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

Abstract

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 l 1 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.

A mesh‐based partitioning algorithm for decreasing conservatism in solving bilinear matrix inequality problems

“…The BMI problem was first used in Reference 3. Since then, the researchers have listed several problems resulting in a BMI problem 4,5 . For example, the applications of the BMI problems in designing control systems 6,7 such as guaranteed cost control, 8,9 static output feedback controller for spectral abscissa optimization, 10 uncertain fractional order system, 11,12 $$$ {H}_2 $$$ and $$$ {H}_{\infty } $$$ optimization, 13 observer‐based robust controller design, 14 model predictive control, 15 fuzzy controller design, 16 Self Optimizing Control, 17 non‐quadratic controller design, 18 and power flow problem in power systems 19 are reported in the literature.…”