Pareto-critical equilibrium condition for non-linear multiobjective optimization problems with applications

Abstract

The new (necessary) Pareto-critical equilibrium condition (PCEC) discussed in this study is based on the Pareto-critical condition for non-linear optimization problems. The proposed condition is evaluated by defining a multiobjective optimization problems (MOOPs) subject to a set of constraints. Then this MOOPs is converted to a single objective optimization problem by using the available optimization techniques, for instance, the weighted sum method, and solved by the non-linear Nelder-Mead simplex method. The obtained solution points satisfy the proposed PCEC of an unconstrained multiobjective optimization problem (MOOPs) under the convexity assumption. This PCEC is evaluated in a search that either results in a point inside the feasible region or a point for which the PCEC holds. The resulting Pareto-critical equilibrium condition is globally valid for convex problems. Numerical experiments are carried out in order to clarify the proposed condition's validity. Moreover, the proposed PCEC is used to place a downlink transmit beamforming system base station in an ideal position as a first application and is also employed in situations where there are multiple agents working towards a common goal as a second application.

Keywords

• Multiobjective optimization
• Pareto-critical equilibrium condition
• Downlink transmit beamforming
• Multi-agent system