Partially Augmented Lagrangian Method For Matrix Inequality Constraints

A large number of design problems in robust control theory may be cast as mini-mizing a system gain under convex linear matrix inequality constraints in tandem with nonlinear algebraic equalities,such as the fixed or reduced-order H2 and H∞ synthe-sis problems、robust control of nonlinear systems with integral quadratic constraints-defined components、robust control design with parameter-dependent Lyapunov func-tions.Those realistic problems have a common feature:some of the constraints are simple and are satisfied easily.In this paper,we develop a partially augmented La-grangian method for a class of linear matrix inequality-constrained problems in robust control theory,where only nonlinear equality constraints are included in the augment-ed Lagrangian while linear matrix inequality constraints are kept explicitly in order to exploit currently available semidefinite programming codes.The partially augment-ed Lagrangian function is adopted only as a merit function.At every iteration,our proposed algorithm firstly solve a convex quadratic programming subproblem with a feasible linear semidefinite constraint,which will generate a direction that improves the measurement of constraint violation.And then another convex quadratic programming subproblem with a feasible linear semidefinite constraint is solved,which will generate a direction that improves the optimality.A regular term is added to the first subprob-lem.Even if the gradients of the equality constraints are linear dependent,it is still ensured that the solution of the first subproblem exists and is bounded.The solution of the first subproblem provides a measurement scale for adjusting the penalty factor in the second subproblem.In other words,when we augment the penalty factor suitably,the search direction,i.e.,the solution of the second subproblem,will give consideration to both the measurement of feasibility and one of optimality.Finally,combining with line search technique,the trial step is determined to guarantee the sufficient decrease in the partially augmented Lagrangian function.Global convergence of the algorithm is analyzed even if the gradients of the equality constraints are linear dependent.The preliminary numerical results are reported.