Some constraint qualifications for quasiconvex vector-valued systems

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, <CitationRef CitationID="CR7">2002</CitationRef>), and Suzuki and Kuroiwa (J Optim Theory Appl 149:554–563, <CitationRef CitationID="CR4">2011</CitationRef>), and (Nonlinear Anal 74:1279–1285, <CitationRef CitationID="CR5">2011</CitationRef>), 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. Copyright Springer Science+Business Media, LLC. 2013