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Extended SQP Methods in Nonsmooth Difference Programming Applied to Problems with Variational Inequality Constraints

Boris S. Mordukhovich, Yixia Song, Shangzhi Zeng, Jin Zhang

TL;DR

This work extends DC programming to constrained difference programming, formulating problems with φ0(x)=g0(x)−h0(x) and φ1(x)=g1(x)−h1(x)≤0 where g_i are C^{1,1} and h_i are prox-regular Lipschitz. It introduces extended SQP variants (CDP-ESQM and CDP-AESQM) that iteratively solve strongly convex quadratic subproblems built from gradients and limiting subgradients, with global convergence justified via the Polyak–Łojasiewicz–Kurdyka property and semialgebraicity. The VI-constrained case is handled by regularizing VI constraints with a gap function μ_γ, embedding the problem into the difference-programming framework, and applying the ESQM family to CNDP instances. Numerical results on continuous network design show improved objective values and stable convergence compared with MPCC and classical NLP benchmarks, indicating practical effectiveness. The work lays groundwork for convergence-rate analysis and extensions to non-smooth g_i, expanding the scope of difference programming in nonsmooth optimization.

Abstract

This paper explores a new class of constrained difference programming problems, where the objective and constraints are formulated as differences of functions, without requiring their convexity. To investigate such problems, novel variants of the extended sequential quadratic method are introduced. These algorithms iteratively solve strongly convex quadratic subproblems constructed via linear approximations of the given data by using their gradients and subgradients. The convergence of the proposed methods is rigorously analyzed by employing, in particular, the Polyak-Łojasiewicz-Kurdyka property that ensures global convergence for various classes of functions in the problem formulation, e.g., semialgebraic ones. The original framework is further extended to address difference programming problems with variational inequality (VI) constraints. By reformulating VI constraints via regularized gap functions, such problems are naturally embedded into constrained difference programming that leads us to direct applications of the proposed algorithms. Numerical experiments for the class of continuous network design problems demonstrate the efficiency of the new methods.

Extended SQP Methods in Nonsmooth Difference Programming Applied to Problems with Variational Inequality Constraints

TL;DR

This work extends DC programming to constrained difference programming, formulating problems with φ0(x)=g0(x)−h0(x) and φ1(x)=g1(x)−h1(x)≤0 where g_i are C^{1,1} and h_i are prox-regular Lipschitz. It introduces extended SQP variants (CDP-ESQM and CDP-AESQM) that iteratively solve strongly convex quadratic subproblems built from gradients and limiting subgradients, with global convergence justified via the Polyak–Łojasiewicz–Kurdyka property and semialgebraicity. The VI-constrained case is handled by regularizing VI constraints with a gap function μ_γ, embedding the problem into the difference-programming framework, and applying the ESQM family to CNDP instances. Numerical results on continuous network design show improved objective values and stable convergence compared with MPCC and classical NLP benchmarks, indicating practical effectiveness. The work lays groundwork for convergence-rate analysis and extensions to non-smooth g_i, expanding the scope of difference programming in nonsmooth optimization.

Abstract

This paper explores a new class of constrained difference programming problems, where the objective and constraints are formulated as differences of functions, without requiring their convexity. To investigate such problems, novel variants of the extended sequential quadratic method are introduced. These algorithms iteratively solve strongly convex quadratic subproblems constructed via linear approximations of the given data by using their gradients and subgradients. The convergence of the proposed methods is rigorously analyzed by employing, in particular, the Polyak-Łojasiewicz-Kurdyka property that ensures global convergence for various classes of functions in the problem formulation, e.g., semialgebraic ones. The original framework is further extended to address difference programming problems with variational inequality (VI) constraints. By reformulating VI constraints via regularized gap functions, such problems are naturally embedded into constrained difference programming that leads us to direct applications of the proposed algorithms. Numerical experiments for the class of continuous network design problems demonstrate the efficiency of the new methods.

Paper Structure

This paper contains 7 sections, 19 theorems, 134 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries from Variational Analysis and Nonsmooth Optimization
  3. ESQM for Constrained Difference Programming
  4. CDP-ESQM Algorithm for Solving Problems of Difference Programming
  5. VI-Constrained Difference Programming
  6. Applications to Continuous Network Design with Numerical Experiments
  7. Concluding Remarks and Future Research

Key Result

Lemma 2.3

Let $f_i: \mathbb{R}^n \to (-\infty,\infty ]$, $i \in \mathcal{I}$, be a finite collection of proper functions, and let $\bar{x} \in \bigcap_{i \in \mathcal{I}} \text{\rm dom}(f_i)$. Suppose that $f_i$ is locally Lipschitzian around this point for all $i \in \mathcal{I}$. Then the maximum function $ where $\mathcal{I}_{\max}(\bar{x}) := \left\{ i \in \mathcal{I} \mid f_i(\bar{x}) = f(\bar{x}) \rig

Figures (2)

  • Figure 1: Performance comparison of objective values and feasibility for demand scenario $r=(10.0,20.0)^T$.
  • Figure 2: Evolution of the penalty parameter over iterations for demand scenario $r=(10.0,20.0)^T$.

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 28 more