Abstract
We propose a new scalarization method which consists in constructing, for a given multiobjective optimization problem, a single scalarization function, whose global minimum points are exactly vector critical points of the original problem. This equivalence holds globally and enables one to use global optimization algorithms (for example, classical genetic algorithms with “roulette wheel” selection) to produce multiple solutions of the multiobjective problem. In this article we prove the mentioned equivalence and show that, if the ordering cone is polyhedral and the function being optimized is piecewise differentiable, then computing the values of a scalarization function reduces to solving a quadratic programming problem. We also present some preliminary numerical results pertaining to this new method.
Highlights
Scalarization is one of the most commonly used methods of solving multiobjective optimization problems
We propose a new scalarization method which consists in constructing, for a given multiobjective optimization problem, a single scalarization function, whose global minimum points are exactly vector critical points of the original problem
We present some preliminary numerical results pertaining to this new method
Summary
Scalarization is one of the most commonly used methods of solving multiobjective optimization problems. We propose a new scalarization method different from the above-mentioned ones It consists in constructing, for a given multiobjective optimization problem, a single scalarization function, whose global minimum points are exactly vector critical points in the sense of [8] for the original problem. 170) A convex set D in p is called polyhedral if it can be expressed as the intersection of some finite collection of closed halfspaces, that is, there exist vectors bi ∈ p and numbers βi such that. It follows from ([14], Thm. 19.1) that a convex set D in p is polyhedral if and only if it is finitely generated, which means that there exist vectors a1, , al such that, for a fixed integer k,. There are specialized algorithms designed for computing such projections (see [17] [18])
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