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To find a global optimal solution to the quadratically constrained quadratic programming problem, we explore the relationship between its Lagrangian multipliers and related linear conic programming problems. This study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature. Moreover, we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems.

This paper presents a canonical duality approach to solve an integer quadratic programming problem, in which the objective function is quadratic and each variable may assume the value of one of p (≥ 3) integers. We first transform the problem into a {−1, 1} integer quadratic programming problem and then derive its "canonical dual". It is shown that, under certain conditions, this nonconvex multi-integer programming problem is equivalent to a concave maximization dual problem over a convex feasible domain. A global optimality condition is derived and some computational examples are provided to illustrate this approach.

Dedication: This paper is dedicated to Professor Jiye Han on the occasion of his 80th birthday.A quadratic optimization problem with one nonconvex quadratic constraint is studied using the canonical dual approach. Under the dual Slater's condition, we show that the canonical dual has a smooth concave objective function over a convex feasible domain, and this dual has a finite supremum unless the original quadratic optimization problem is infeasible. This supremum, when it exists, always equals to the minimum value of the primal problem. Moreover, a global minimizer of the primal problem can be provided by a dual-to-primal conversion plus a "boundarification" technique. Application to solving * Corresponding author 1540007-1 Asia Pac. J. Oper. Res. 2015.32. Downloaded from www.worldscientific.com by THE UNIVERSITY OF WESTERN ONTARIO on 04/13/15. For personal use only. W. Xing et al.a quadratic programming problem over a ball is included and an error bound estimation is provided.