Maximal vectors and multi-objective optimization

Maximal vector andweak-maximal vector are the two basic notions underlying the various broader definitions (like efficiency, admissibility, vector maximum, noninferiority, Pareto's optimum, etc.) for optimal solutions of multi-objective optimization problems. Moreover, the understanding and characterization of maximal and weak-maximal vectors on the space of index vectors (vectors of values of the multiple objective functions) is fundamental and useful to the understanding and characterization of Pareto-optimal and weak-optimal solutions on the space of solutions.This paper is concerned with various characterizations of maximal and weak-maximal vectors in a general subset of the EuclideanN-space, and with necessary conditions for Pareto-optimal and weak-optimal solutions to a generalN-objective optimization problem having inequality, equality, and open-set constraints on then-space. A geometric method is described; the validity of scalarization by linear combination is studied, and weak conditioning by directional convexity is considered; local properties and a fundamental necessary condition are given. A necessary and sufficient condition for maximal vectors in a simplex or a polyhedral cone is derived. Necessary conditions for Pareto-optimal and weak-optimal solutions are given in terms of Lagrange multipliers, linearly independent gradients, Jacobian and Gramian matrices, and Jacobian determinants.Several advantages in approaching the multi-objective optimization problem in two steps (investigate optimal index vectors on the space of index vectors first, and study optimal solutions on the specific space of solutions next) are demonstrated in this paper.This work was supported by the National Science Foundation under Grant No. GK-32701.