J Optim Theory Appl (2008) 139: 1–16 DOI 10.1007/s10957-008-9431-1 Quasiconvex Minimization on a Locally Finite Union of Convex Sets D. Aussel · J.J. Ye Published online: 6 June 2008 © Springer Science+Business Media, LLC 2008 Abstract Extending the approach initiated in Aussel and Hadjisavvas (SIAM J. Op- tim. 16:358–367, 2005) and Aussel and Ye (Optimization 55:433–457, 2006), we obtain the existence of a local minimizer of a quasiconvex function on the locally fi- nite union of closed convex subsets of a Banach space. We apply the existence result to some difficult nonconvex optimization problems such as the disjunctive program- ming problem and the bilevel programming problem. Keywords Quasiconvex programming · Existence results · Nonconvex constraint set 1 Introduction Let us consider the following mathematical programming problem: min f (x), s.t. min j ∈ J g j (x) ≤ 0, Dedicated to Jean-Pierre Crouzeix on the occasion of his 65th birthday. Communicated by T. Rapcsak. The authors thank two anonymous referees for careful reading of the paper and helpful suggestions. The research of the second author was partially supported by NSERC/Canada. D. Aussel ( ) Département de Mathématiques, Université de Perpignan, Perpignan, France e-mail: aussel@univ-perp.fr J.J. Ye Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada