ON DUALITY FOR NONCONVEX QUADRATIC OPTIMIZATION PROBLEMS
Moon Hee Kim · East Asian Mathematical Journal · 2011
In this paper, we consider an optimization problem which consists a nonconvex quadratic objective function and two nonconvex quadratic constraint functions. We formulate its dual problem with semi- denite constraints, and we establish weak and strong duality theorems which hold between these two problems. And we give an example to il- lustrate our duality results. It is worth while noticing that our weak and strong duality theorems hold without convexity assumptions. 1. Introduction and Preliminaries We begin the notations and denitions that will be used throughout this paper. The real line is denoted by R and the n-dimensional Euclidean space is denoted by R n . The space of all (n n) symmetric matrices is denoted by S n . The notation A B means that the matrix A B is positive semidenite. Moreover, the notation A B means that the matrix A B is positive denite. Consider the following quadratic optimization problems: (P) Minimize f(x) subject to g(x) 0; h(x) 0; where f; g; h : R n ! R (n 3) are dened by f(x) = 1 x T Afx +b T x +cf , g(x) = 1 x T Agx +b T x +cg and h(x) = 1 x T Ahx +b T x +ch, Af;Ag;Ah2 S n , bf;bg;bh2 R n and cf;cg;ch2 R. Denote FP =fx2 R n : g(x) 0; h(x) 0g The problem (P) is said to be regular whenever there exist 1; 22 R such that