Controlled perturbations for quadratically constrained quadratic programs

Abstract

Consider a minimization problem of a convex quadratic function of several variables over a set of inequality constraints of the same type of function. The duel program is a maximization problem with a concave objective function and a set of constrains that are essentially linear. However, the objective function is not differentiable over the constraint region. In this paper, we study a general theory of dual perturbations and derive a fundamental relationship between a perturbed dual program and the original problem. Based on this relationship, we establish a perturbation theory to display that a well-controlled perturbation on the dual program can overcome the nondifferentiability issue and generate an ź-optimal dual solution for an arbitrarily small number ź. A simple linear program is then constructed to make an easy conversion from the dual solution to a corresponding ź-optimal primal solution. Moreover, a numerical example is included to illustrate the potential of this controlled perturbation scheme.