A Lifting-Penalty Method for Quadratic Programming with a Quadratic Matrix Inequality Constraint

In this paper, a lifting-penalty method for solving the quadratic programming with a quadratic matrix inequality constraint is proposed. Additional variables are introduced to represent the quadratic terms. The quadratic programming is reformulated as a minimization problem having a linear objective function, linear conic constraints and a quadratic equality constraint. A majorization-minimization method is used to solve instead a <mml:semantics>l1</mml:semantics> penalty reformulation of the minimization problem. The subproblems arising in the method can be solved by using the current semidefinite programming software packages. Global convergence of the method is proven under some suitable assumptions. Some examples and numerical results are given to show that the proposed method is feasible and efficient.