An efficient outer approximations algorithm for solving infinite sets of inequalities

Many control design constraints may be formulated as semi-infinite constraints. Examples include hard constraints on time and frequency responses and robustness constraints. A useful algorithm for solving such inequalities is the outer approximations algorithm. The standard outer approximations algorithm for solving an infinite set of inequalities φ(x,y)⩽0 for all y∈ Y proceeds by solving at iteration i of the master algorithm, a finite set of inequalities (φ(x,y)⩽0 for all y∈Y _i⊂Y) to yield x_i and then updating Y_i to Y_i+1=Y _i∪{y_i} where y_i∈arg max {φ(x_i,y )|y∈Y}. Since global optimization is computationally extremely expensive, it is desirable to reduce the number of such optimizations. A modified version of the outer approximations algorithm which achieves this objective is presented