A finite nonadjacent extreme-point search algorithm for optimization over the efficient set
Journal of Optimization Theory and Applications
Global optimality criterion and a duality with a zero gap in nonconvex optimization
SIAM Journal on Mathematical Analysis
Minimization of a quasi-concave function over an efficient set
Mathematical Programming: Series A and B
Optimization over the efficient set: four special cases
Journal of Optimization Theory and Applications
Optimizing a linear function over an efficient set
Journal of Optimization Theory and Applications
Dual approach to minimization on the set of Pareto-optimal solutions
Journal of Optimization Theory and Applications
Journal of Optimization Theory and Applications
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
An Inner Approximation Method for Optimization over the Weakly Efficient Set
Journal of Global Optimization
Conical Algorithm in Global Optimization for Optimizing over Efficient Sets
Journal of Global Optimization
Optimizing Over the Efficient Set Using a Top-Down Search of Faces
Operations Research
Numerical solution for optimization over the efficient set by d.c. optimization algorithms
Operations Research Letters
An Evolutionary Algorithm to Estimate the Nadir Point in MOLP
EMO '09 Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization
Criteria and dimension reduction of linear multiple criteria optimization problems
Journal of Global Optimization
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Over the past several decades, the optimization over the efficient set has seen a substantial development. The aim of this paper is to provide a state-of-the-art survey of the development. Given p linear criteria c1x,ċċċ,cp x and a feasible region X of Rn, the linear multicriteria problem is to find a point x of X such that no point x' of X satisfies (c1 x',ċċċ,cp x')≥(c1 x,ċċċ,cp x) and (c1x',ċċċ,cp x')≠q (c1 x ,ċċċ,cp x). Such a point is called an efficient point. The optimization over the efficient set is the maximization of a given function φ over the set of efficient points. The difficulty of this problem is mainly due to the nonconvexity of this set. The existing algorithms for solving this problem could be classified into several groups such as adjacent vertex search algorithm, nonadjacent vertex search algorithm, branch-and-bound based algorithm, Lagrangian relaxation based algorithm, dual approach and bisection algorithm. In this paper we review a typical algorithm from each group and compare them from the computational point of view.