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 phi(x, y) less-than-or-equal-to 0 for all y is-an-element-of Y proceeds by solving, at iteration i of the master algorithm, a finite set of inequalities (phi(x, y) less-than-or-equal-to 0 for all y is-an-element-of Y(i)) to yield x(i) and then updating Y(i) to Y(i+1) = Y(i) or {y(i)} where y(i) is-an-element-of arg max {phi(x(i), y)y is-an-element-of 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.