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Lagrange Multipliers and Calmness Conditions of Order p

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China
College of Business and Administration, Zhejiang University of Technology, Zhejiang, China
mayangxq{at}polyu.edu.hk, http://myweb.polyu.edu.hk/mayangxq/index.htm
mengzhiqing{at}zjut.edu.cn

In this paper, by assuming that a non-Lipschitz penalty function is exact, new conditions for the existence of Lagrange multipliers are established for an inequality and equality-constrained continuously differentiable optimization problem. This is done by virtue of a first-order necessary optimality condition of the penalty problem, which is obtained by estimating Dini upper-directional derivatives of the penalty function in terms of Taylor expansions, and a Farkas lemma. Relations among the obtained results and some well-known constraint qualifications are discussed.

Key Words: non-Lipschitz penalty function; generalized calmness condition; Dini directional derivative; Lagrange multiplier
History: Received: August 18, 2005; revision received: December 30, 2005;