An elementary proof of linear programming optimality conditions without using Farkas’ lemma

Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas’ lemma, but proofs of Farkas’ lemma typically involve “nonlinear” topics such as separating hyperplanes between disjoint convex sets, or else more … Read more

Convergence of the restricted Nelder-Mead algorithm in two dimensions

The Nelder-Mead algorithm, a longstanding direct search method for unconstrained optimization published in 1965, is designed to minimize a scalar-valued function $f$ of $n$ real variables using only function values, without any derivative information. Each Nelder–Mead iteration is associated with a nondegenerate simplex defined by $n + 1$ vertices and their function values; a typical … Read more