<p>Extremum problems with vanishing constraints are models several applications in structural and topology optimization. In this paper, a class of nonsmooth vector optimization problems with both inequality, equality and vanishing constraints is considered. The Abadie regularity condition and the modified Abadie regularity condition are introduced for the aforesaid multicriteria optimization problems if the functions constituting them are Hadamard differentiable. Under the mentioned regularity conditions, the Karush-Kuhn-Tucker type necessary optimality conditions are established for vector optimization problems with vanishing constraints in which the involved functions are Gàteaux differentiable. Further, the sufficient optimality conditions are proved for such nondifferentiable multiobjective programming problems with vanishing constraints under assumptions that the objective functions are pseudo-convex and constraint functions are quasi-convex. Thus, the fundamental results from optimization theory, that is, optimality conditions are proved for a new class of structural and topological optimization problems for which the aforesaid multicriteria optimization problems with vanishing constraints are models.</p>