Lecture 11: October 8 11.1 Primal and Dual Problems

j=1 vj`j(x). Lagrange multipliers u ∈ R, v ∈ R. Lemma 11.1 At each feasible x, f(x) = supu≥0,v L(x, u, v), and the supremum is taken iff u ≥ 0 satisfying uihi(x) = 0, i = 1, · · · ,m. Proof: At each feasible x, we have hi(x) ≤ 0 and `(x) = 0, thus L(x, u, v) = f(x) + ∑m i=1 uihi(x) + ∑r j=1 vj`j(x) ≤ f(x). The last inequality becomes equality iff uihi(x) = 0, i = 1, · · · ,m. Proposition 11.2 The optimal value of the primal problem, named as f, satisfies: f = inf x sup u≥0,v L(x, u, v).