JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 106, No. 2, pp. 373–398, AUGUST 2000 Constraint Qualifications in Nonsmooth Multiobjective Optimization 1,2 X. F. LI 3 Communicated by H. P. Benson Abstract. For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary con- ditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem. Key Words. Nonsmooth multiobjective optimization, constraint quali- fications, KT necessary conditions, Lipschitz continuity, Clarke subdifferential. 1. Introduction For differentiable optimization problems with scalar-valued objective functions, necessary conditions of KT-type for optimality can be established under various constraint qualifications; see Refs. 1–6 and references therein. Seven of the well-known qualifications are the Guignard qualification, 1 The paper is dedicated to J. L. Dong on the occasion of his 62nd birthday. 2 This work was supported by the National Natural Science Foundation of P.R. China. The author is grateful to the referees for valuable comments and suggestions which greatly improved the original paper. The author also thanks Professor G. H. Tian for helpful dis- cussions on topological properties of cones. 3 Associate Professor, Department of Applied Mathematics, Jilin University of Technology, Changchun, P.R. China. Ph.D. student of Department of Mathematics, City University of Hong Kong. 373 0022-3239000800-0373$18.000  2000 Plenum Publishing Corporation