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. (2002), and Suzuki and Kuroiwa (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.