The Saddle Point Problem of Polynomials

Nie, Jiawang; Yang, Zi; Guang-ming, Zhou

doi:10.1007/s10208-021-09526-8

Cited by 7 publications

(5 citation statements)

References 54 publications

(70 reference statements)

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“…Recently, LMEs have been widely used in various problems given by polynomial functions, such as bilevel programmings, Nash equilibrium problems, tensor computation, etc. We refer to [21,22,23,24,25] for applications of LMEs. One wonders when LMEs exist, i.e., there exist matrices L(x), D(x) such that (2.18) holds.…”

Section: Optimality Conditions and Lagrange Multiplier Expressionsmentioning

confidence: 99%

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A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

Qin¹,

Tang²

2022

Preprint

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In this paper, we consider polynomial optimization with correlative sparsity. For such problems, one may apply the correlative sparse sum of squares (CS-SOS) relaxations to exploit the correlative sparsity, which produces semidefinite programs of smaller sizes, compared with the classical moment-SOS relaxations. The Lagrange multiplier expression (LME) is also a useful tool to construct tight SOS relaxations for polynomial optimization problems, which leads to convergence with a lower relaxation order. However, the correlative sparsity is usually corrupted by the LME based reformulation. We propose a novel parametric LME approach which inherits the original correlative sparsity pattern, that we call correlative sparse LME (CS-LME). The reformulation using CS-LMEs can be solved using the CS-SOS relaxation to obtain sharper lower bound and convergence is guaranteed under mild conditions. Numerical examples are presented to show the efficiency of our new approach.

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Section: Optimality Conditions and Lagrange Multiplier Expressionsmentioning

confidence: 99%

“…Therefore, a natural question that follows is how to apply CS-LMEs to these applications. For example, when a saddle point problem is given by polynomials with correlative sparsity, can we apply CS-LMEs to construct polynomial optimization reformulation similar to the one in [25] for finding saddle points?…”

Section: Numerical Experimentsmentioning

confidence: 99%

A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

Qin¹,

Tang²

2022

Preprint

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“…They are also useful for solving truncated moment problems [21,43] and tensor decompositions [44,45]. We refer to [31,32,34,35,37,42,50] for more references about polynomial optimization and moment problems.…”

Section: Localizing and Moment Matricesmentioning

confidence: 99%

“…For the classical Nash equilibrium problems (NEPs) of polynomials, there exist semidefinite relaxation methods [2,48,50]. Convex GNEPs can be reformulated as variational inequality (VI) or quasi-variational inequality (QVI) problems [15,23,24,36,51].…”

Section: Introductionmentioning

confidence: 99%

Convex generalized Nash equilibrium problems and polynomial optimization

Nie

Tang

2021

Math. Program.

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This paper studies convex generalized Nash equilibrium problems that are given by polynomials. We use rational and parametric expressions for Lagrange multipliers to formulate efficient polynomial optimization for computing generalized Nash equilibria (GNEs). The Moment-SOS hierarchy of semidefinite relaxations are used to solve the polynomial optimization. Under some general assumptions, we prove the method can find a GNE if there exists one, or detect nonexistence of GNEs. Numerical experiments are presented to show the efficiency of the method.

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“…Moreover, various algorithms have been developed for solving saddle point problems; see e.g. [14,15,17,24,25,27].…”

mentioning

confidence: 99%

Convergence rate analysis of the gradient descent-ascent method for convex-concave saddle-point problems

Zamani¹,

Abbaszadehpeivasti²,

Klerk³

2022

Preprint

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In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent-ascent enjoys linear convergence.

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The Saddle Point Problem of Polynomials

Cited by 7 publications

References 54 publications

A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

A correlative sparse Lagrange multiplier expression relaxation for polynomial optimization

Convex generalized Nash equilibrium problems and polynomial optimization

Convergence rate analysis of the gradient descent-ascent method for convex-concave saddle-point problems

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