Constrained Optimization

minimize f(x) subject to: hi(x) = 0 i = 1, . . . , m ≤ n (1) gj(x) ≤ 0 j = 1, . . . , p. hi’s are equality constraints and gj ’s are inequality constraints and usually they are assumed to be within the class C. A point that satisfies all constraints is said to be a feasible point. An inequality constraint is said to be active at a feasible point x if gi(x) = 0 and inactive if gi(x) < 0. Equality constraints are always active at any feasible point. To simplify notation we write h = [h1, . . . , hm] and g = [g1, . . . , gp], and the constraints now become h(x) = 0 and g(x) ≤ 0.