In this paper, several new set quality metrics are introduced that can be used to evaluate the ‘‘goodness’’ of an observed Pareto solution set. These metrics, which are formulated in closed-form and geometrically illustrated, include hyperarea difference, Pareto spread, accuracy of an observed Pareto frontier, number of distinct choices and cluster. The metrics should enable a designer to eithe...

We will present a topological approach to the Arrow impossibility theorem of social choice theory that there exists no binary social choice rule (which we will call a social welfare function) which satisfies the conditions of transitivity, independence of irrelevant alternatives (IIA), Pareto principle and non-existence of dictator. Our research is in line with the studies of topological approa...

Selecting a set of alternatives based on the preferences of agents is an important problem in committee selection and beyond. Among the various criteria put forth for desirability of a committee, Pareto optimality is a minimal and important requirement. As asking agents to specify their preferences over exponentially many subsets of alternatives is practically infeasible, we assume that each ag...

Selecting a set of alternatives based on the preferences of agents is an important problem in committee selection and beyond. Among the various criteria put forth for the desirability of a committee, Pareto optimality is a minimal and important requirement. As asking agents to specify their preferences over exponentially many subsets of alternatives is practically infeasible, we assume that eac...

In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency/minimality. Using advanced tools of variational analysis and generalized differentiation, we estab...

In this paper we address the question of how many objective functions are really needed to decide whether a given point is Pareto optimal. We prove a reduction result for the case of quasi-convex objective functions and a convex feasible set. This result states that in order to decide whether a point x in the decision space is Pareto optimal it suuces to consider at most n + 1 objectives at a t...

In this paper we introduce and study enhanced notions of relative Pareto minimizers to constrained multiobjective problems that are defined via several kinds of relative interiors of ordering cones and occupy intermediate positions between the classical notions of Pareto and weak Pareto efficiency /minimality. Using advanced tools of variational analysis and generalized differentiation, we esta...

We consider a two-user multiple-input multiple-output (MIMO) interference channel (IC), where a single data stream is transmitted and each receiver applies the minimum mean square error (MMSE) filter. In this paper, we study an open topic on the Pareto boundary of the rate region. The Pareto boundary is divided by two turning points into the weak Pareto boundary (including the horizontal part a...

Multi-objective optimisation problems normally have not one but a set of solutions, which are called Pareto-optimal solutions or non-dominated solutions. Once a Pareto-optimal set has been obtained, the decision-maker faces the challenge of analysing a potentially large set of solutions. Selecting one solution over others can be quite a challenging task because the Pareto set can contain an unm...

A multiobjective optimization problem is considered in a general formulation. It is well known that the solution of the problem is not unique and represented in the objective space by a Pareto frontier. In an engineering design it can be important to provide a discrete representation of the entire Pareto surface. Meanwhile, the obtaining of a single Pareto solution may be time-consuming. Moreov...