Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

J Glob Optim (2012) 54:275293

DOI 10.1007/s10898-010-9625-6

Joe-Mei Feng Gang-Xuan Lin Reuy-Lin Sheu

Yong Xia

Received: 10 August 2010 / Accepted: 26 October 2010 / Published online: 17 November 2010 Springer Science+Business Media, LLC. 2010

Abstract This paper extends and completes the discussion by Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted) about the quadratic programming over one quadratic constraint (QP1QC). In particular, we relax the assumption to cover more general cases when the two matrices from the objective and the constraint functions can be simultaneously diagonalizable via congruence. Under such an assumption, the nonconvex (QP1QC) problem can be solved through a dual approach with no duality gap. This is unusual for general nonconvex programming but we can explain by showing that (QP1QC) is indeed equivalent to a linearly constrained convex problem, which happens to be dual of the dual of itself. Another type of hidden convexity can be also found in the boundarication technique developed in Xing et al. (Canonical dual solutions to the quadratic programming over a quadratic constraint, submitted).

Keywords Non-convex quadratic programming Simultaneously diagonalizable via

congruence Slaters condition Duality Hidden convexity

1 Introduction

Let A and B be two n n real symmetric matrices, be a real number and f, g be two n 1

vectors. This paper concerns the (nonconvex) quadratic minimization problem (QP1QC)

P0 = min P(x) = 12 xT Ax f T x

s.t. 1

2 xT Bx gT x ,

J.-M. Feng G.-X. Lin R.-L. Sheu (B)

Department of Mathematics, National Cheng Kung University, Tainan, Taiwan e-mail: [email protected]

Y. Xia

LMIB of the Ministry of Education, School of Mathematics and System Sciences, Beihang University, Beijing, Peoples Republic of China

Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint

(1)

123

276 J Glob Optim (2012) 54:275293

which has a single non-homogeneous quadratic constraint. To make sense the problem, we assume that the problem (QP1QC) has a non-empty feasible domain, and that

(A1) there exists a 0 such that A + B 0,where the notation means positive definiteness. Although the assumption (A1) looks

restrictive, it can be shown that our results indeed cover more general cases when A and B are simultaneously diagonalizable via congruence (SDC in short). However,...