Optimality conditions for quasiconvex programs Nguyen Thi Hong Linh ∗ and Jean-Paul Penot † Abstract We present necessary and sufficient optimality conditions for a problem with a convex set constraint and a quasiconvex objective function. We apply the obtained results to a mathematical programming problem involving quasiconvex functions. Key words : convex programming, normal cone, optimality conditions, quasicon- vex function, quasiconvex programming, subdifferential. Mathematics Subject Classification : 90C26, 26B25, 52A41 1 Introduction It is the purpose of this note to present some optimality conditions for constrained op- timization problems under generalized convexity assumptions. We essentially deal with quasiconvex functions, i.e. functions whose sublevel sets are convex. Such functions form the main class of generalized convex functions and are widely used in mathematical eco- nomics. We do not assume the data of the problem are smooth. Thus, we replace the derivatives appearing in the classical results by subdifferentials. We only use the adapted subdifferentials of quasiconvex analysis, namely the Plastria subdifferential and the in- fradifferential or Guti´ errez subdifferential. Because these subdifferentials are useful for algorithmic purposes ([31], [16]) and have links with duality ([13, Prop. 6.1], [26], [25], [27]), their use in problems in which quasiconvexity properties occur seems to be sensible, albeit their calculus rules are not as rich as in the case of convex analysis ([20], [29]). In [13, Prop. 6.1] Mart´ ınez-Legaz presented a result of the Kuhn-Tucker type using these subdifferentials; in [14, Thm. 4.1] and [15, Prop. 6.3] variants of these subdifferentials are used. In each of these results, a strict quasiconvexity assumption (and a Slater condition) is imposed. Such an assumption lays aside the important convex case. Here we essentially assume the functions are quasiconvex and we deduce the results from optimality conditions for problems with set constraints. We also point out the link with the differentiable case. * Department of Mathematics, University of Natural Sciences, Ho Chi Minh City, Vietnam honglinh98t1@yahoo.com † Laboratoire de Math´ ematiques, CNRS UMR 5142, Facult´ e des Sciences, Av. de l’Universit´ e 64000 PAU, France jean-paul.penot@univ-pau.fr 1