A Lifting-Penalty Method for Quadratic Programming with a Quadratic Matrix Inequality Constraint

Wei Liu,Li Yang,Bo Yu

doi:10.3390/math8020153

Abstract

In this paper, a lifting-penalty method for solving the quadratic programming with a quadratic matrix inequality constraint is proposed. Additional variables are introduced to represent the quadratic terms. The quadratic programming is reformulated as a minimization problem having a linear objective function, linear conic constraints and a quadratic equality constraint. A majorization–minimization method is used to solve instead a l 1 penalty reformulation of the minimization problem. The subproblems arising in the method can be solved by using the current semidefinite programming software packages. Global convergence of the method is proven under some suitable assumptions. Some examples and numerical results are given to show that the proposed method is feasible and efficient.

Highlights

IntroductionWe consider the quadratic programming with a quadratic matrix inequality constraint of the following form: min x s.t
In this paper, we consider the quadratic programming with a quadratic matrix inequality constraint of the following form: min x s.t.x T Qx + q T x, d n ∑i,j=1 xi x j Aij + ∑i =1 xi Ai0 + A00 0, A x − b ∈ K, (1)where q, x ∈ Rn, Q, Aij ∈ Sm, A is a linear operator, the cone K is a product of semidefinite cones, second-order cones, nonnegative orthants and Euclidean spaces, Sm is the space of m × m symmetric matrices, and A 0 indicates that A is positive semidefinite
The examples and some preliminary numerical results taken by the lifting-penalty method (LPM) and a modified augmented Lagrangian method (MALM) [46] are given below

Summary

Introduction

We consider the quadratic programming with a quadratic matrix inequality constraint of the following form: min x s.t. Following the same line of the work in [1], an iterative procedure to search a local optimum of more general nonconvex problem was developed in [27], based on a positive semidefinite convex overestimate of a positive semidefinite nonconvex matrix mapping. These local methods may not be able to obtain a global optimum. Though the approaches based on the generalized Benders decomposition and branch-and-bound algorithms are global methods, it is in general impractical to solve large-scale problems.

The Lifting-Penalty Method

Numerical Experiments

Conclusions