Some constraint qualifications for quasiconvex vector-valued systems

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In this paper, we consider minimization problems with a quasiconvex vector-valued inequality constraint. We propose two constraint quali12;cations, the closed cone constraint quali12;cation for vector-valued quasiconvex programming (the VQ-CCCQ) and the basic constraint quali12;cation for vector-valued quasicon-vex programming (the VQ-BCQ). Based on previous results by Benoist, Borwein, and Popovici (Proc. Amer. Math. Soc. 13: 1109-1113, 2002), and the authors (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 quali12;cation 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.