Narrow Results

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

Vector constrained extremum problems and Vector Variational Inequalities are considered, and a separation scheme is introduced. Starting from such a scheme, several theoretical aspects can be developed as well as methods of solution; for instance, scalarization of Vector Optimization is analysed and a method is described which does not require any assumption on the problem. The analysis is extended to a Vector Variational Inequality; this turns out to be equivalent to a scalar Quasi–Variational Inequality.

Problems in chemical engineering, like most real-world optimization problems, typically, have several conflicting performance criteria or objectives and they often are computationally demanding, which sets special requirements on the optimization methods used. In this paper, we point out some shortcomings of some widely used basic methods of multi-objective optimization. As an alternative, we suggest using interactive approaches where the role of a decision maker or a designer is emphasized. Interactive multi-objective optimization has been shown to suit well for chemical process design problems because it takes the preferences of the decision maker into account in an iterative manner that enables a focused search for the best Pareto optimal solution, that is, the best compromise between the conflicting objectives. For this reason, only those solutions that are of interest to the decision maker need to be generated making this kind of an approach computationally efficient. Besides, the decision maker does not have to compare many solutions at a time which makes interactive approaches more usable from the cognitive point of view. Furthermore, during the interactive solution process the decision maker can learn about the interrelationships among the objectives. In addition to describing the general philosophy of interactive approaches, we discuss the possibilities of interactive multi-objective optimization in chemical process design and give some examples of interactive methods to illustrate the ideas. Finally, we demonstrate the usefulness of interactive approaches in chemical process design by summarizing some reported studies related to, for example, paper making and sugar industries. Let us emphasize that the approaches described are appropriate for problems with more than two objective functions.

Problems in chemical engineering, like most real-world optimization problems, typically, have several conflicting performance criteria or objectives and they often are computationally demanding, which sets special requirements on the optimization methods used. In this chapter, we point out some shortcomings of some widely used basic methods of multi-objective optimization. As an alternative, we suggest using interactive approaches where the role of a decision maker or a designer is emphasized. Interactive multi-objective optimization has been shown to suit well for chemical process design problems because it takes the preferences of the decision maker into account in an iterative manner that enables a focused search for the best Pareto optimal solution, that is, the best compromise between the conflicting objectives. For this reason, only those solutions that are of interest to the decision maker need to be generated making this kind of an approach computationally efficient. Besides, the decision maker does not have to compare many solutions at a time which makes interactive approaches more usable from the cognitive point of view. Furthermore, during the interactive solution process the decision maker can learn about the interrelationships among the objectives. In addition to describing the general philosophy of interactive approaches, we discuss the possibilities of interactive multi-objective optimization in chemical process design and give some examples of interactive methods to illustrate the ideas. Finally, we demonstrate the usefulness of interactive approaches in chemical process design by summarizing some reported studies related to, for example, paper making and sugar industries. Let us emphasize that the approaches described are appropriate for problems with more than two objective functions.