This paper examines normal cones of the feasible set for mathematical programming problems with switching constraints (MPSC)‎. ‎Functions involved are assumed to be continuously differentiable‎. ‎The primary focus is on providing the upper estimate of the Mordukhovich normal cone for the feasible set of MPSCs‎. ‎First‎, ‎a constraint qualification‎, ‎called the ``MPSC-No Nonzero Abnormal Multiplier Constraint Qualification''‎, ‎is considered for the problem‎. ‎Based on this qualification‎, ‎the main result of the paper is presented‎. ‎Finally‎, ‎an optimality condition‎, ‎called the ``necessary M-stationarity condition'' is proposed for optimal solutions of the considered problems‎. ‎Since other optimization problems with multiplicative constraints can be rewritten in the form of MPSCs‎, ‎results obtained in this paper can be extended to a wider class of problems involving multiplicative constraints‎.

This paper addresses a non-smooth multi-objective semi-infinite programming problem that involves a feasible set defined by inequality constraints‎. ‎Our focus is on introducing a new weak Slater constraint qualification and deriving the necessary and sufficient conditions for (weakly‎, ‎properly) efficient solutions to the problem using (weak and strong) Karush-Kuhn-Tucker types‎. ‎Additionally‎, ‎we present two duals of the Mond-Weir type for the problem and provide (weak and strong) duality results for them‎. ‎All of the results are given in terms of Clarke subdifferential‎.

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.

This paper introduces several Abadie-type constraint qualifications and derives necessary optimality conditions in the Karush-Kuhn-Tucker‎ ‎for both weakly efficient solutions and efficient solutions of a nonsmooth multi-objective semi-infinite programming problem characterized by locally Lipschitz data‎. ‎The findings are expressed in terms of the Micheal-Penot subdifferential‎.

In this paper an uncertain multi objective closed-loop supply chain is developed. The first objective function is maximizing the total profit. The second objective function is minimizing the use of row materials. In the other word, the second objective function is maximizing the amount of remanufacturing and recycling. Genetic algorithm is used for optimization; and for finding the pareto optimal line, Epsilon-constraint method is used. Finally a numerical example is solved with proposed approach and performance of the model is evaluated in different sizes. The results show that this approach is effective and useful for managerial decisions.

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.

Constrain is the condition of driven by necessity. The principle of constraint determines the duties of the person constrained. In jurisprudence, it includes a range of activities from cleanness to blood compensation. However, jurisprudents have discussed it mainly with regard to those that can be eaten and drunk, and presented three different interpretations: fear of losing one"s life; fear in general, including fear of losing one"s life, or one"s limb, etc.; and fear of things not related to oneself but to others. The roots of constraint are compulsion, precautionary concealment, loss and necessity. The present article deals with a kind of constraint whose source is necessiy. The Qur"an has explicitly excused the person constrained from duties provided that the oppressor shows no inclination to sin. Tradition says that "there is so unlawful act that God Almighty has not made it permissible under constraint." The intellect also knows it permissible to give up necessary duties and to do the unlawful acts under constraint. Finally, the writer presents the conditions under which the application of the principle of constrain is made permissible.

The concept of design is associated with the notion of constraint as a limiting factor, and the search for the constraint in the design process is essential in design research. This study applies the ontological method to define the domain and scope of constraints, their classes and subclasses, and how to apply them. We concluded that there are two types of constraints in design: inside the designer’, s mind or internal constraint and outside it or external constraint. Each of these is classified into two subclasses: internal constraints can be divided into deduced and subject constraints, and external constraints are categorised into imposed and intrinsic constraints. In the end, we proposed a model that shows that designers can identify, choose, manage, and even generate constraints in the design process to facilitate the restructuring of the design problem, which helps them solve the design problem.