@MISC{Zhang_constrainedoptimization, author = {Jian Zhang}, title = {CONSTRAINED OPTIMIZATION}, year = {} }

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Abstract

Here we introduce some basic concepts of constrained optimization and illustrate with very simple examples. More details can be found in [1] and other references. Consider the minimization problem of the form (1) minimize f(x) subject to: hi(x) = 0 i = 1,...,m ≤ n 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 2. 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. Karush-Kuhn-Tucker (KKT) Conditions KKT conditions (a.k.a. Kuhn-Tucker conditions) are necessary conditions for the local minimum solutions of problem (1). Theorem 1.1. Let x ∗ be a local minimum point for Problem (1) and suppose x ∗ is a regular point for the constraints. Then there is a vector λ ∈ R m and a vector µ ∈ R p with µ ≥ 0 such that