An efficient outer approximations algorithm for solving infinite sets of inequalities

摘要

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 phi (x,y)or=0 for all y in Y proceeds by solving at iteration i of the master algorithm, a finite set of inequalities ( phi (x,y)or=0 for all y in Y/sub i/ contained in/implied by Y) to yield x/sub i/ and then updating Y/sub i/ to Y/sub i+1/=Y/sub i/ union (y/sub i/) where y/sub i/ in arg max ( phi (x/sub i/,y) mod y in 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.