kkt - CONSTRAINT QUALIFICATIONS FOR NONLINEAR PROGRAMMING...

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CONSTRAINT QUALIFICATIONS FOR NONLINEAR PROGRAMMING RODRIGO G. EUST ´ AQUIO § , ELIZABETH W. KARAS , AND ADEMIR A. RIBEIRO Abstract. This paper deals with optimality conditions to solve nonlinear programming problems. The classical Karush-Kuhn-Tucker (KKT) optimality conditions are demonstrated through a cone approach, using the well known Farkas’ Lemma. These conditions are valid at a minimizer of a nonlinear programming problem if a constraint qualification is satisfied. First we prove the KKT theorem supposing the equality between the polar of the tangent cone and the polar of the first order feasible variations cone. Although this condition is the weakest assumption, it is extremely difficult to be verified. Therefore, other constraints qualifications, which are easier to be verified, are studied, as: Slater’s, linear independence of gradients, Mangasarian-Fromovitz’s and quasiregularity. The relations among them are discussed. Key words. Optimality conditions, Karush-Kuhn-Tucker, constraint qualifications. 1. Introduction. We shall study the nonlinear programming problem ( P ) minimize f ( x ) subject to h ( x ) = 0 g ( x ) 0 , where the functions f : IR n IR, g : IR n IR p and h : IR n IR m are continuously differentiable. The feasible set is Ω = { x IR n | h ( x ) = 0 , g ( x ) 0 } . Given x * Ω, the classical Karush-Kuhn-Tucker (KKT) conditions say that there exist La- grangian multipliers λ * IR m and μ * IR p such that: -∇ f ( x * ) = m X i =1 λ * i h i ( x * ) + p X j =1 μ * j g j ( x * ) , μ * j 0 , j = 1 , . . . , p, μ * j g j ( x * ) = 0 , j = 1 , . . . , p. In nonlinear programming we would like that KKT are necessary conditions for a given point to be a solution to the problem. When the problem is unconstrained (Ω = IR n ), the KKT conditions reduce to f ( x * ) = 0 which is a necessary optimality condition. However, this not always happens, as shown in the following example. Example 1. Consider the problem ( P ) with f : IR 2 IR and g : IR 2 IR 2 defined by f ( x ) = x 1 and g ( x ) = ( x 2 - (1 - x 1 ) 3 , - x 2 ) T . Note that x * = (1 , 0) T is a minimizer of the problem but the KKT conditions do not hold. In this paper we shall discuss assumptions on the constraints in order to ensure that the KKT conditions hold at a minimizer. Such an assumption is called constraint qualification (CQ) . Formally, we say that the constraints h ( x ) = 0 and g ( x ) 0 satisfy a constraint qualification at x * Ω when, given any differentiable function f minimized at x * with respect to Ω, the KKT conditions are valid. Several authors have obtained different constraint qualifications. In this work, we will discuss many of them as well as some relations between them. A special interest is devoted to show the weakest such qualification. In this context, the concept of cones and their polars will be useful. § Master Program in Numerical Methods in Engineering, Federal University of Paran´a, Cx. Postal 19081, 81531-
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This note was uploaded on 01/11/2012 for the course ECON 200 taught by Professor Riley during the Fall '10 term at UCLA.

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kkt - CONSTRAINT QUALIFICATIONS FOR NONLINEAR PROGRAMMING...

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