An interior point sequential quadratic programming-type method for log-determinant semi-infinite programs

TL;DR

This paper introduces an interior point sequential quadratic programming method for efficiently solving semi-infinite programs involving log-determinant functions and positive definite matrix constraints, with proven convergence and numerical validation.

Contribution

It develops a novel SQP-type algorithm tailored for SIPLOG problems, incorporating Monteiro-Zhang directions and proving convergence to KKT points.

Findings

The method converges weakly* to a KKT point under mild assumptions.

Numerical experiments demonstrate the efficiency of the proposed algorithm.

The approach effectively handles semi-infinite constraints with positive definite matrices.

Abstract

In this paper, we consider a nonlinear semi-infinite program that minimizes a function including a log-determinant (logdet) function over positive definite matrix constraints and infinitely many convex inequality constraints, called SIPLOG for short. The main purpose of the paper is to develop an algorithm for computing a Karush-Kuhn-Tucker (KKT) point for the SIPLOG efficiently. More specifically, we propose an interior point sequential quadratic programming-type method that inexactly solves a sequence of semi-infinite quadratic programs approximating the SIPLOG. Furthermore, to generate a search direction in the dual matrix space associated with the semi-definite constraint, we solve scaled Newton equations {that yield} the family of Monteiro-Zhang directions. We prove that the proposed method weakly* converges to a KKT point under some mild assumptions. Finally, we conduct some numerical experiments to demonstrate the efficiency of the proposed method.