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Optimization with Parametric Variational Inequality Constraints on a Moving Set

Xiaojun Chen, Jin Zhang, Yixuan Zhang

TL;DR

The Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI and it is proved that the metric regularity of the constraints holds automatically, which allow to characterize stationary points without any additional assumptions.

Abstract

This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have parameters not only in the function but also in the related set. We show that the solution function of the PVI is Lipschitz continuous with respect to the upper-level decision variables and the solution set of the optimization problem is nonempty and bounded. Moreover, we prove that the metric regularity of the constraints holds automatically, which allow us to characterize stationary points without any additional assumptions. A Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI. We prove the convergence of SIGA to a stationary point of the optimization problem and numerically validate the efficiency of SIGA by portfolio management problems with real data.

Optimization with Parametric Variational Inequality Constraints on a Moving Set

TL;DR

The Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI and it is proved that the metric regularity of the constraints holds automatically, which allow to characterize stationary points without any additional assumptions.

Abstract

This paper focuses on optimization problems constrained by Parametric Variational Inequalities (PVI) defined on a moving set. Unlike most existing works on mathematical programs with equilibrium constraints, the equilibrium constraints have parameters not only in the function but also in the related set. We show that the solution function of the PVI is Lipschitz continuous with respect to the upper-level decision variables and the solution set of the optimization problem is nonempty and bounded. Moreover, we prove that the metric regularity of the constraints holds automatically, which allow us to characterize stationary points without any additional assumptions. A Smoothing Implicit Gradient Algorithm (SIGA) is proposed based on the smoothing approximation of the PVI. We prove the convergence of SIGA to a stationary point of the optimization problem and numerically validate the efficiency of SIGA by portfolio management problems with real data.
Paper Structure (12 sections, 28 theorems, 125 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 12 sections, 28 theorems, 125 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Properties of problem \ref{['eq:1equation constraint']}
  3. Solution existence to \ref{['eq:1equation constraint']}
  4. Optimality condition under metric regularity
  5. Smoothing approximation
  6. SIGA: algorithmic framework
  7. Applications in portfolio management
  8. Conclusions
  9. PVI on a moving box
  10. Lemmas for convergence analysis in Section \ref{['sec: SIGA']}
  11. Lemmas of properties of $y_{\mu}(\mathbf{x})$, $v_{\mu}(\mathbf{x})$, and $h_{\mu}(\mathbf{x})$
  12. Lemmas of descent formula

Key Result

Proposition 2.2

If $\Omega(\mathbf{x}) = \{ \mathbf{y} \in \mathbb{R}^n : A \mathbf{y} \ge b(\mathbf{x}), \mathbf{y} \ge \mathbf{c}\}$ is a nonempty polyhedron for any $\mathbf{x} \in \mathcal{X}$, where $A \in \mathbb{R}^{r \times n}, \mathbf{c}\in \mathbb{R}^n$ and $b:\mathcal{X} \rightarrow \mathbb{R}^r$ is Lips

Figures (1)

  • Figure 1: Convergence curves and SR and CR on the testing set under Data 3.

Theorems & Definitions (35)

  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7: MR
  • Proposition 2.8
  • Definition 2.9: Stationary point of problem \ref{['eq:1equation constraint']}
  • Theorem 2.10: Optimality condition under the MR
  • Remark 2.11
  • ...and 25 more