We provide optimality conditions in terms of Michel–Penot’s directional derivatives in locally Lipschitz vector equilibrium problem with set, inequality and equality constraints. Using this derivatives, we introduce three constraint qualifications (CQ1), (CQ1-s) and (CQ2-s), and then, we establish primal and dual Karush–Kuhn–Tucker necessary optimality conditions for a local weak efficient solution and a local efficient solution. Under suitable assumptions on the MP-pseudoconvexity and strict MP-pseudoconvexity, sufficient optimality conditions, which are very near to dual Karush–Kuhn–Tucker necessary optimality conditions, are presented. Some examples to demonstrate for our findings are also provided.

我们根据具有集合、不等式和等式约束的局部 Lipschitz 向量平衡问题中的 Michel-Penot 方向导数提供最优条件。使用这个导数，我们引入了三个约束条件（CQ1）、（CQ1-s）和（CQ2-s），然后，我们为局部弱有效解和局部弱有效解建立了原始和对偶 Karush-Kuhn-Tucker 必要最优条件。有效的解决方案。在对 MP 伪凸性和严格 MP 伪凸性的适当假设下，提出了充分的最优性条件，非常接近于对偶 Karush-Kuhn-Tucker 必要最优性条件。还提供了一些示例来证明我们的发现。