Optimizing over the properly efficient set of convex multi-objective optimization problems
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Optimizing over the properly efficient set of convex multi-objective optimization problems Kahina Ghazli1,2
· Nicolas Gillis3 · Mustapha Moulaï1
Accepted: 29 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Optimizing over the efficient set of a multi-objective optimization problem is among the difficult problems in global optimization because of its nonconvexity, even in the linear case. In this paper, we consider only properly efficient solutions which are characterized through weighted sum scalarization. We propose a numerical method to tackle this problem when the objective functions and the feasible set of the multi-objective optimization problem are convex. This algorithm penalizes progressively iterates that are not properly efficient and uses a sequence of convex nonlinear subproblems that can be solved efficiently. The proposed algorithm is shown to perform well on a set of standard problems from the literature, as it allows to obtain optimal solutions in all cases. Keywords Multi-objective programming · Convex optimization · Properly efficient set · Penalty approach
1 Introduction Multi-objective optimization (also referred to as vector or multi-criteria optimization) deals with the situations where one wishes to minimize several conflicting objectives. Formally, a multi-objective optimization problem (MOP) can be formulated as follows min f 1 (x), . . . , f p (x) such that x ∈ X , (1) x
where p ≥ 2, x = (x1 , . . . , xn ) is the vector of decision variables, X ⊆ Rn represent the feasible set and f k : Rn → R (k = 1, . . . , p) are the objective functions. A solution of (1) is efficient if it is impossible to improve the value of one objective function without deteriorating at least one of the others, and it is weakly efficient if it is impossible to simultaneously improve all the values of the objective functions. Solving (1) requires to find the set of all efficient
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Kahina Ghazli [email protected]
1
LaROMad, Faculty of Mathematics (USTHB), Algiers, Algeria
2
University of Bejaia, Béjaïa, Algeria
3
Department of Mathematics and Operational Research, Faculté Polytechnique, University of Mons, Mons, Belgium
123
Annals of Operations Research
solutions (Pareto optimal solutions) or the set of all weakly efficient solutions (weak Pareto optimal solutions). The problem (1) has been widely studied in the literature both theoretically and practically; see for example Steuer (1986), Luc (1989), Miettinen (1999), Ehrgott (2005) and Chinchuluun and Pardalos (2007). In many situations, the decision-making process does not require explicit enumeration of all efficient solutions, but only efficient solutions achieving the optimum of some scalar function expressing the decision maker’s preferences within the set of (weak) Pareto optimal solutions. This is the problem of optimizing over the (weakly) efficient set introduced by Phillip (1972). This variant of multi-objective optimization avoids numerical difficulties related to generating all efficient sol
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