofs and assumptions are left to Appendix A due the space constraint.
The training and inference algorithms are detailed in Algorithm 1 and Algorithm 2, respectively. Additional implementation details, such as data preprocessing are included in Appendix C. We believe these details are suf... | 65 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Reproducibility Statement
On the theoretical part, all proofs and assumptions are left to Appendix A due the space constraint.
The training and inference algorithms are detailed in Algorithm 1 and Algorithm 2, respectively. Addition... | 91 |
30 | In this work, we proposed a method for modeling irregular time series for practical applications, suggesting that ACSSM does not directly influence ethical or societal issues in a positive or negative way. However, because ACSSM can be applied to health-care datasets, we believe it has the potential to benefit society ... | 69 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Ethics Statement
In this work, we proposed a method for modeling irregular time series for practical applications, suggesting that ACSSM does not directly influence ethical or societal issues in a positive or negative way. However, ... | 93 |
31 | In this section, we present the proofs and derivations for all relevant theorems, lemmas, and corollaries We first restate the core concepts of stochastic calculus, which will be used without further explanation.
Assumptions. Throughout the paper, we work with a probability space (Ω,ℱ,{ℱt}t∈[0,T],ℙ)Ωℱsubscriptsubscript... | 1,230 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
In this section, we present the proofs and derivations for all relevant theorems, lemmas, and corollaries We first restate the core concepts of stochastic calculus, which will be used without furthe... | 1,260 |
32 | Let us consider an Itô diffusion process of the form:
, 1 = d𝐗t=b(t,𝐗t)dt+σ(t)⊤d𝐖t,𝑑subscript𝐗𝑡𝑏𝑡subscript𝐗𝑡𝑑𝑡𝜎superscript𝑡top𝑑subscript𝐖𝑡d\mathbf{X}_{t}=b(t,\mathbf{X}_{t})dt+\sigma(t)^{\top}d\mathbf{W}_{t},. , 2 = . , 3 = (29)
Then, an infinitesimal generator of the above diffusion process is ... | 459 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
Definition A.1 (Infinitesimal Generator).
Let us consider an Itô diffusion process of the form:
, 1 = d𝐗t=b(t,𝐗t)dt+σ(t)⊤d𝐖t,𝑑subscript𝐗𝑡𝑏𝑡subscript𝐗𝑡𝑑𝑡𝜎superscript𝑡top𝑑subscri... | 499 |
33 | Let us consider a path-sequence of random variables {𝐗ti}i∈[1:N]subscriptsubscript𝐗subscript𝑡𝑖𝑖delimited-[]:1𝑁\{\mathbf{X}_{t_{i}}\}_{i\in[1:N]} over an interval 0=ti≤⋯≤tN=T0subscript𝑡𝑖⋯subscript𝑡𝑁𝑇0=t_{i}\leq\cdots\leq t_{N}=T, where each 𝐗tisubscript𝐗subscript𝑡𝑖\mathbf{X}_{t_{i}} taking values in mea... | 1,830 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
Definition A.2 (Path Measure).
Let us consider a path-sequence of random variables {𝐗ti}i∈[1:N]subscriptsubscript𝐗subscript𝑡𝑖𝑖delimited-[]:1𝑁\{\mathbf{X}_{t_{i}}\}_{i\in[1:N]} over an interv... | 1,868 |
34 | Let v(t,x)𝑣𝑡𝑥v(t,x) be C1superscript𝐶1C^{1} in t𝑡t and C2superscript𝐶2C^{2} in x𝑥x and let 𝐗tsubscript𝐗𝑡\mathbf{X}_{t} be the Itô diffusion process of the form in equation (29).
Then, the stochastic process v(t,𝐗t)𝑣𝑡subscript𝐗𝑡v(t,\mathbf{X}_{t}) is also an Itô diffusion process satisfying:
, 1 = dv(... | 432 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
Lemma A.3 (Itô’s formula).
Let v(t,x)𝑣𝑡𝑥v(t,x) be C1superscript𝐶1C^{1} in t𝑡t and C2superscript𝐶2C^{2} in x𝑥x and let 𝐗tsubscript𝐗𝑡\mathbf{X}_{t} be the Itô diffusion process of the form ... | 472 |
35 | Consider the two Itô diffusion processes of form
, 1 = d𝐗t=b(t,𝐗t)dt+σ(t,𝐗t)⊤d𝐖t,t∈[0,T],formulae-sequence𝑑subscript𝐗𝑡𝑏𝑡subscript𝐗𝑡𝑑𝑡𝜎superscript𝑡subscript𝐗𝑡top𝑑subscript𝐖𝑡𝑡0𝑇\displaystyle d\mathbf{X}_{t}=b(t,\mathbf{X}_{t})dt+\sigma(t,\mathbf{X}_{t})^{\top}d\mathbf{W}_{t},\quad t\in[0,T],.... | 1,421 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
Theorem A.4 (Girsanov Theorem).
Consider the two Itô diffusion processes of form
, 1 = d𝐗t=b(t,𝐗t)dt+σ(t,𝐗t)⊤d𝐖t,t∈[0,T],formulae-sequence𝑑subscript𝐗𝑡𝑏𝑡subscript𝐗𝑡𝑑𝑡𝜎superscript... | 1,464 |
36 | We start the section by showing the normalizing property of {fi}i∈[1:k]subscriptsubscript𝑓𝑖𝑖delimited-[]:1𝑘\{f_{i}\}_{i\in[1:k]} in (3). By definition, it satisfied that
, 1 = ∏i=1k𝐋i(gi)=∏i=1k∫ℝdgi(𝐲ti|𝐱ti)𝑑ℙ(𝐱0:T)=(i)𝔼ℙ[∏i=1kgi(𝐲ti|𝐱ti)]=𝐙(ℋtk),superscriptsubscriptproduct𝑖1𝑘subscript𝐋𝑖subscr... | 1,151 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
We start the section by showing the normalizing property of {fi}i∈[1:k]subscriptsubscript𝑓𝑖𝑖delimited-[]:1𝑘\{f_{i}\}_{i\in[1:k]} in (3). By definition, it satisfied th... | 1,193 |
37 | Let us define a sequence of functions {hi}i∈[1:k]subscriptsubscriptℎ𝑖𝑖delimited-[]:1𝑘\{h_{i}\}_{i\in[1:k]}, where each hi:[ti−1,ti)×ℝd→ℝ+:subscriptℎ𝑖→subscript𝑡𝑖1subscript𝑡𝑖superscriptℝ𝑑subscriptℝh_{i}:[t_{i-1},t_{i})\times\mathbb{R}^{d}\to\mathbb{R}_{+}, for all i∈[1:k]i\in[1:k], is a conditional expectatio... | 1,150 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Theorem 3.1 (Multi-marginal Doob’s hℎh-transform).
Let us define a sequence of functions {hi}i∈[1:k]subscriptsubscriptℎ𝑖𝑖delimited-[]:1𝑘\{h_{i}\}_{i\in[1:k]}, where eac... | 1,212 |
38 | We start with the interval [ti−1,ti)subscript𝑡𝑖1subscript𝑡𝑖[t_{i-1},t_{i}) without loss of generality. For all t∈[ti−1,ti)𝑡subscript𝑡𝑖1subscript𝑡𝑖t\in[t_{i-1},t_{i}) and for any Ati⊂ℝdsubscript𝐴subscript𝑡𝑖superscriptℝ𝑑A_{t_{i}}\subset\mathbb{R}^{d}, the transition kernel of the conditioned process is defin... | 647 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
We start with the interval [ti−1,ti)subscript𝑡𝑖1subscript𝑡𝑖[t_{i-1},t_{i}) without loss of generality. For all t∈[ti−1,ti)𝑡subscript𝑡𝑖1subscript𝑡𝑖t\in[t_{i-1... | 691 |
39 | , 1 = ∫ℝd𝐏ihi(𝐱t,d𝐱ti)subscriptsuperscriptℝ𝑑subscriptsuperscript𝐏subscriptℎ𝑖𝑖subscript𝐱𝑡𝑑subscript𝐱subscript𝑡𝑖\displaystyle\int_{\mathbb{R}^{d}}\mathbf{P}^{h_{i}}_{i}(\mathbf{x}_{t},d\mathbf{x}_{t_{i}}). , 2 = =∫ℝdhi(ti,𝐗ti)hi(t,𝐱t)𝐏i(𝐱t,d𝐱ti)absentsubscriptsuperscriptℝ𝑑subscriptℎ𝑖subscript𝑡... | 1,758 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = ∫ℝd𝐏ihi(𝐱t,d𝐱ti)subscriptsuperscriptℝ𝑑subscriptsuperscript𝐏subscriptℎ𝑖𝑖subscript𝐱𝑡𝑑subscript𝐱subscript𝑡𝑖\displaystyle\int_{\mathbb{R}^{d}}\mathbf... | 1,802 |
40 | , 1 = 𝒜thiφtsubscriptsuperscript𝒜subscriptℎ𝑖𝑡subscript𝜑𝑡\displaystyle\mathcal{A}^{h_{i}}_{t}\varphi_{t}. , 2 = =lims↓0𝔼ℙh[φ(ts,𝐗t+s)|𝐗t=𝐱]−φ(t,𝐱)sabsentsubscript↓𝑠0subscript𝔼superscriptℙℎdelimited-[]conditional𝜑subscript𝑡𝑠subscript𝐗𝑡𝑠subscript𝐗𝑡𝐱𝜑𝑡𝐱𝑠\displaystyle=\lim_{s\downarrow 0}\frac{... | 1,822 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = 𝒜thiφtsubscriptsuperscript𝒜subscriptℎ𝑖𝑡subscript𝜑𝑡\displaystyle\mathcal{A}^{h_{i}}_{t}\varphi_{t}. , 2 = =lims↓0𝔼ℙh[φ(ts,𝐗t+s)|𝐗t=𝐱]−φ(t,𝐱)sabsen... | 1,866 |
41 | , 1 = φt+shi,t+s=φ0hi,0+∫0t+sφu𝑑hi,u+∫0t+shi,u𝑑φu+[φ,hi]t+ssubscript𝜑𝑡𝑠subscriptℎ𝑖𝑡𝑠subscript𝜑0subscriptℎ𝑖0superscriptsubscript0𝑡𝑠subscript𝜑𝑢differential-dsubscriptℎ𝑖𝑢superscriptsubscript0𝑡𝑠subscriptℎ𝑖𝑢differential-dsubscript𝜑𝑢subscript𝜑subscriptℎ𝑖𝑡𝑠\displaystyle\varphi_{t+s}h_{i,t+s}=\var... | 1,682 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = φt+shi,t+s=φ0hi,0+∫0t+sφu𝑑hi,u+∫0t+shi,u𝑑φu+[φ,hi]t+ssubscript𝜑𝑡𝑠subscriptℎ𝑖𝑡𝑠subscript𝜑0subscriptℎ𝑖0superscriptsubscript0𝑡𝑠subscript𝜑𝑢differe... | 1,726 |
42 | , 1 = dφt=𝒜tφtdt+(∇𝐱φ)⊤d𝐖t,dhi,t=𝒜thi,tdt+(∇𝐱hi,t)⊤d𝐖t.formulae-sequence𝑑subscript𝜑𝑡subscript𝒜𝑡subscript𝜑𝑡𝑑𝑡superscriptsubscript∇𝐱𝜑top𝑑subscript𝐖𝑡𝑑subscriptℎ𝑖𝑡subscript𝒜𝑡subscriptℎ𝑖𝑡𝑑𝑡superscriptsubscript∇𝐱subscriptℎ𝑖𝑡top𝑑subscript𝐖𝑡d\varphi_{t}=\mathcal{A}_{t}\varphi_{t}d... | 1,521 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = dφt=𝒜tφtdt+(∇𝐱φ)⊤d𝐖t,dhi,t=𝒜thi,tdt+(∇𝐱hi,t)⊤d𝐖t.formulae-sequence𝑑subscript𝜑𝑡subscript𝒜𝑡subscript𝜑𝑡𝑑𝑡superscriptsubscript∇𝐱𝜑top𝑑s... | 1,565 |
43 | , 1 = 𝔼ℙt,𝐱[∫tt+s(φu−φt)𝒜uhi,u𝑑u]subscriptsuperscript𝔼𝑡𝐱ℙdelimited-[]superscriptsubscript𝑡𝑡𝑠subscript𝜑𝑢subscript𝜑𝑡subscript𝒜𝑢subscriptℎ𝑖𝑢differential-d𝑢\displaystyle\mathbb{E}^{t,\mathbf{x}}_{\mathbb{P}}\left[\int_{t}^{t+s}(\varphi_{u}-\varphi_{t})\mathcal{A}_{u}h_{i,u}du\right]. , 2 = ≤(𝔼ℙt,𝐱... | 1,647 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = 𝔼ℙt,𝐱[∫tt+s(φu−φt)𝒜uhi,u𝑑u]subscriptsuperscript𝔼𝑡𝐱ℙdelimited-[]superscriptsubscript𝑡𝑡𝑠subscript𝜑𝑢subscript𝜑𝑡subscript𝒜𝑢subscriptℎ𝑖𝑢differe... | 1,691 |
44 | for any u∈[ti−1,ti)𝑢subscript𝑡𝑖1subscript𝑡𝑖u\in[t_{i-1},t_{i}) and ℙℙ\mathbb{P} almost surely. In other words, supu∈[t,t+s]𝔼ℙt,𝐱[|𝒜uhi,u|p]<∞,∀t∈[0,T],s>0,andp>1formulae-sequencesubscriptsupremum𝑢𝑡𝑡𝑠subscriptsuperscript𝔼𝑡𝐱ℙdelimited-[]superscriptsubscript𝒜𝑢subscriptℎ𝑖𝑢𝑝formulae-sequencefor-all𝑡0... | 1,935 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
for any u∈[ti−1,ti)𝑢subscript𝑡𝑖1subscript𝑡𝑖u\in[t_{i-1},t_{i}) and ℙℙ\mathbb{P} almost surely. In other words, supu∈[t,t+s]𝔼ℙt,𝐱[|𝒜uhi,u|p]<∞,∀t∈[0,T],s>0,a... | 1,979 |
45 | Therefore, the infinitesimal generator for 𝐏ihisubscriptsuperscript𝐏subscriptℎ𝑖𝑖\mathbf{P}^{h_{i}}_{i} is defined by:
, 1 = 𝒜thiφt=𝒜tφt+(∇𝐱φt)⊤∇𝐱loghi,tsubscriptsuperscript𝒜subscriptℎ𝑖𝑡subscript𝜑𝑡subscript𝒜𝑡subscript𝜑𝑡superscriptsubscript∇𝐱subscript𝜑𝑡topsubscript∇𝐱subscriptℎ𝑖𝑡\mathcal{A}^{h_{... | 1,358 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
Therefore, the infinitesimal generator for 𝐏ihisubscriptsuperscript𝐏subscriptℎ𝑖𝑖\mathbf{P}^{h_{i}}_{i} is defined by:
, 1 = 𝒜thiφt=𝒜tφt+(∇𝐱φt)⊤∇𝐱loghi,tsu... | 1,402 |
46 | , 1 = dℙh(𝐱0:T)𝑑superscriptℙℎsubscript𝐱:0𝑇\displaystyle d\mathbb{P}^{h}(\mathbf{x}_{0:T}). , 2 = =dμ0⋆(𝐱0)∏i=1k[∏j=1N𝐏i(j)hi(𝐱ti(j−1),d𝐱ti(j))]absent𝑑subscriptsuperscript𝜇⋆0subscript𝐱0superscriptsubscriptproduct𝑖1𝑘delimited-[]superscriptsubscriptproduct𝑗1𝑁superscriptsubscript𝐏𝑖𝑗subscriptℎ𝑖s... | 1,863 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = dℙh(𝐱0:T)𝑑superscriptℙℎsubscript𝐱:0𝑇\displaystyle d\mathbb{P}^{h}(\mathbf{x}_{0:T}). , 2 = =dμ0⋆(𝐱0)∏i=1k[∏j=1N𝐏i(j)hi(𝐱ti(j−1),d𝐱ti(j))]absen... | 1,907 |
47 | , 1 = dℙh(𝐱0:T)𝑑superscriptℙℎsubscript𝐱:0𝑇\displaystyle d\mathbb{P}^{h}(\mathbf{x}_{0:T}). , 2 = =∏i=1kfi(𝐲ti|𝐱ti)dℙ(𝐱0:T)absentsuperscriptsubscriptproduct𝑖1𝑘subscript𝑓𝑖conditionalsubscript𝐲subscript𝑡𝑖subscript𝐱subscript𝑡𝑖𝑑ℙsubscript𝐱:0𝑇\displaystyle=\prod_{i=1}^{k}f_{i}(\mathbf{y}_{t_{i}}|\ma... | 534 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.1 Proof of Theorem 3.1
Proof.
, 1 = dℙh(𝐱0:T)𝑑superscriptℙℎsubscript𝐱:0𝑇\displaystyle d\mathbb{P}^{h}(\mathbf{x}_{0:T}). , 2 = =∏i=1kfi(𝐲ti|𝐱ti)dℙ(𝐱0:T)absentsuperscriptsubscriptprodu... | 578 |
48 | Let us consider a sequence of left continuous functions {𝒱i}i∈[1:k+1]subscriptsubscript𝒱𝑖𝑖delimited-[]:1𝑘1\{\mathcal{V}_{i}\}_{i\in[1:k+1]}, where each 𝒱i∈C1,2([ti−1,ti)×ℝd)subscript𝒱𝑖superscript𝐶12subscript𝑡𝑖1subscript𝑡𝑖superscriptℝ𝑑\mathcal{V}_{i}\in C^{1,2}([t_{i-1},t_{i})\times\mathbb{R}^{d})
, 1 =... | 1,337 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Theorem 3.2 (Dynamic Programming Principle).
Let us consider a sequence of left continuous functions {𝒱i}i∈[1:k+1]subscriptsubscript𝒱𝑖𝑖delimited-[]:1𝑘1\{\mathcal{V}_{... | 1,390 |
49 | Following the approach used in the proof of the standard dynamic programming principle with the flow property induced by Markov control (Van Handel, 2007), we can apply similar methods to our cost function. We start the proof by establishing the recursion of 𝒥𝒥\mathcal{J}. Let us define the sequence of left continuou... | 1,393 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Proof.
Following the approach used in the proof of the standard dynamic programming principle with the flow property induced by Markov control (Van Handel, 2007), we can app... | 1,437 |
50 | , 1 = 𝒱(t,𝐱)+ϵ≥𝒥(t,𝐱,α′)𝒱𝑡𝐱italic-ϵ𝒥𝑡𝐱superscript𝛼′\displaystyle\mathcal{V}(t,\mathbf{x})+\epsilon\geq\mathcal{J}(t,\mathbf{x},\alpha^{\prime}). , 2 = . , 3 = (83). , 1 = =𝔼ℙα′t,𝐱t[∫ttI(u)12∥αs′∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα′)+𝒥I(u)+1(tI(u),𝐗tI(u)α′,α′)]absentsubscriptsuperscript𝔼𝑡s... | 1,697 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Proof.
, 1 = 𝒱(t,𝐱)+ϵ≥𝒥(t,𝐱,α′)𝒱𝑡𝐱italic-ϵ𝒥𝑡𝐱superscript𝛼′\displaystyle\mathcal{V}(t,\mathbf{x})+\epsilon\geq\mathcal{J}(t,\mathbf{x},\alpha^{\prime}). , 2 = . ... | 1,741 |
51 | , 1 = 𝒱(t,𝐱)≥infα′∈𝔸[t,T]𝔼ℙα′t,𝐱t[∫ttI(u)12∥αs′∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα′)+𝒱I(u)+1(tI(u),𝐗tI(u)α′)].𝒱𝑡𝐱subscriptinfimumsuperscript𝛼′𝔸𝑡𝑇subscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙsuperscript𝛼′delimited-[]superscriptsubscript𝑡subscript𝑡𝐼𝑢12superscriptdelimited-∥∥subscript... | 819 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Proof.
, 1 = 𝒱(t,𝐱)≥infα′∈𝔸[t,T]𝔼ℙα′t,𝐱t[∫ttI(u)12∥αs′∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα′)+𝒱I(u)+1(tI(u),𝐗tI(u)α′)].𝒱𝑡𝐱subscriptinfimumsuperscript�... | 863 |
52 | , 1 = . , 2 = 𝒥(t,𝐱,α~)≥infα2∈𝔸[tI(u),T]𝒥(t,𝐱,α~)𝒥𝑡𝐱~𝛼subscriptinfimumsuperscript𝛼2𝔸subscript𝑡𝐼𝑢𝑇𝒥𝑡𝐱~𝛼\displaystyle\mathcal{J}(t,\mathbf{x},\tilde{\alpha})\geq\inf_{\alpha^{2}\in\mathbb{A}[t_{I(u)},T]}\mathcal{J}(t,\mathbf{x},\tilde{\alpha}). , 3 = . , 4 = (89). , 1 = . , 2 = =𝔼ℙα1t,𝐱t[∫ttI(u... | 1,862 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Proof.
, 1 = . , 2 = 𝒥(t,𝐱,α~)≥infα2∈𝔸[tI(u),T]𝒥(t,𝐱,α~)𝒥𝑡𝐱~𝛼subscriptinfimumsuperscript𝛼2𝔸subscript𝑡𝐼𝑢𝑇𝒥𝑡𝐱~𝛼\displaystyle\mathcal{J}(t,\mathbf{x},\ti... | 1,906 |
53 | , 1 = 𝒱(t,𝐱)=minα∈𝔸𝔼ℙα[∫ttI(u)12∥αs∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα)+𝒱I(u)+1(tI(u),𝐗tI(u)α)|𝐗tα=𝐱t].𝒱𝑡𝐱subscript𝛼𝔸subscript𝔼superscriptℙ𝛼delimited-[]superscriptsubscript𝑡subscript𝑡𝐼𝑢12superscriptdelimited-∥∥subscript𝛼𝑠2differential-d𝑠subscript:𝑖𝑡subscript𝑡𝑖subscript𝑡𝐼𝑢subs... | 547 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.2 Proof of Theorem 3.2
Proof.
, 1 = 𝒱(t,𝐱)=minα∈𝔸𝔼ℙα[∫ttI(u)12∥αs∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα)+𝒱I(u)+1(tI(u),𝐗tI(u)α)|𝐗tα=𝐱t].𝒱𝑡𝐱subscript𝛼𝔸subscript𝔼superscrip... | 591 |
54 | Suppose there exist a sequence of left continuous functions 𝒱i(t,𝐱)∈C1,2([ti−1,ti),ℝd)subscript𝒱𝑖𝑡𝐱superscript𝐶12subscript𝑡𝑖1subscript𝑡𝑖superscriptℝ𝑑\mathcal{V}_{i}(t,\mathbf{x})\in C^{1,2}([t_{i-1},t_{i}),\mathbb{R}^{d}), for all i∈[1:k]i\in[1:k], satisfying the following Hamiltonian-Jacobi-Bellman (HJB)... | 1,320 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Theorem 3.3 (Verification Theorem).
Suppose there exist a sequence of left continuous functions 𝒱i(t,𝐱)∈C1,2([ti−1,ti),ℝd)subscript𝒱𝑖𝑡𝐱superscript𝐶12subscript𝑡𝑖1s... | 1,373 |
55 | Without loss of generality, consider t∈[ti−1,ti)𝑡subscript𝑡𝑖1subscript𝑡𝑖t\in[t_{i-1},t_{i}). By applying the Itô’s formula to the value function 𝒱𝒱\mathcal{V} and taking expectation with respect to ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha}, we obtain
, 1 = 𝔼ℙαt,𝐱t[𝒱i(ti,𝐗tiα)]=𝒱i(t,𝐱)+𝔼ℙαt,𝐱t[∫tti(∂t𝒱i,s+... | 890 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Proof.
Without loss of generality, consider t∈[ti−1,ti)𝑡subscript𝑡𝑖1subscript𝑡𝑖t\in[t_{i-1},t_{i}). By applying the Itô’s formula to the value function 𝒱𝒱\mathcal{V} ... | 934 |
56 | , 1 = LHS. , 2 = =𝔼ℙαt,𝐱t[𝒱i(ti,𝐗tiα)]+𝔼ℙαt,𝐱t[∫tti12∥αi,s∥2𝑑s]absentsubscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-[]subscript𝒱𝑖subscript𝑡𝑖subscriptsuperscript𝐗𝛼subscript𝑡𝑖subscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-[]superscriptsubscript𝑡subscript𝑡𝑖12superscriptde... | 1,131 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Proof.
, 1 = LHS. , 2 = =𝔼ℙαt,𝐱t[𝒱i(ti,𝐗tiα)]+𝔼ℙαt,𝐱t[∫tti12∥αi,s∥2𝑑s]absentsubscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-[]subscript𝒱𝑖subscript... | 1,175 |
57 | , 1 = RHS=𝒱i(t,𝐱)+𝔼ℙαt,𝐱t[∫tti(∂t𝒱i,s+𝒜t𝒱i,s+(∇𝐱𝒱i,s)⊤αi,s)𝑑s]+𝔼ℙαt,𝐱t[∫tti12∥αi,s∥2𝑑s]RHSsubscript𝒱𝑖𝑡𝐱subscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-[]superscriptsubscript𝑡subscript𝑡𝑖subscript𝑡subscript𝒱𝑖𝑠subscript𝒜𝑡subscript𝒱𝑖𝑠superscriptsubscript∇𝐱subscript𝒱𝑖𝑠to... | 1,497 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Proof.
, 1 = RHS=𝒱i(t,𝐱)+𝔼ℙαt,𝐱t[∫tti(∂t𝒱i,s+𝒜t𝒱i,s+(∇𝐱𝒱i,s)⊤αi,s)𝑑s]+𝔼ℙαt,𝐱t[∫tti12∥αi,s∥2𝑑s]RHSsubscript𝒱𝑖𝑡𝐱subscriptsuperscript𝔼𝑡subscript𝐱𝑡s... | 1,541 |
58 | , 1 = 𝒥i(t,𝐱,α)=𝒱i(t,𝐱)subscript𝒥𝑖𝑡𝐱𝛼subscript𝒱𝑖𝑡𝐱\displaystyle\mathcal{J}_{i}(t,\mathbf{x},\alpha)=\mathcal{V}_{i}(t,\mathbf{x}). , 2 = . , 3 = . , 1 = +𝔼ℙαt,𝐱t[∫tti([(∇𝐱𝒱i,s)⊤αi,s+12∥αi,s∥2]−minα∈𝔸[(∇𝐱𝒱i,s)⊤αi,s+12∥αi,s∥2])𝑑s].subscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-... | 1,558 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Proof.
, 1 = 𝒥i(t,𝐱,α)=𝒱i(t,𝐱)subscript𝒥𝑖𝑡𝐱𝛼subscript𝒱𝑖𝑡𝐱\displaystyle\mathcal{J}_{i}(t,\mathbf{x},\alpha)=\mathcal{V}_{i}(t,\mathbf{x}). , 2 = . , 3 = . , 1 ... | 1,602 |
59 | , 1 = 𝒱(t,𝐱t)𝒱𝑡subscript𝐱𝑡\displaystyle\mathcal{V}(t,\mathbf{x}_{t}). , 2 = =minα∈𝔸𝔼ℙαt,𝐱t[∫ttI(u)12∥αs∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα)+𝒱I(u)+1(tI(u),𝐗tI(u)α)]absentsubscript𝛼𝔸subscriptsuperscript𝔼𝑡subscript𝐱𝑡superscriptℙ𝛼delimited-[]superscriptsubscript𝑡subscript𝑡𝐼𝑢12superscrip... | 1,506 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.3 Proof of Theorem 3.3
Proof.
, 1 = 𝒱(t,𝐱t)𝒱𝑡subscript𝐱𝑡\displaystyle\mathcal{V}(t,\mathbf{x}_{t}). , 2 = =minα∈𝔸𝔼ℙαt,𝐱t[∫ttI(u)12∥αs∥2𝑑s−∑i:{t≤ti≤tI(u)}logfi(𝐲ti|𝐗tiα)+𝒱I(u... | 1,550 |
60 | We first restate the Feynman-Kac formula (Oksendal, 1992; Baldi, 2017), which gives a probabilistic representation of the solution to certain PDEs using expectations of stochastic processes. It relies on the fact that the conditional expectation is martingale process. | 63 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
We first restate the Feynman-Kac formula (Oksendal, 1992; Baldi, 2017), which gives a probabilistic representation of the solution to certain PDEs using expectations of stocha... | 104 |
61 | Let us define f∈C2(ℝd)𝑓superscript𝐶2superscriptℝ𝑑f\in C^{2}(\mathbb{R}^{d}) and g∈C(ℝd)𝑔𝐶superscriptℝ𝑑g\in C(\mathbb{R}^{d}). Then, a function h(t,𝐱t)=𝔼ℙ[e−∫tTf(s,𝐗s)𝑑sg(𝐗T)|𝐗t=𝐱t]ℎ𝑡subscript𝐱𝑡subscript𝔼ℙdelimited-[]conditionalsuperscript𝑒superscriptsubscript𝑡𝑇𝑓𝑠subscript𝐗𝑠differential-d... | 534 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
Lemma A.5 (The Feynman-Kac formula).
Let us define f∈C2(ℝd)𝑓superscript𝐶2superscriptℝ𝑑f\in C^{2}(\mathbb{R}^{d}) and g∈C(ℝd)𝑔𝐶superscriptℝ𝑑g\in C(\mathbb{R}^{d}). Then... | 588 |
62 | Define the process 𝐘t=e−∫tTf(s,𝐗s)𝑑sh(t,𝐗t)subscript𝐘𝑡superscript𝑒superscriptsubscript𝑡𝑇𝑓𝑠subscript𝐗𝑠differential-d𝑠ℎ𝑡subscript𝐗𝑡\mathbf{Y}_{t}=e^{-\int_{t}^{T}f(s,\mathbf{X}_{s})ds}h(t,\mathbf{X}_{t}). Since hℎh is a conditional expectation with respect to ℙℙ\mathbb{P}, implies that 𝐘tsubscript𝐘... | 1,550 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
Proof.
Define the process 𝐘t=e−∫tTf(s,𝐗s)𝑑sh(t,𝐗t)subscript𝐘𝑡superscript𝑒superscriptsubscript𝑡𝑇𝑓𝑠subscript𝐗𝑠differential-d𝑠ℎ𝑡subscript𝐗𝑡\mathbf{Y}_{t}=e^{... | 1,593 |
63 | The hℎh function satisfying the following linear PDE:
, 1 = . , 2 = ∂thi,t+𝒜thi,t=0,ti−1≤t<tiformulae-sequencesubscript𝑡subscriptℎ𝑖𝑡subscript𝒜𝑡subscriptℎ𝑖𝑡0subscript𝑡𝑖1𝑡subscript𝑡𝑖\displaystyle\partial_{t}h_{i,t}+\mathcal{A}_{t}h_{i,t}=0,\quad t_{i-1}\leq t<t_{i}. , 3 = . , 4 = (117). , 1 = . , 2 = hi(ti,... | 344 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
Lemma 3.4 (Hopf-Cole Transformation).
The hℎh function satisfying the following linear PDE:
, 1 = . , 2 = ∂thi,t+𝒜thi,t=0,ti−1≤t<tiformulae-sequencesubscript𝑡subscriptℎ𝑖𝑡... | 397 |
64 | The linaer PDE presented in (117-118) can be directly derived from the function hi(t,𝐱t)=𝔼ℙ[∏j≥i−1kfj(𝐗tj)|𝐗t=𝐱t]subscriptℎ𝑖𝑡subscript𝐱𝑡subscript𝔼ℙdelimited-[]conditionalsuperscriptsubscriptproduct𝑗𝑖1𝑘subscript𝑓𝑗subscript𝐗subscript𝑡𝑗subscript𝐗𝑡subscript𝐱𝑡h_{i}(t,\mathbf{x}_{t})=\mathbb{E}_{\mat... | 1,941 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
Proof.
The linaer PDE presented in (117-118) can be directly derived from the function hi(t,𝐱t)=𝔼ℙ[∏j≥i−1kfj(𝐗tj)|𝐗t=𝐱t]subscriptℎ𝑖𝑡subscript𝐱𝑡subscript𝔼ℙdelimite... | 1,984 |
65 | where (i)𝑖(i) follows from (117). Now, we can simplify (123) by dividing both sides with h>0ℎ0h>0:
, 1 = ∂t𝒱i,tsubscript𝑡subscript𝒱𝑖𝑡\displaystyle\partial_{t}\mathcal{V}_{i,t}. , 2 = =(−∇𝐱𝒱i,t)⊤bt+12Trace[∥∇𝐱𝒱i,t∥2]−12Trace[∇𝐱𝐱𝒱i,t]absentsuperscriptsubscript∇𝐱subscript𝒱𝑖𝑡topsubscript𝑏𝑡12Tracedel... | 1,148 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.4 Proof of Lemma 3.4
Proof.
where (i)𝑖(i) follows from (117). Now, we can simplify (123) by dividing both sides with h>0ℎ0h>0:
, 1 = ∂t𝒱i,tsubscript𝑡subscript𝒱𝑖𝑡\displaystyle\partial_{t}\mat... | 1,191 |
66 | For optimal control α⋆superscript𝛼⋆\alpha^{\star} induced by the cost function (7) with dynamics (6), it satisfies α⋆=∇𝐱loghsuperscript𝛼⋆subscript∇𝐱ℎ\alpha^{\star}=\nabla_{\mathbf{x}}\log h. In other words, we can simulate the conditional SDEs in (5) by simulating the controlled SDE (6) with optimal control α⋆supe... | 138 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.5 Proof of Corollary 3.5
Corollary 3.5 (Optimal Control).
For optimal control α⋆superscript𝛼⋆\alpha^{\star} induced by the cost function (7) with dynamics (6), it satisfies α⋆=∇𝐱loghsuperscript... | 193 |
67 | Lemma 3.4 implies that we can obtain the relation −𝒱i(t,𝐱)=loghi(t,𝐱)subscript𝒱𝑖𝑡𝐱subscriptℎ𝑖𝑡𝐱-\mathcal{V}_{i}(t,\mathbf{x})=\log h_{i}(t,\mathbf{x}). Moreover, by following the definition of the optimal control α⋆superscript𝛼⋆\alpha^{\star} in (12) and the value function 𝒱𝒱\mathcal{V} in (110), it sug... | 468 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.5 Proof of Corollary 3.5
Proof.
Lemma 3.4 implies that we can obtain the relation −𝒱i(t,𝐱)=loghi(t,𝐱)subscript𝒱𝑖𝑡𝐱subscriptℎ𝑖𝑡𝐱-\mathcal{V}_{i}(t,\mathbf{x})=\log h_{i}(t,\mathbf{x}).... | 513 |
68 | The Donsker–Varadhan variational principle provides a variational formula for the large deviations of functionals of Brownian motion, often related to free-energy minimization problems (Boué & Dupuis, 1998). Moreover, through Girsanov’s theorem, this principle extends to a wide range of Markov processes, including Itô ... | 97 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
The Donsker–Varadhan variational principle provides a variational formula for the large deviations of functionals of Brownian motion, often related to free-energy minimizati... | 139 |
69 | For a bounded and measurable functions 𝒲:C([0,T],ℝd)→ℝ:𝒲→𝐶0𝑇superscriptℝ𝑑ℝ\mathcal{W}:C([0,T],\mathbb{R}^{d})\to\mathbb{R}, following relation holds:
, 1 = −log𝔼𝐗∼ℙ[e−𝒲(𝐗0:T)]=minℚ≪ℙ[𝔼𝐘∼ℚ[𝒲(𝐘[0,T])]+DKL(ℚ|ℙ)]subscript𝔼similar-to𝐗ℙdelimited-[]superscript𝑒𝒲subscript𝐗:0𝑇subscriptmuch-less-thanℚℙ... | 374 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Lemma A.6 (Donsker–Varadhan Variational Principle).
For a bounded and measurable functions 𝒲:C([0,T],ℝd)→ℝ:𝒲→𝐶0𝑇superscriptℝ𝑑ℝ\mathcal{W}:C([0,T],\mathbb{R}^{d})\to\ma... | 432 |
70 | The proof relies on the change of measure and the Jensen’s inequality:
, 1 = −log𝔼𝐗∼ℙ[e−𝒲(𝐗0:T)]subscript𝔼similar-to𝐗ℙdelimited-[]superscript𝑒𝒲subscript𝐗:0𝑇\displaystyle-\log\mathbb{E}_{\mathbf{X}\sim\mathbb{P}}\left[e^{-\mathcal{W}(\mathbf{X}_{0:T})}\right]. , 2 = =−log𝔼𝐘∼ℚ[e−𝒲(𝐘0:T)dℙdℚ(𝐘0:T)... | 926 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
The proof relies on the change of measure and the Jensen’s inequality:
, 1 = −log𝔼𝐗∼ℙ[e−𝒲(𝐗0:T)]subscript𝔼similar-to𝐗ℙdelimited-[]superscript𝑒𝒲subscript𝐗:... | 970 |
71 | Let us assume that the path measure ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha} induced by (6) for any α∈𝔸𝛼𝔸\alpha\in\mathbb{A} satisfies DKL(ℙα|ℙ⋆)<∞subscript𝐷KLconditionalsuperscriptℙ𝛼superscriptℙ⋆D_{\text{KL}}(\mathbb{P}^{\alpha}|\mathbb{P}^{\star})<\infty. Then, for a cost function 𝒥𝒥\mathcal{J} in (7) and μ0⋆subsc... | 1,117 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Theorem 3.6 (Tight Variational Bound).
Let us assume that the path measure ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha} induced by (6) for any α∈𝔸𝛼𝔸\alpha\in\mathbb{A} satisfies D... | 1,172 |
72 | We begin by deriving the KL-divergence between ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha} and ℙ⋆superscriptℙ⋆\mathbb{P}^{\star}:
, 1 = DKL(ℙα|ℙ⋆)subscript𝐷KLconditionalsuperscriptℙ𝛼superscriptℙ⋆\displaystyle D_{\text{KL}}(\mathbb{P}^{\alpha}|\mathbb{P}^{\star}). , 2 = =(i)DKL(μ0|μ0⋆)+𝔼𝐱0∼μ0[DKL(ℙα(⋅|𝐗0α)|ℙ⋆(⋅|𝐗0α))|𝐗0... | 1,749 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
We begin by deriving the KL-divergence between ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha} and ℙ⋆superscriptℙ⋆\mathbb{P}^{\star}:
, 1 = DKL(ℙα|ℙ⋆)subscript𝐷KLconditionalsup... | 1,793 |
73 | , 1 = 𝔼ℙα0,𝐱0[logdℙdℙ⋆(𝐗0:Tα)]=𝔼ℙα0,𝐱0[−∑i=1klogfi(𝐲ti|𝐗tiα)].subscriptsuperscript𝔼0subscript𝐱0superscriptℙ𝛼delimited-[]𝑑ℙ𝑑superscriptℙ⋆subscriptsuperscript𝐗𝛼:0𝑇subscriptsuperscript𝔼0subscript𝐱0superscriptℙ𝛼delimited-[]superscriptsubscript𝑖1𝑘subscript𝑓𝑖conditionalsubscript𝐲subscript𝑡𝑖su... | 1,434 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
, 1 = 𝔼ℙα0,𝐱0[logdℙdℙ⋆(𝐗0:Tα)]=𝔼ℙα0,𝐱0[−∑i=1klogfi(𝐲ti|𝐗tiα)].subscriptsuperscript𝔼0subscript𝐱0superscriptℙ𝛼delimited-[]𝑑ℙ𝑑superscriptℙ⋆subscripts... | 1,478 |
74 | , 1 = 𝒥(0,𝐱0,α)𝒥0subscript𝐱0𝛼\displaystyle\mathcal{J}(0,\mathbf{x}_{0},\alpha). , 2 = =𝔼ℙα[∫0T12∥α(t,𝐗tα)∥2𝑑t−∑i=1klogfi(𝐲ti|𝐗ti)|𝐗0α=𝐱0]absentsubscript𝔼superscriptℙ𝛼delimited-[]superscriptsubscript0𝑇12superscriptdelimited-∥∥𝛼𝑡subscriptsuperscript𝐗𝛼𝑡2differential-d𝑡conditionalsuperscriptsubs... | 1,439 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
, 1 = 𝒥(0,𝐱0,α)𝒥0subscript𝐱0𝛼\displaystyle\mathcal{J}(0,\mathbf{x}_{0},\alpha). , 2 = =𝔼ℙα[∫0T12∥α(t,𝐗tα)∥2𝑑t−∑i=1klogfi(𝐲ti|𝐗ti)|𝐗0α=𝐱0]absentsubs... | 1,483 |
75 | , 1 = DKL(μ0|μ0⋆)subscript𝐷KLconditionalsubscript𝜇0subscriptsuperscript𝜇⋆0\displaystyle D_{\text{KL}}(\mu_{0}|\mu^{\star}_{0}). , 2 = =𝔼𝐱0∼μ0[logdμ0dμ0⋆(𝐱0)]=𝔼𝐱0∼μ0[−logh1(0,𝐱0)]absentsubscript𝔼similar-tosubscript𝐱0subscript𝜇0delimited-[]𝑑subscript𝜇0𝑑subscriptsuperscript𝜇⋆0subscript𝐱0subscript... | 1,810 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
, 1 = DKL(μ0|μ0⋆)subscript𝐷KLconditionalsubscript𝜇0subscriptsuperscript𝜇⋆0\displaystyle D_{\text{KL}}(\mu_{0}|\mu^{\star}_{0}). , 2 = =𝔼𝐱0∼μ0[logdμ0dμ0⋆(𝐱... | 1,854 |
76 | Thus, we obtain α⋆=argminα∈𝔸𝒥(0,𝐱0,α)=argminα∈𝔸𝒥~(0,𝐱0,α)superscript𝛼⋆subscriptargmin𝛼𝔸𝒥0subscript𝐱0𝛼subscriptargmin𝛼𝔸~𝒥0subscript𝐱0𝛼\alpha^{\star}=\operatorname*{arg\,min}_{\alpha\in\mathbb{A}}\mathcal{J}(0,\mathbf{x}_{0},\alpha)=\operatorname*{arg\,min}_{\alpha\in\mathbb{A}}\tilde{\mathcal{J}}(... | 1,926 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.6 Proof of Theorem 3.6
Proof.
Thus, we obtain α⋆=argminα∈𝔸𝒥(0,𝐱0,α)=argminα∈𝔸𝒥~(0,𝐱0,α)superscript𝛼⋆subscriptargmin𝛼𝔸𝒥0subscript𝐱0𝛼subscriptargmin𝛼𝔸~𝒥0subscript𝐱0𝛼\alpha^{\s... | 1,970 |
77 | Let 𝐨0:Tsubscript𝐨:0𝑇\mathbf{o}_{0:T} is given time-series data. Then, for an auxiliary variable 𝐲0:T∼qϕ(𝐲0:T|𝐨0:T)similar-tosubscript𝐲:0𝑇subscript𝑞italic-ϕconditionalsubscript𝐲:0𝑇subscript𝐨:0𝑇\mathbf{y}_{0:T}\sim q_{\phi}(\mathbf{y}_{0:T}|\mathbf{o}_{0:T}), the ELBO is given as
, 1 = logpψ(𝐨0:T)≥𝔼qϕ... | 1,902 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.7 Derivation of Amortized ELBO in (28).
Let 𝐨0:Tsubscript𝐨:0𝑇\mathbf{o}_{0:T} is given time-series data. Then, for an auxiliary variable 𝐲0:T∼qϕ(𝐲0:T|𝐨0:T)similar-tosubscript𝐲:0𝑇subscript... | 1,948 |
78 | where (i)𝑖(i) follows from 𝔼qϕ(𝐲0:T|𝐨0:T)[−∑i=1Klogqϕ(𝐲ti|𝐨ti)]=Csubscript𝔼subscript𝑞italic-ϕconditionalsubscript𝐲:0𝑇subscript𝐨:0𝑇delimited-[]superscriptsubscript𝑖1𝐾subscript𝑞italic-ϕconditionalsubscript𝐲subscript𝑡𝑖subscript𝐨subscript𝑡𝑖𝐶\mathbb{E}_{q_{\phi}(\mathbf{y}_{0:T}|\mathbf{o}_{0:T})}\... | 248 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.7 Derivation of Amortized ELBO in (28).
where (i)𝑖(i) follows from 𝔼qϕ(𝐲0:T|𝐨0:T)[−∑i=1Klogqϕ(𝐲ti|𝐨ti)]=Csubscript𝔼subscript𝑞italic-ϕconditionalsubscript𝐲:0𝑇subscript𝐨:0𝑇delimited-... | 294 |
79 | Let us consider sequences of SPD matrices {𝐀i}i∈[1:k]subscriptsubscript𝐀𝑖𝑖delimited-[]:1𝑘\{\mathbf{A}_{i}\}_{i\in[1:k]} and vectors {αi}i∈[1:k]subscriptsubscript𝛼𝑖𝑖delimited-[]:1𝑘\{\alpha_{i}\}_{i\in[1:k]} and the following control-affine SDEs for all i∈[1:k]i\in[1:k]:
, 1 = d𝐗t=[−𝐀i𝐗t+αi]dt+σd𝐖t... | 1,403 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.8 Proof of Theorem 3.8
Theorem 3.8 (Simulation-free estimation).
Let us consider sequences of SPD matrices {𝐀i}i∈[1:k]subscriptsubscript𝐀𝑖𝑖delimited-[]:1𝑘\{\mathbf{A}_{i}\}_{i\in[1:k]} and ... | 1,456 |
80 | Note that since {𝐀i}i∈[1:k]subscriptsubscript𝐀𝑖𝑖delimited-[]:1𝑘\{\mathbf{A}_{i}\}_{i\in[1:k]} are SPD matrices, we can apply the transformation outlined in Remark 3.7. For simplicity, we will treat {𝐀i}i∈[1:k]subscriptsubscript𝐀𝑖𝑖delimited-[]:1𝑘\{\mathbf{A}_{i}\}_{i\in[1:k]} as diagonal matrices, and expr... | 1,791 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.8 Proof of Theorem 3.8
Proof.
Note that since {𝐀i}i∈[1:k]subscriptsubscript𝐀𝑖𝑖delimited-[]:1𝑘\{\mathbf{A}_{i}\}_{i\in[1:k]} are SPD matrices, we can apply the transformation outlined in Rem... | 1,835 |
81 | , 1 = 𝚺t=𝔼ℙα[e−2Δi(t)𝐀i(𝐗ti−𝐦ti+𝐌i(t))(𝐗ti−𝐦ti+𝐌i(t))⊤]subscript𝚺𝑡subscript𝔼superscriptℙ𝛼delimited-[]superscript𝑒2subscriptΔ𝑖𝑡subscript𝐀𝑖subscript𝐗subscript𝑡𝑖subscript𝐦subscript𝑡𝑖subscript𝐌𝑖𝑡superscriptsubscript𝐗subscript𝑡𝑖subscript𝐦subscript𝑡𝑖subscript𝐌𝑖𝑡top\displaystyle\mat... | 1,874 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.8 Proof of Theorem 3.8
Proof.
, 1 = 𝚺t=𝔼ℙα[e−2Δi(t)𝐀i(𝐗ti−𝐦ti+𝐌i(t))(𝐗ti−𝐦ti+𝐌i(t))⊤]subscript𝚺𝑡subscript𝔼superscriptℙ𝛼delimited-[]superscript𝑒2subscriptΔ𝑖𝑡subscript𝐀𝑖sub... | 1,918 |
82 | Furthermore, given recurrence forms of mean (163) and covariance (166), the first two moments of Gaussian distribution for each time steps tisubscript𝑡𝑖t_{i} can be computed sequentially. For simplicity, assume that 𝐀=𝐀i=𝐀j𝐀subscript𝐀𝑖subscript𝐀𝑗\mathbf{A}=\mathbf{A}_{i}=\mathbf{A}_{j} for all i,j∈[1:k]i,j\in... | 1,274 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.8 Proof of Theorem 3.8
Proof.
Furthermore, given recurrence forms of mean (163) and covariance (166), the first two moments of Gaussian distribution for each time steps tisubscript𝑡𝑖t_{i} can be... | 1,318 |
83 | , 1 = 𝚺t1subscript𝚺subscript𝑡1\displaystyle\mathbf{\Sigma}_{t_{1}}. , 2 = =e−2Δ0(t1)𝐀𝚺t0−12𝐀−1(e−2Δ0(t1)𝐀−𝐈)absentsuperscript𝑒2subscriptΔ0subscript𝑡1𝐀subscript𝚺subscript𝑡012superscript𝐀1superscript𝑒2subscriptΔ0subscript𝑡1𝐀𝐈\displaystyle=e^{-2\Delta_{0}(t_{1})\mathbf{A}}\mathbf{\Sigma}_{t_{0}}... | 1,158 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix A Proofs and Derivations
A.8 Proof of Theorem 3.8
Proof.
, 1 = 𝚺t1subscript𝚺subscript𝑡1\displaystyle\mathbf{\Sigma}_{t_{1}}. , 2 = =e−2Δ0(t1)𝐀𝚺t0−12𝐀−1(e−2Δ0(t1)𝐀−𝐈)absentsuperscript𝑒2subscriptΔ0subscript�... | 1,202 |
84 | The parallelization of scan operation so called all-prefix-sums algorithms (Blelloch, 1990) have been well studied. Given an associative operator ⊗tensor-product\otimes and a sequence of elements [st1,⋯stK]subscript𝑠subscript𝑡1⋯subscript𝑠subscript𝑡𝐾[s_{t_{1}},\cdots s_{t_{K}}], the parallel scan algorithm compute... | 1,805 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix B Parallel Scan
The parallelization of scan operation so called all-prefix-sums algorithms (Blelloch, 1990) have been well studied. Given an associative operator ⊗tensor-product\otimes and a sequence of elements [st1,⋯stK]... | 1,831 |
85 | , 1 = 𝐌i⊗𝐌i+1=(𝐀^i∘𝐀^i−1,𝐀^i∘𝐁^i−1αi−1+𝐁^iαi)tensor-productsubscript𝐌𝑖subscript𝐌𝑖1subscript^𝐀𝑖subscript^𝐀𝑖1subscript^𝐀𝑖subscript^𝐁𝑖1subscript𝛼𝑖1subscript^𝐁𝑖subscript𝛼𝑖\displaystyle\mathbf{M}_{i}\otimes\mathbf{M}_{i+1}=(\hat{\mathbf{A}}_{i}\circ\hat{\mathbf{A}}_{i-1},\hat{\mathbf{A}}_{i}\circ\... | 1,864 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix B Parallel Scan
, 1 = 𝐌i⊗𝐌i+1=(𝐀^i∘𝐀^i−1,𝐀^i∘𝐁^i−1αi−1+𝐁^iαi)tensor-productsubscript𝐌𝑖subscript𝐌𝑖1subscript^𝐀𝑖subscript^𝐀𝑖1subscript^𝐀𝑖subscript^𝐁𝑖1subscript𝛼𝑖1subscript^𝐁𝑖subscript𝛼𝑖\displaystyle... | 1,890 |
86 | Now, the operator ⊗tensor-product\otimes yields similar results for 𝐒isubscript𝐒𝑖\mathbf{S}_{i}, both 𝐌isubscript𝐌𝑖\mathbf{M}_{i} and 𝐒isubscript𝐒𝑖\mathbf{S}_{i} can be computed using a parallel scan algorithm. In other words, we can access the marginal distributions for ℙαsuperscriptℙ𝛼\mathbb{P}^{\alpha} at ... | 158 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix B Parallel Scan
Now, the operator ⊗tensor-product\otimes yields similar results for 𝐒isubscript𝐒𝑖\mathbf{S}_{i}, both 𝐌isubscript𝐌𝑖\mathbf{M}_{i} and 𝐒isubscript𝐒𝑖\mathbf{S}_{i} can be computed using a parallel sca... | 184 |
87 | The Human Activity dataset666https://doi.org/10.24432/C57G8X under CC BY 4.0 consists of time series data collected from five individuals performing different activities. Following the preprocessing steps described in (Rubanova et al., 2019), we obtained 6,55465546,554 sequences, each with 211211211 time points and a f... | 182 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.1 Datasets
Human Activity
The Human Activity dataset666https://doi.org/10.24432/C57G8X under CC BY 4.0 consists of time series data collected from five individuals performing different activities.... | 217 |
88 | The pendulum images were algorithmically generated through numerical simulation as outlined in (Becker et al., 2019). We followed the setup described in (Schirmer et al., 2022), where 4,000 image sequences were generated. Each sequence consists of 50 time stamps, irregularly sampled from T=100𝑇100T=100, with each imag... | 145 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.1 Datasets
Pendulum
The pendulum images were algorithmically generated through numerical simulation as outlined in (Becker et al., 2019). We followed the setup described in (Schirmer et al., 2022)... | 181 |
89 | The USHCN dataset (Menne et al., 2015)777https://data.ess-dive.lbl.gov/view/doi:10.3334/CDIAC/CLI.NDP019 under CC BY 4.0 includes daily measurements from 1,21812181,218 weather stations across the US, covering five variables: precipitation, snowfall, snow depth, and minimum and maximum temperature. We follow the pre-pr... | 245 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.1 Datasets
USHCN
The USHCN dataset (Menne et al., 2015)777https://data.ess-dive.lbl.gov/view/doi:10.3334/CDIAC/CLI.NDP019 under CC BY 4.0 includes daily measurements from 1,21812181,218 weather st... | 280 |
90 | The Physionet dataset (Silva et al., 2012)888https://physionet.org/content/challenge-2012/1.0.0/ under ODC-BY 1.0 contains 800080008000 multivariate clinical time-series obtained from the intensive care unit (ICU). Each time-series includes various clinical features recorded during the first 484848 hours after the pati... | 251 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.1 Datasets
Physionet
The Physionet dataset (Silva et al., 2012)888https://physionet.org/content/challenge-2012/1.0.0/ under ODC-BY 1.0 contains 800080008000 multivariate clinical time-series obtai... | 287 |
91 | The masked attention mechanism for the assimilation schemes introduced in Sec 3.3, is illustrated in detail in Figure 4. | 26 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.2 Masking Scheme
The masked attention mechanism for the assimilation schemes introduced in Sec 3.3, is illustrated in detail in Figure 4. | 59 |
92 | For all experiments, except for human activity classification, we followed the same experimental setup as CRU (Schirmer et al., 2022)999https://github.com/boschresearch/Continuous-Recurrent-Units under AGPL-3.0 license. For the human activity classification task, we used the setup described in mTAND (Shukla & Marlin, 2... | 1,135 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.3 Training details
Training
For all experiments, except for human activity classification, we followed the same experimental setup as CRU (Schirmer et al., 2022)999https://github.com/boschresearch... | 1,169 |
93 | In all experiments except for the Pendulum dataset, the time series 𝐨0,Tsubscript𝐨0𝑇\mathbf{o}_{0,T} was provided with the observation mask concatenated. We was used a dropout rate of 0.2 for a Human Activity, while no dropout rate was used for the other experiments. For our method, the networks used for each datase... | 1,282 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.3 Training details
Architecture
In all experiments except for the Pendulum dataset, the time series 𝐨0,Tsubscript𝐨0𝑇\mathbf{o}_{0,T} was provided with the observation mask concatenated. We was ... | 1,316 |
94 | Figure 1: Conceptual illustration. Given the observed time stamps 𝒯={ti}i∈[1:4]𝒯subscriptsubscript𝑡𝑖𝑖delimited-[]:14\mathcal{T}=\{t_{i}\}_{i\in[1:4]} and the unseen time stamps 𝒯usubscript𝒯𝑢{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathcal{T}_{u}}(×\color[rgb]{1,0,0}\definecolor[named... | 1,484 | Amortized Control of Continuous State Space Feynman-Kac Model for Irregular Time Series
Appendix C Implementation Details
C.3 Training details
Architecture
Figure 1: Conceptual illustration. Given the observed time stamps 𝒯={ti}i∈[1:4]𝒯subscriptsubscript𝑡𝑖𝑖delimited-[]:14\mathcal{T}=\{t_{i}\}_{i\in[1:4]} and the... | 1,518 |
0 | Consistent Approximations in Composite Optimization
Johannes O. Royset
Operations Research Department
Naval Postgraduate School
joroyset@nps.edu
Abstract. Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, wit... | 214 | Consistent Approximations in Composite Optimization
Johannes O. Royset
Operations Research Department
Naval Postgraduate School
joroyset@nps.edu
Abstract. Approximations of optimization problems arise in computational procedures and sensitivity analysis. The resulting effect on solutions can be significant, wit... | 214 |
1 | A fundamental approach to optimization is to replace an actual problem by an approximating one, which is then solved using an existing algorithm. In sensitivity analysis, the actual problem of interest is also replaced by approximating ones for the purpose of identifying the effect of perturbations. These situations ra... | 1,904 | 1 Introduction
A fundamental approach to optimization is to replace an actual problem by an approximating one, which is then solved using an existing algorithm. In sensitivity analysis, the actual problem of interest is also replaced by approximating ones for the purpose of identifying the effect of perturbations. Thes... | 1,907 |
2 | Approximations of (1.1) may stem from the smoothing of hℎh and/or F𝐹F. The specific function γ↦max{0,γ}maps-to𝛾0𝛾\gamma\mapsto\max\{0,\gamma\} can be approximated by smooth functions via convolution [15]. The approximations exhibit both epi-convergence as well as certain convergence of gradients to the subgradients... | 967 | 1 Introduction
Approximations of (1.1) may stem from the smoothing of hℎh and/or F𝐹F. The specific function γ↦max{0,γ}maps-to𝛾0𝛾\gamma\mapsto\max\{0,\gamma\} can be approximated by smooth functions via convolution [15]. The approximations exhibit both epi-convergence as well as certain convergence of gradients to t... | 970 |
3 | Terminology. A ball under norm ∥⋅∥\|\cdot\| is denoted by 𝔹(x¯,ρ)={x∈ℝn|‖x−x¯‖≤ρ}𝔹¯𝑥𝜌conditional-set𝑥superscriptℝ𝑛norm𝑥¯𝑥𝜌\mathbb{B}(\bar{x},\rho)=\{x\in\mathbb{R}^{n}~{}|~{}\|x-\bar{x}\|\leq\rho\}. A function f:ℝn→ℝ¯:𝑓→superscriptℝ𝑛¯ℝf:\mathbb{R}^{n}\to\overline{\mathbb{R}} has a domain domf={x∈ℝn|f(x)<∞... | 1,630 | 1 Introduction
Terminology. A ball under norm ∥⋅∥\|\cdot\| is denoted by 𝔹(x¯,ρ)={x∈ℝn|‖x−x¯‖≤ρ}𝔹¯𝑥𝜌conditional-set𝑥superscriptℝ𝑛norm𝑥¯𝑥𝜌\mathbb{B}(\bar{x},\rho)=\{x\in\mathbb{R}^{n}~{}|~{}\|x-\bar{x}\|\leq\rho\}. A function f:ℝn→ℝ¯:𝑓→superscriptℝ𝑛¯ℝf:\mathbb{R}^{n}\to\overline{\mathbb{R}} has a domain dom... | 1,633 |
4 | Sequences of points, sets, and so forth are usually indexed by superscript ν∈ℕ={1,2,…}𝜈ℕ12…\nu\in\mathbb{N}=\{1,2,\dots\}. The collection of subsequences of ℕℕ\mathbb{N} is denoted by 𝒩∞#superscriptsubscript𝒩#{\cal N}_{\infty}^{\scriptscriptstyle\#}, with convergence of {xν,ν∈ℕ}superscript𝑥𝜈𝜈ℕ\{x^{\nu},\nu\in\mat... | 1,577 | 1 Introduction
Sequences of points, sets, and so forth are usually indexed by superscript ν∈ℕ={1,2,…}𝜈ℕ12…\nu\in\mathbb{N}=\{1,2,\dots\}. The collection of subsequences of ℕℕ\mathbb{N} is denoted by 𝒩∞#superscriptsubscript𝒩#{\cal N}_{\infty}^{\scriptscriptstyle\#}, with convergence of {xν,ν∈ℕ}superscript𝑥𝜈𝜈ℕ\{x^{... | 1,580 |
5 | A set-valued mapping T:ℝn→→ℝm:𝑇superscriptℝ𝑛→→superscriptℝ𝑚T:\mathbb{R}^{n}\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-12.0pt\hbox{\raise 2.5pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m} has subsets of ℝmsuperscriptℝ𝑚\mathbb{R}^{m} as its “values” and a graph written as gphT={(x,y)∈ℝn+m|y∈T(x)}gph𝑇conditional-set𝑥𝑦s... | 1,090 | 1 Introduction
A set-valued mapping T:ℝn→→ℝm:𝑇superscriptℝ𝑛→→superscriptℝ𝑚T:\mathbb{R}^{n}\;{\lower 1.0pt\hbox{$\rightarrow$}}\kern-12.0pt\hbox{\raise 2.5pt\hbox{$\rightarrow$}}\;\mathbb{R}^{m} has subsets of ℝmsuperscriptℝ𝑚\mathbb{R}^{m} as its “values” and a graph written as gphT={(x,y)∈ℝn+m|y∈T(x)}gph𝑇condi... | 1,093 |
6 | Parallel to the actual problem (1.1), we define the approximating problems
, 1 = {minimizex∈ℝnφν(x)=ιXν(x)+hν(Fν(x)),ν∈ℕ}formulae-sequencesubscriptminimize𝑥superscriptℝ𝑛superscript𝜑𝜈𝑥subscript𝜄superscript𝑋𝜈𝑥superscriptℎ𝜈superscript𝐹𝜈𝑥𝜈ℕ\bigg{\{}\mathop{\rm minimize}_{x\in\mathbb{R}^{n}}~{}\varphi^{\nu... | 1,895 | 2 Consistent Approximations
Parallel to the actual problem (1.1), we define the approximating problems
, 1 = {minimizex∈ℝnφν(x)=ιXν(x)+hν(Fν(x)),ν∈ℕ}formulae-sequencesubscriptminimize𝑥superscriptℝ𝑛superscript𝜑𝜈𝑥subscript𝜄superscript𝑋𝜈𝑥superscriptℎ𝜈superscript𝐹𝜈𝑥𝜈ℕ\bigg{\{}\mathop{\rm minimize}_{x\in\m... | 1,901 |
7 | , 1 = S(x,y,z)𝑆𝑥𝑦𝑧\displaystyle S(x,y,z). , 2 = ={F(x)−z}×{∂h(z)−y}×(∑i=1myicon∂fi(x)+NX(x))absent𝐹𝑥𝑧ℎ𝑧𝑦superscriptsubscript𝑖1𝑚subscript𝑦𝑖consubscript𝑓𝑖𝑥subscript𝑁𝑋𝑥\displaystyle=\big{\{}F(x)-z\big{\}}\times\big{\{}\partial h(z)-y\big{\}}\times\Big{(}\mathop{\sum}\nolimits_{i=1}^{m}y_{i}\opera... | 1,887 | 2 Consistent Approximations
, 1 = S(x,y,z)𝑆𝑥𝑦𝑧\displaystyle S(x,y,z). , 2 = ={F(x)−z}×{∂h(z)−y}×(∑i=1myicon∂fi(x)+NX(x))absent𝐹𝑥𝑧ℎ𝑧𝑦superscriptsubscript𝑖1𝑚subscript𝑦𝑖consubscript𝑓𝑖𝑥subscript𝑁𝑋𝑥\displaystyle=\big{\{}F(x)-z\big{\}}\times\big{\{}\partial h(z)-y\big{\}}\times\Big{(}\mathop{\sum}\n... | 1,893 |
8 | (optimality condition).
Suppose that the following qualification holds at x⋆superscript𝑥⋆x^{\star}:
, 1 = y∈Ndomh(F(x⋆)) and 0∈∑i=1myicon∂fi(x⋆)+NX(x⋆)⟹y=0.𝑦subscript𝑁domℎ𝐹superscript𝑥⋆ and 0superscriptsubscript𝑖1𝑚subscript𝑦𝑖consubscript𝑓𝑖superscript𝑥⋆subscript𝑁𝑋superscript𝑥⋆⟹𝑦0y\in N_{\operato... | 1,840 | 2 Consistent Approximations
2.1 Proposition
(optimality condition).
Suppose that the following qualification holds at x⋆superscript𝑥⋆x^{\star}:
, 1 = y∈Ndomh(F(x⋆)) and 0∈∑i=1myicon∂fi(x⋆)+NX(x⋆)⟹y=0.𝑦subscript𝑁domℎ𝐹superscript𝑥⋆ and 0superscriptsubscript𝑖1𝑚subscript𝑦𝑖consubscript𝑓𝑖superscript𝑥⋆sub... | 1,851 |
9 | as long as h~~ℎ\tilde{h} is epi-regular at F~(x⋆)~𝐹superscript𝑥⋆\tilde{F}(x^{\star}) and lF~(⋅,(w,y))subscript𝑙~𝐹⋅𝑤𝑦l_{\tilde{F}}(\cdot\,,(w,y)) is epi-regular at x⋆superscript𝑥⋆x^{\star} for all (w,y)∈∂h~(F~(x⋆))𝑤𝑦~ℎ~𝐹superscript𝑥⋆(w,y)\in\partial\tilde{h}(\tilde{F}(x^{\star})). A closer examination sho... | 1,993 | 2 Consistent Approximations
2.1 Proposition
as long as h~~ℎ\tilde{h} is epi-regular at F~(x⋆)~𝐹superscript𝑥⋆\tilde{F}(x^{\star}) and lF~(⋅,(w,y))subscript𝑙~𝐹⋅𝑤𝑦l_{\tilde{F}}(\cdot\,,(w,y)) is epi-regular at x⋆superscript𝑥⋆x^{\star} for all (w,y)∈∂h~(F~(x⋆))𝑤𝑦~ℎ~𝐹superscript𝑥⋆(w,y)\in\partial\tilde{h}(\ti... | 2,004 |
10 | Generally, the advantage of the optimality condition 0∈S(x,y,z)0𝑆𝑥𝑦𝑧0\in S(x,y,z) is its explicit form in terms of the “primitives” of the actual problem (1.1). It is therefore much more computationally accessible than 0∈∂φ(x)0𝜑𝑥0\in\partial\varphi(x), which can be checked numerically only in special cases.
Pro... | 226 | 2 Consistent Approximations
2.1 Proposition
Generally, the advantage of the optimality condition 0∈S(x,y,z)0𝑆𝑥𝑦𝑧0\in S(x,y,z) is its explicit form in terms of the “primitives” of the actual problem (1.1). It is therefore much more computationally accessible than 0∈∂φ(x)0𝜑𝑥0\in\partial\varphi(x), which can be ch... | 237 |
11 | (consistent approximations).
The pairs {(φν,Sν),ν∈ℕ}superscript𝜑𝜈superscript𝑆𝜈𝜈ℕ\{(\varphi^{\nu},S^{\nu}),\nu\in\mathbb{N}\} are consistent approximations of (φ,S)𝜑𝑆(\varphi,S) when
, 1 = φν→e φ and Sν→g S.superscript𝜑𝜈→e 𝜑 and superscript𝑆𝜈→g 𝑆\varphi^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.... | 1,043 | 2 Consistent Approximations
2.2 Definition
(consistent approximations).
The pairs {(φν,Sν),ν∈ℕ}superscript𝜑𝜈superscript𝑆𝜈𝜈ℕ\{(\varphi^{\nu},S^{\nu}),\nu\in\mathbb{N}\} are consistent approximations of (φ,S)𝜑𝑆(\varphi,S) when
, 1 = φν→e φ and Sν→g S.superscript𝜑𝜈→e 𝜑 and superscript𝑆𝜈→g 𝑆\varphi^{\nu}\,... | 1,054 |
12 | (consequences of consistency). Suppose that {(φν,Sν),ν∈ℕ}superscript𝜑𝜈superscript𝑆𝜈𝜈ℕ\{(\varphi^{\nu},S^{\nu}),\nu\in\mathbb{N}\} are weakly consistent approximations of (φ,S)𝜑𝑆(\varphi,S), the tolerances ενsuperscript𝜀𝜈\varepsilon^{\nu} and δνsuperscript𝛿𝜈\delta^{\nu} vanish, and α<β𝛼𝛽\alpha<\beta. Then, ... | 1,407 | 2 Consistent Approximations
2.3 Proposition
(consequences of consistency). Suppose that {(φν,Sν),ν∈ℕ}superscript𝜑𝜈superscript𝑆𝜈𝜈ℕ\{(\varphi^{\nu},S^{\nu}),\nu\in\mathbb{N}\} are weakly consistent approximations of (φ,S)𝜑𝑆(\varphi,S), the tolerances ενsuperscript𝜀𝜈\varepsilon^{\nu} and δνsuperscript𝛿𝜈\delta^{... | 1,418 |
13 | Item (c) confirms that any cluster point (x¯,y¯,z¯)¯𝑥¯𝑦¯𝑧(\bar{x},\bar{y},\bar{z}) of a sequence {(xν,yν,zν),ν∈ℕ}superscript𝑥𝜈superscript𝑦𝜈superscript𝑧𝜈𝜈ℕ\{(x^{\nu},y^{\nu},z^{\nu}),\,\nu\in\mathbb{N}\}, with 0∈Sν(xν,yν,zν)0superscript𝑆𝜈superscript𝑥𝜈superscript𝑦𝜈superscript𝑧𝜈0\in S^{\nu}(x^{\nu},y^{\... | 1,930 | 2 Consistent Approximations
2.3 Proposition
Item (c) confirms that any cluster point (x¯,y¯,z¯)¯𝑥¯𝑦¯𝑧(\bar{x},\bar{y},\bar{z}) of a sequence {(xν,yν,zν),ν∈ℕ}superscript𝑥𝜈superscript𝑦𝜈superscript𝑧𝜈𝜈ℕ\{(x^{\nu},y^{\nu},z^{\nu}),\,\nu\in\mathbb{N}\}, with 0∈Sν(xν,yν,zν)0superscript𝑆𝜈superscript𝑥𝜈superscript... | 1,941 |
14 | (sufficient conditions for consistent approximations).
Suppose that hν→e hsuperscriptℎ𝜈→e ℎh^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pth, Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\s... | 662 | 2 Consistent Approximations
2.4 Theorem
(sufficient conditions for consistent approximations).
Suppose that hν→e hsuperscriptℎ𝜈→e ℎh^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pth, Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\... | 674 |
15 | (a)
(real-valuedness): hℎh is real-valued.
(b)
(pointwise convergence): hν(z)→h(z)→superscriptℎ𝜈𝑧ℎ𝑧h^{\nu}(z)\to h(z) for all z∈bdry(domh)𝑧bdrydomℎz\in\operatorname{bdry}(\operatorname{dom}h) and, in addition, Fν(x)=F(x)superscript𝐹𝜈𝑥𝐹𝑥F^{\nu}(x)=F(x) for all x∈X𝑥𝑋x\in X and X⊂Xν𝑋superscript𝑋𝜈X\su... | 1,938 | 2 Consistent Approximations
2.4 Theorem
(a)
(real-valuedness): hℎh is real-valued.
(b)
(pointwise convergence): hν(z)→h(z)→superscriptℎ𝜈𝑧ℎ𝑧h^{\nu}(z)\to h(z) for all z∈bdry(domh)𝑧bdrydomℎz\in\operatorname{bdry}(\operatorname{dom}h) and, in addition, Fν(x)=F(x)superscript𝐹𝜈𝑥𝐹𝑥F^{\nu}(x)=F(x) for all x∈X... | 1,950 |
16 | If Xν=Xsuperscript𝑋𝜈𝑋X^{\nu}=X and fiν=fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu}=f_{i}, then each of (a)-(e) is sufficient for consistent approximations.
If fiν,fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu},f_{i} are smooth, then (2.3) is equivalent to fiν(xν)→fi(x)→superscriptsubscript𝑓𝑖𝜈supe... | 1,903 | 2 Consistent Approximations
2.4 Theorem
If Xν=Xsuperscript𝑋𝜈𝑋X^{\nu}=X and fiν=fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu}=f_{i}, then each of (a)-(e) is sufficient for consistent approximations.
If fiν,fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu},f_{i} are smooth, then (2.3) is equivalent to fiν(x... | 1,915 |
17 | First, suppose that (a) holds. Since Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle s$}}\hskip 7.0ptX, there exists xν∈Xν→xsuperscript𝑥𝜈superscript𝑋𝜈→𝑥x^{\nu}\in X^{\nu}\to x. Moreover, hνsuperscriptℎ𝜈h^{\nu} converges uniformly to hℎh on... | 928 | 2 Consistent Approximations
2.4 Theorem
First, suppose that (a) holds. Since Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle s$}}\hskip 7.0ptX, there exists xν∈Xν→xsuperscript𝑥𝜈superscript𝑋𝜈→𝑥x^{\nu}\in X^{\nu}\to x. Moreover, hνsuperscript... | 940 |
18 | Third, suppose that (d) holds. We consider two cases. Suppose that F(x)∈int(domh)𝐹𝑥intdomℎF(x)\in\operatorname{int}(\operatorname{dom}h). Then, construct {xν=x,ν∈ℕ}formulae-sequencesuperscript𝑥𝜈𝑥𝜈ℕ\{x^{\nu}=x,\nu\in\mathbb{N}\} and note that hν(Fν(x))→h(F(x))→superscriptℎ𝜈superscript𝐹𝜈𝑥ℎ𝐹𝑥h^{\nu}(F^{... | 1,932 | 2 Consistent Approximations
2.4 Theorem
Third, suppose that (d) holds. We consider two cases. Suppose that F(x)∈int(domh)𝐹𝑥intdomℎF(x)\in\operatorname{int}(\operatorname{dom}h). Then, construct {xν=x,ν∈ℕ}formulae-sequencesuperscript𝑥𝜈𝑥𝜈ℕ\{x^{\nu}=x,\nu\in\mathbb{N}\} and note that hν(Fν(x))→h(F(x))→supersc... | 1,944 |
19 | , 1 = limsup(ιXν(xν)+∑k=1rhkν(Fkν(xν)))≤limsupιXν(x)+∑k=1rlimsuphkν(Fkν(x)).limsupsubscript𝜄superscript𝑋𝜈superscript𝑥𝜈superscriptsubscript𝑘1𝑟superscriptsubscriptℎ𝑘𝜈superscriptsubscript𝐹𝑘𝜈superscript𝑥𝜈limsupsubscript𝜄superscript𝑋𝜈𝑥superscriptsubscript𝑘1𝑟limsupsuperscriptsubscriptℎ𝑘𝜈subscripts... | 1,560 | 2 Consistent Approximations
2.4 Theorem
, 1 = limsup(ιXν(xν)+∑k=1rhkν(Fkν(xν)))≤limsupιXν(x)+∑k=1rlimsuphkν(Fkν(x)).limsupsubscript𝜄superscript𝑋𝜈superscript𝑥𝜈superscriptsubscript𝑘1𝑟superscriptsubscriptℎ𝑘𝜈superscriptsubscript𝐹𝑘𝜈superscript𝑥𝜈limsupsubscript𝜄superscript𝑋𝜈𝑥superscriptsubscript𝑘1𝑟l... | 1,572 |
20 | We next turn to the relation between gphSνgphsuperscript𝑆𝜈\operatorname{gph}S^{\nu} and gphSgph𝑆\operatorname{gph}S and start by showing LimOut(gphSν)⊂gphSLimOutgphsuperscript𝑆𝜈gph𝑆\mathop{\rm LimOut}\nolimits(\operatorname{gph}S^{\nu})\subset\operatorname{gph}S. Let (x¯,y¯,z¯,u¯,v¯,w¯)∈LimOut(gphSν)¯𝑥¯𝑦¯�... | 1,939 | 2 Consistent Approximations
2.4 Theorem
We next turn to the relation between gphSνgphsuperscript𝑆𝜈\operatorname{gph}S^{\nu} and gphSgph𝑆\operatorname{gph}S and start by showing LimOut(gphSν)⊂gphSLimOutgphsuperscript𝑆𝜈gph𝑆\mathop{\rm LimOut}\nolimits(\operatorname{gph}S^{\nu})\subset\operatorname{gph}S. Let (x... | 1,951 |
21 | , 1 = aiν∈con∂fiν(xν),i=1,…,m, such that wν−∑i=1myiνaiν∈NXν(xν).formulae-sequencesuperscriptsubscript𝑎𝑖𝜈consuperscriptsubscript𝑓𝑖𝜈superscript𝑥𝜈formulae-sequence𝑖1…𝑚 such that superscript𝑤𝜈superscriptsubscript𝑖1𝑚superscriptsubscript𝑦𝑖𝜈superscriptsubscript𝑎𝑖𝜈subscript𝑁superscript𝑋𝜈superscript�... | 1,705 | 2 Consistent Approximations
2.4 Theorem
, 1 = aiν∈con∂fiν(xν),i=1,…,m, such that wν−∑i=1myiνaiν∈NXν(xν).formulae-sequencesuperscriptsubscript𝑎𝑖𝜈consuperscriptsubscript𝑓𝑖𝜈superscript𝑥𝜈formulae-sequence𝑖1…𝑚 such that superscript𝑤𝜈superscriptsubscript𝑖1𝑚superscriptsubscript𝑦𝑖𝜈superscriptsubscript𝑎𝑖... | 1,717 |
22 | Via Attouch’s theorem [50, Thm. 12.35], ιXν→e ιXsubscript𝜄superscript𝑋𝜈→e subscript𝜄𝑋\iota_{X^{\nu}}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pt\iota_{X} implies NXν→g NXsubscript𝑁superscript𝑋𝜈→g subscript𝑁𝑋N_{X^{\nu}}\,{\lower 1.0pt\hbox{$\right... | 1,975 | 2 Consistent Approximations
2.4 Theorem
Via Attouch’s theorem [50, Thm. 12.35], ιXν→e ιXsubscript𝜄superscript𝑋𝜈→e subscript𝜄𝑋\iota_{X^{\nu}}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pt\iota_{X} implies NXν→g NXsubscript𝑁superscript𝑋𝜈→g subscript𝑁�... | 1,987 |
23 | , 1 = ∂f(x)=argminv{f∗(v)−⟨v,x⟩} and ∂fν(x)=argminv{fν∗(v)−⟨v,x⟩}𝑓𝑥subscriptargmin𝑣superscript𝑓𝑣𝑣𝑥 and superscript𝑓𝜈𝑥subscriptargmin𝑣superscript𝑓𝜈𝑣𝑣𝑥\partial f(x)=\mathop{\rm argmin}\nolimits_{v}\big{\{}f^{*}(v)-\langle v,x\rangle\big{\}}~{}~{}~{}\mbox{ and }~{}~{}~{}\partial f^{\nu}(x)=\mathop{\... | 1,407 | 2 Consistent Approximations
2.4 Theorem
, 1 = ∂f(x)=argminv{f∗(v)−⟨v,x⟩} and ∂fν(x)=argminv{fν∗(v)−⟨v,x⟩}𝑓𝑥subscriptargmin𝑣superscript𝑓𝑣𝑣𝑥 and superscript𝑓𝜈𝑥subscriptargmin𝑣superscript𝑓𝜈𝑣𝑣𝑥\partial f(x)=\mathop{\rm argmin}\nolimits_{v}\big{\{}f^{*}(v)-\langle v,x\rangle\big{\}}~{}~{}~{}\mbox{ and... | 1,419 |
24 | Since ∂f(x¯)𝑓¯𝑥\partial f(\bar{x}) is nonempty by [56, Prop. 2.25], there is v¯∈argming¯𝑣argmin𝑔\bar{v}\in\mathop{\rm argmin}\nolimits g. From the definition of epi-convergence, gν→e gsuperscript𝑔𝜈→e 𝑔g^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0... | 1,906 | 2 Consistent Approximations
2.4 Theorem
Since ∂f(x¯)𝑓¯𝑥\partial f(\bar{x}) is nonempty by [56, Prop. 2.25], there is v¯∈argming¯𝑣argmin𝑔\bar{v}\in\mathop{\rm argmin}\nolimits g. From the definition of epi-convergence, gν→e gsuperscript𝑔𝜈→e 𝑔g^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4... | 1,918 |
25 | Under the assumption that Xν=Xsuperscript𝑋𝜈𝑋X^{\nu}=X and fiν=fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu}=f_{i}, we show that LimInn(gphSν)⊃gphSgph𝑆LimInngphsuperscript𝑆𝜈\mathop{\rm LimInn}\nolimits(\operatorname{gph}S^{\nu})\supset\operatorname{gph}S. Let (x¯,y¯,z¯,u¯,v¯,w¯)∈gphS¯𝑥¯𝑦¯𝑧¯𝑢¯𝑣¯𝑤gph... | 1,815 | 2 Consistent Approximations
2.4 Theorem
Under the assumption that Xν=Xsuperscript𝑋𝜈𝑋X^{\nu}=X and fiν=fisuperscriptsubscript𝑓𝑖𝜈subscript𝑓𝑖f_{i}^{\nu}=f_{i}, we show that LimInn(gphSν)⊃gphSgph𝑆LimInngphsuperscript𝑆𝜈\mathop{\rm LimInn}\nolimits(\operatorname{gph}S^{\nu})\supset\operatorname{gph}S. Let (x¯,y¯... | 1,827 |
26 | with a similar expression for the approximating functions. Let (x¯,y¯,z¯,u¯,v¯,w¯)∈gphS¯𝑥¯𝑦¯𝑧¯𝑢¯𝑣¯𝑤gph𝑆(\bar{x},\bar{y},\bar{z},\bar{u},\bar{v},\bar{w})\in\operatorname{gph}S. This means that u¯=F(x¯)−z¯¯𝑢𝐹¯𝑥¯𝑧\bar{u}=F(\bar{x})-\bar{z}, v¯∈∂h(z¯)−y¯¯𝑣ℎ¯𝑧¯𝑦\bar{v}\in\partial h(\bar{z})-\bar{y}, and w¯∈... | 1,756 | 2 Consistent Approximations
2.4 Theorem
with a similar expression for the approximating functions. Let (x¯,y¯,z¯,u¯,v¯,w¯)∈gphS¯𝑥¯𝑦¯𝑧¯𝑢¯𝑣¯𝑤gph𝑆(\bar{x},\bar{y},\bar{z},\bar{u},\bar{v},\bar{w})\in\operatorname{gph}S. This means that u¯=F(x¯)−z¯¯𝑢𝐹¯𝑥¯𝑧\bar{u}=F(\bar{x})-\bar{z}, v¯∈∂h(z¯)−y¯¯𝑣ℎ¯𝑧¯𝑦\bar{v... | 1,768 |
27 | (approximations without full epi-convergence).
Suppose that hν→e hsuperscriptℎ𝜈→e ℎh^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pth, Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptsty... | 924 | 2 Consistent Approximations
2.5 Corollary
(approximations without full epi-convergence).
Suppose that hν→e hsuperscriptℎ𝜈→e ℎh^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-10.0pt\hbox{\raise 4.0pt\hbox{$\,\scriptstyle e$}}\hskip 7.0pth, Xν→s Xsuperscript𝑋𝜈→s 𝑋X^{\nu}\,{\lower 1.0pt\hbox{$\rightarrow$}}\kern-1... | 937 |
28 | An array of examples illustrate the breadth of the framework, but numerous possibilities are omitted including those involving polyhedral approximations of X𝑋X as in [6]. We recall that X,Xν𝑋superscript𝑋𝜈X,X^{\nu} and fi,fiνsubscript𝑓𝑖superscriptsubscript𝑓𝑖𝜈f_{i},f_{i}^{\nu} are nonconvex unless specified othe... | 106 | 3 Examples
An array of examples illustrate the breadth of the framework, but numerous possibilities are omitted including those involving polyhedral approximations of X𝑋X as in [6]. We recall that X,Xν𝑋superscript𝑋𝜈X,X^{\nu} and fi,fiνsubscript𝑓𝑖superscriptsubscript𝑓𝑖𝜈f_{i},f_{i}^{\nu} are nonconvex unless spe... | 109 |
29 | (goal optimization).
For parameters αi∈[0,∞)subscript𝛼𝑖0\alpha_{i}\in[0,\infty) and τi∈ℝsubscript𝜏𝑖ℝ\tau_{i}\in\mathbb{R}, we consider the problem
, 1 = minimizex∈X∑i=1mαimax{0,fi(x)−τi},subscriptminimize𝑥𝑋superscriptsubscript𝑖1𝑚subscript𝛼𝑖0subscript𝑓𝑖𝑥subscript𝜏𝑖\mathop{\rm minimize}_{x\in X}\mathop{... | 1,739 | 3 Examples
3.1 Example
(goal optimization).
For parameters αi∈[0,∞)subscript𝛼𝑖0\alpha_{i}\in[0,\infty) and τi∈ℝsubscript𝜏𝑖ℝ\tau_{i}\in\mathbb{R}, we consider the problem
, 1 = minimizex∈X∑i=1mαimax{0,fi(x)−τi},subscriptminimize𝑥𝑋superscriptsubscript𝑖1𝑚subscript𝛼𝑖0subscript𝑓𝑖𝑥subscript𝜏𝑖\mathop{\rm min... | 1,747 |
30 | Similarly, the optimality condition 0∈S(x,y,z)0𝑆𝑥𝑦𝑧0\in S(x,y,z) specializes to F(x)=z𝐹𝑥𝑧F(x)=z, y∈∂h(z)𝑦ℎ𝑧y\in\partial h(z), −∇F(x)⊤y∈NX(x)∇𝐹superscript𝑥top𝑦subscript𝑁𝑋𝑥-\nabla F(x)^{\top}y\in N_{X}(x). Since ∂h(z)=C1×⋯×Cmℎ𝑧subscript𝐶1⋯subscript𝐶𝑚\partial h(z)=C_{1}\times\cdots\times C_{m}, w... | 691 | 3 Examples
3.1 Example
Similarly, the optimality condition 0∈S(x,y,z)0𝑆𝑥𝑦𝑧0\in S(x,y,z) specializes to F(x)=z𝐹𝑥𝑧F(x)=z, y∈∂h(z)𝑦ℎ𝑧y\in\partial h(z), −∇F(x)⊤y∈NX(x)∇𝐹superscript𝑥top𝑦subscript𝑁𝑋𝑥-\nabla F(x)^{\top}y\in N_{X}(x). Since ∂h(z)=C1×⋯×Cmℎ𝑧subscript𝐶1⋯subscript𝐶𝑚\partial h(z)=C_{1}\tim... | 699 |
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