THE FM AND BCQ QUALIFICATIONS FOR INEQUALITY SYSTEMS OF CONVEX FUNCTIONS IN NORMED LINEAR SPACES

For an inequality system defined by an infinite family of proper lower semicontinuous convex functions in normed linear space, we consider the Farkas-Minkowski (FM for short) type qualification and the basic constraint qualification (BCQ for short). By employing a new approach based on some new results established here on the SECQ (sum of epigraphs constraint qualification) for families of closed convex sets, some sufficient conditions involving further relaxing Slater type conditions for ensuring the FM qualification are provided. As applications, new sufficient conditions for ensuring the BCQ are given. These results significantly improve the corresponding ones in [C. Li and K. F. Ng, SIAM T. Optim., 15 (2005), pp. 488-512] and [C. Li, X. P. Zhao, and Y. H. Hu, SIAM J. Optim., 23 (2013), pp. 2208-2230], and they are obtained without the key continuity assumption on the sup-function of the inequality system which the previous works depend heavily on. Some examples are also presented to illustrate the applicability of our results.