Overlapping Schwarz Decomposition for Constrained Quadratic Programs

Shin, Sungho; Anitescu, Mihai; Zavala, Victor M.

doi:10.48550/arxiv.2003.07502

Cited by 2 publications

(3 citation statements)

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“…Question (Q1) has been recently addressed in specific settings such as nonlinear dynamic optimization [22,24,31,34] and graph-structured quadratic programs [30,33]. Our work generalizes such results.…”

Section: Our Work Is Motivated By the Following Questionsupporting

confidence: 60%

“…Addressing this question is crucial for understanding solution stability of a wide range of problem classes, for designing approximation schemes (often cast as parametric perturbations) [6,13,23,32,34], and for designing solution algorithms [24,30,31]. For instance, it has been recently shown that EDS plays a central role in assessing the impact of coarsening schemes [19,32] for dynamic optimization and in establishing convergence of overlapping Schwarz algorithms for graph-structured problems [24,30,31]. From an application stand-point, our results seek to provide new insights on how perturbations propagate through graphs and on how the problem formulation influences such propagation.…”

Section: Our Work Is Motivated By the Following Questionmentioning

confidence: 99%

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Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

Shin¹,

Anitescu²,

Zavala³

2021

Preprint

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We study solution sensitivity for nonlinear programs (NLPs) whose structure is induced by a graph G = (V, E). These graph-structured NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that the sensitivity of the primal-dual solution at node i ∈ V against a data perturbation at node j ∈ V is bounded by Υρ d G (i,j) for constants Υ > 0 and ρ ∈ (0, 1) and where d G (i, j) is the distance between i and j on G. In other words, the sensitivity of the solution decays exponentially with the distance to the perturbation point. This result, which we call exponential decay of sensitivity (EDS), holds under fairly standard assumptions used in classical NLP sensitivity theory: the strong second-order sufficiency condition and the linear independence constraint qualification. We also present conditions under which the constants (Υ, ρ) remain uniformly bounded; this allows us to characterize behavior for NLPs defined over subgraphs of infinite graphs (e.g., as those arising in problems with unbounded domains). Our results provide new insights on how perturbations propagate through the NLP graph and on how the problem formulation influences such propagation. Specifically, we provide empirical evidence that positive objective curvature and constraint flexibility tend to dampen propagation. The developments are illustrated with numerical examples.

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Section: Our Work Is Motivated By the Following Questionsupporting

confidence: 60%

Section: Our Work Is Motivated By the Following Questionmentioning

confidence: 99%

Exponential Decay of Sensitivity in Graph-Structured Nonlinear Programs

Shin¹,

Anitescu²,

Zavala³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…algebraic modeling capabilities of JuMP.jl (Dunning et al, 2017) and facilitates access to infrastructure modeling tools such as GasModels.jl and PowerModels.jl (Bent et al, 2020;Coffrin et al, 2018). Another key benefit of Plasmo.jl is that it can communicate model structures and this facilitates the implementation of different decomposition strategies such as the alternating direction method of multipliers (Boyd et al, 2011), overlapping Schwarz (Shin et al, 2020a), and parallel interior-point (IP) methods (Chiang et al, 2014;Rodriguez et al, 2020).…”

Section: Introductionmentioning

confidence: 99%