An efficient algorithm for solving semi-infinite inequality problems with box constraints

Many design objectives may be formulated as semi-infinite constraints. Examples in control design, for example, include hard constraints on time and frequency responses and robustness constraints. A useful algorithm for solving such inequalities is the outer approximations algorithm. One version of an outer approximations algorithm for solving an infinite set of inequalities(x, y) 0 for allyY proceeds by solving, at iterationi of the master algorithm, a finite set of inequalities ((x, y) 0 for allyY _i) to yieldx _i and then updatingY _i toY _i+1=Y _i {y_i } wherey _i arg max {(x _i,y)¦y Y}. Since global optimization is computationally extremely expensive, it is desirable to reduce the number of such optimizations. We present, in this paper, a modified version of the outer approximations algorithm which achieves this objective.The research reported herein was sponsored by the National Science Foundation Grants ECS-9024944, ECS-8816168, the Air Force Office of Scientific Research Contract AFOSR-90-0068, and the NSERC of Canada under Grant OGPO-138352.