SIAM Journal on Optimization, Volume 31, Issue 2, Page 1410-1432, January 2021.
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 J. 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.

中文翻译：

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赋范线性空间中凸函数不等式系统的FM和BCQ资格

SIAM优化杂志，第31卷，第2期，第1410-1432页，2021年1月。
对于由范数线性空间中的无限个适当的下半连续凸函数的无穷系列定义的不等式系统，我们考虑了Farkas-Minkowski型（简称FM）类型限定和基本约束限定（简称BCQ）。通过基于在SECQ（题词约束条件的总和）上在此处建立的一些新结果，采用新方法来处理封闭凸集的族，提供了一些条件，其中包括进一步放宽Slater类型条件以确保FM资格。作为应用，给出了确保BCQ的新的充分条件。这些结果显着改善了[C. Li和KF Ng，SIAM J. Optim。，15（2005），第488--512页]和[C. Li XP Zhao，YH Hu，SIAM J. Optim。，2013年第23期，第2208--2230页]，它们是在没有对先前工作严重依赖的不平等系统的功能保持关键连续性假设的情况下获得的。还提供了一些示例以说明我们的结果的适用性。