| # Local and Global Optima in Convex Optimization Theorem: Global vs. local optima in convex optimization For convex problems, any locally optimal point is globally optimal. In addition, the optimal set is convex. Proof: Let be a local minimizer of on the set , and let . By definition, . We need to prove that . There is nothing to prove if , so let us assume that . By convexity of and , we have , and: Since is a local minimizer, the left-hand side in this inequality is nonnegative for all small enough values of . We conclude that the right hand side is nonnegative, i.e., , as claimed. Also, the optimal set is convex, since it can be written This ends our proof.