Global optimality conditions for nonconvex minimization problems with quadratic constraints

Guoquan Li,Zhiyou Wu,Jing Quan

doi:10.1186/s13660-015-0776-3

Guoquan Li, Zhiyou Wu + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0776-3

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Abstract

In this paper, some global optimality conditions for nonconvex minimization problems subject to quadratic inequality constraints are presented. Then some sufficient and necessary global optimality conditions for nonlinear programming problems with box constraints are derived. We also establish a sufficient global optimality condition for a nonconvex quadratic minimization problem with box constraints, which is expressed in a simple way in terms of the problem’s data. In addition, a sufficient and necessary global optimality condition for a class of nonconvex quadratic programming problems with box constraints is discussed. We also present some numerical examples to illustrate the significance of our optimality conditions.

Highlights

Consider the following nonconvex minimization problem (QCNP): (QCNP) min f (x) s.t. fi(x) = xT Aix + xT ai ci ≤, i =, . . . , m, where f : Rn → R is a twice continuously differentiable function, ci ∈ R, ai ∈ Rn, Ai ∈ Sn, i =, . . . , m, and Sn is the set of all symmetric n × n matrices
Some sufficient global optimality conditions for nonconvex quadratic minimization problems with box constraints were obtained by using abstract subdifferentials in [ ]
We study the global optimality conditions for general nonconvex minimization problems with inequality quadratic constraints, which can be viewed as a generalization of Jeyakumar et al [ ] for a single quadratic constraint

Summary

Introduction

Consider the following nonconvex minimization problem (QCNP):. min f (x) s.t. In [ ], Moré studied the problem of minimizing a quadratic function subject to one general quadratic constraint and obtained some sufficient and necessary global optimality conditions. In [ ], Jeyakumar et al established by a Lagrange multiplier sufficient as well as necessary conditions for global optimality of general quadratic minimization problems with quadratic constraints. Some sufficient global optimality conditions for nonconvex quadratic minimization problems with box constraints were obtained by using abstract subdifferentials in [ ]. We study the global optimality conditions for general nonconvex minimization problems with inequality quadratic constraints, which can be viewed as a generalization of Jeyakumar et al [ ] for a single quadratic constraint. In Section , we present some necessary and sufficient global optimality conditions for nonlinear programming problems with inequality quadratic constraints (QCNP).

This shows us that

The feasible set B can be written as

Then a direct calculation shows that λ

Let us consider another point y

If xi