In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

在本文中，我们研究了在具有非凸约束函数的约束集上最小化凸函数的优化问题（P）。为此，我们给出了Robinson约束条件的新特征，将其简化为在参考点具有几乎凸的可行集的非凸规划问题的广义Slater条件和广义锐化非退化条件的组合。接下来，使用强CHIP的版本，我们提出了约束限定条件，这对于优化问题（P）是必不可少的。最后，使用鲁滨逊约束条件的新特征，我们给出了问题（P）最优的必要条件和充分条件。