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Penalty Interior-Point Method Fails to Converge

Sven Leyffer

TL;DR

The paper examines convergence of the Penalty Interior-Point Method (PIPA) for mathematical programs with complementarity constraints (MPCCs) and provides a counterexample where PIPA converges to a feasible but nonstationary point, challenging previous convergence results. The counterexample uses a simple MPCC with objective $f(x,w)=x+w$ and constraints including $-1 \le x \le 1$, $-1+x+y=0$, and $0 \le y \perp w \ge 0$, showing a limit point with $w^\infty=0$ and $y^\infty>0$ that is not stationary. The root cause is identified as the trust-region constraint $\|d_x\|_2^2 \le \Delta_k$, which tends to shrink as feasibility is approached, impeding progress. A remedy is proposed via model-based trust-region control and a mixed trust-region/line-search scheme, with convergence recoverable under assumptions such as nonsingularity and boundedness of the Jacobian of the linearized constraints.

Abstract

Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interior-point algorithm (PIPA). This paper presents a small example for which PIPA converges to a nonstationary point, providing a counterexample to the established theory. The reasons for this adverse behavior are discussed.

Penalty Interior-Point Method Fails to Converge

TL;DR

The paper examines convergence of the Penalty Interior-Point Method (PIPA) for mathematical programs with complementarity constraints (MPCCs) and provides a counterexample where PIPA converges to a feasible but nonstationary point, challenging previous convergence results. The counterexample uses a simple MPCC with objective and constraints including , , and , showing a limit point with and that is not stationary. The root cause is identified as the trust-region constraint , which tends to shrink as feasibility is approached, impeding progress. A remedy is proposed via model-based trust-region control and a mixed trust-region/line-search scheme, with convergence recoverable under assumptions such as nonsingularity and boundedness of the Jacobian of the linearized constraints.

Abstract

Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interior-point algorithm (PIPA). This paper presents a small example for which PIPA converges to a nonstationary point, providing a counterexample to the established theory. The reasons for this adverse behavior are discussed.

Paper Structure

This paper contains 4 sections, 2 theorems, 38 equations, 1 table.

Table of Contents

  1. Introduction
  2. The Penalty Interior-Point Algorithm (PIPA)
  3. A Counterexample
  4. Discussion and Remedy

Key Result

Theorem 2.1

LuoPanRal:96 Suppose that the Hessian matrices $W^k$ are bounded and that $\sigma > 0$. If the penalty parameter $\alpha$ is bounded, then every limit point of $(x^k,y^k,w^k,z^k)$ that satisfies [SC] and [NS] is a stationary point of (MPEC).

Theorems & Definitions (2)

  • Theorem 2.1
  • Lemma 3.1