The aim of this paper is to establish Sequential necessary and sufficient approximate optimality conditions for a constrained convex vector mini-mization problem without any constraint qualifications, characterizing the approximate proper and weak efficient solutions. The constraints are de-scribed by mappings taking values in different preorder vector spaces. Our approach is based essentially on the Sequential approximate subdifferential calculus rule for the sums of a finite family of cone convex mappings. To illustrate our main result, an application to multiobjective fractional pro-gramming problem is given. Finally, we present an important subclass of such problems showing the applicability of the obtained conditions.

Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These Sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constraint qualifications. It is proved that nonsmooth complementary approximate Karush-Kuhn-Tucker conditions are stronger than nonsmooth approximate gradient projection conditions. Sufficiency for differentiable generalized convex programming is established.

This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems‎. ‎These problems involve inequality constraints whose index set depends on the decision vector‎, ‎and all emerging functions are assumed to be locally Lipschitz‎. ‎We introduce a new constraint qualification for these problems‎. ‎Building upon this qualification‎, ‎we derive an upper estimate for the Clarke sub-differential of the value function of the problem‎. ‎Furthermore‎, ‎we demonstrate the necessary optimality conditions for properly efficient solutions to the problem‎.

In this paper, we consider fuzzy optimization problems more general than all those that exist in the literature by using the concept of metric-based differentiability of fuzzy-valued functions. We get necessary optimality conditions based on fuzzy stationary point definition, and we prove these conditions are also sufficient under new fuzzy invexity notions. We illustrate these results with numerical examples.

In this paper, we investigate relation between weak subdifferential and augmented normal cone. We define augmented normal cone via weak subdifferential and vice versa. The necessary conditions for the global maximum are also stated. We produce preliminary properties of augmented normal cones and discuss them via the distance function. Then we obtain the augmented normal cone for the indicator function. Relation between weak subifferential and augmented normal cone and epigraph is also explored. We also obtain optimality conditions via weak subdifferential and augmented normal cone. Finally, we define the Stampacchia and Minty solution via weak subdifferential and investigate the relation between Stampacchia and Minty solution and the minimal point.

The primary objective of this paper is to enhance several well-known geometric constraint qualifications and necessary optimality conditions for nonsmooth semi-infinite optimization problems (SIPs). We focus on defining novel algebraic Mangasarian-Fromovitz type constraint qualifications, and on presenting two Karush-Kuhn-Tucker type necessary optimality conditions for nonsmooth SIPs defined by locally Lipschitz functions. Then, by employing a new type of generalized invex functions, we present sufficient conditions for the optimality of a feasible point of the considered problems. It is noteworthy that the new class of invex functions we considered encompasses several classes of invex functions introduced previously. Our results are based on the Michel-Penot subdifferential.

In the present paper, we show the some properties of the fuzzy solution of the control linear fuzzy differential equations and research the optimal time problems for it.