Some constraint qualifications for quasiconvex vector-valued systems

Abstract

In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint qualifications, the closed cone constraint qualification for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint qualification for vector-valued quasiconvex programming (the VQ-BCQ). Based on previous results by Benoist et al. (Proc Am Math Soc 13:1109---1113, 2002), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554---563, 2011), and (Nonlinear Anal 74:1279---1285, 2011), we show that the VQ-CCCQ (resp. the VQ-BCQ) is the weakest constraint qualification for Lagrangian-type strong (resp. min---max) duality. As consequences of the main results, we study semi-definite quasiconvex programming problems in our scheme, and we observe the weakest constraint qualifications for Lagrangian-type strong and min---max dualities. Finally, we summarize the characterizations of the weakest constraint qualifications for convex and quasiconvex programming.