A Dual Method for Quadratic Programs with Quadratic Constraints

In this paper, a dual method is developed for minimizing a convex quadratic function of several variables subject to inequality constraints on the same type of function. The dual program is a concave maximization problem with constraints that are essentially linear. However, the dual objective function is not differentiable over the dual constraint region. In particular, if the primal constraints are not all active, the dual objective function is not differentiable at the optimal point. The numerical difficulties associated with this nondifferentiability are circumvented by considering a sequence of dual programs via a modified penalty function technique that does not eliminate the dual constraints but does insure that they will all be active at optimality. A numerical example is included.