Skip to main content
SpringerLink
Log in

Optimization over the efficient set: overview

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

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 c 1x,ċċċ,cp x and a feasible region X of R n, 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Al-Khayyal, F.A. and Falk, J.E. (1983), Jointly constrained biconvex programming. Mathematics of Operations Research 8, 273–286.

    Google Scholar 

  • An, L.T.H., Tao, P.D. and Muu, L.D. (1996), Numerical solution for optimization over the efficient set by d.c. optimization algorithms. Operations Research Letters 19, 117–128.

    Google Scholar 

  • Atteia, M. and El Qortobi, Quasi-convex duality. In Optimization and Optimal Control,Lecture Notes in Control and Information Science 35, Springer, Berlin, 1981, pp. 3–8.

    Google Scholar 

  • Benson, H.P. (1991), An all-linear programming relaxation algorithm for optimizing over the efficient set. Journal of Global Optimization 1, 83–104.

    Google Scholar 

  • Benson, H.P. (1992), A finite nonadjacent extreme-point search algorithm for optimization over the efficient set. Journal of Optimization Theory and Applications 73, 47–64.

    Google Scholar 

  • Benson, H.P. (1995), A geometric analysis of the efficient outcome set in multiple objective convex program with linear criteria functions. Journal of Global Optimization 6, 213–251.

    Google Scholar 

  • Benson, H.P. and Lee, D. (1996), Outcome-based algorithm for optimizing over the efficient set of a bicriteria linear programming. Journal of Optimization Theory and Applications 88, 77–105.

    Google Scholar 

  • Benson, H.P. and Sayin, S. (1994), Optimization over the efficient set: four special case. Journal of Optimization Theory and Applications 80, 3–18.

    Google Scholar 

  • Bolintineanu, S. (1993), Minimization of a quasi-concave function over an efficient set. Mathematical Programming 61, 89–110.

    Google Scholar 

  • Chen, P.C., Hansen, P. and Jaumard, B. (1991), On-line and off-line vertex enumeration by adjacency lists. Operations Research Letters 10, 403–409.

    Google Scholar 

  • Dauer, J.P. and Fosnaugh, T.A. (1995), Optimization over the efficient set. Journal of Global Optimization 7, 261–277.

    Google Scholar 

  • Ecker, J.G. and Song, J.H. (1994), Optimizing a linear function over an efficient set. Journal of Optimization Theory and Applications 83, 541–563.

    Google Scholar 

  • Fülöp, J. (1994), A cutting plane algorithm for linear optimization over the efficient set. In S. Komlösi, T. Rapcsàk and S. Shaible (eds.), Generalized Convexity, Lecture Notes in Economics and Mathematical Systems 405, Springer, Berlin, pp. 374–385.

    Google Scholar 

  • Garay, M.R. and Johnson, D.S. (1979), Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco.

    Google Scholar 

  • Horst, R., de Vries, J. and Thoai, N.V. (1988), On finding new vertices and redundant constraints in cutting plane algorithms for global optimization. Operations Research Letters 7, 85–90.

    Google Scholar 

  • Horst, R. and Thoai, N.V. (1999), Maximizing a concave function over the efficient or weaklyefficient set. European Journal of Operational Research 117, 239–252.

    Google Scholar 

  • Horst, R. and Tuy, H. (1996), Global Optimization: Deterministic Approach. Springer-Verlag, Berlin.

    Google Scholar 

  • Iri, M. (1994), An essay in the theory of uncontrollable flows and congestion. Technical Report, Department of Information and System Engineering, Faculty of Science and Engineering, Chuo University, TRISE 94–03.

    Google Scholar 

  • Iri, M. (1996), Network flow-theory and applications with practical impact. In J. Doležal and J. Fidler (eds.), System Modelling and Optimization, Chapman & Hall, London, 1996, pp. 24–36.

    Google Scholar 

  • Isermann, H. and Steuer, E. (1987), Computational experience concerning payoff tables and minimum criterion values over the efficient set. European Journal of Operational Research 33, 91–97.

    Google Scholar 

  • Konno, H., Thach, P.T. and Tuy, H. (1997), Optimization on Low Rank Nonconvex Structures. Kluwer, Dordrecht.

    Google Scholar 

  • Muu, L.D. (2000), A convex-concave programming method for optimizing over the efficient set. Acta Mathematica Vietnamica 25(1), 67–85.

    Google Scholar 

  • Naccache, P.H. (1978), Connectedness of the set of nondominated outcomes in multicriteria optimization. Journal of Optimization Theory and Applications 25, 459–467.

    Google Scholar 

  • Philip, J. (1972), Algorithms for the vector maximization problem. Mathematical Programming 2, 207–229.

    Google Scholar 

  • Phong, T.Q. and Tuyen, J.Q. (2000), Bisection search algorithm for optimizing over the efficient set. to appear in Vietman Journal of Mathematics.

  • Reeves, G.R. and Reid, C. (1988), Minimum values over the efficient set in multiple objective decision making. European Journal of Operational Research 36, 334–338.

    Google Scholar 

  • Rockafellar, R.T. and Wets, R.J-B. (1998), Variational Analysis. Springer, Berlin.

    Google Scholar 

  • Sayin, S. (1996), An algorithm based on facial decomposition for finding the efficient set in multiple objective linear programming. Operations Research Letters 19, 87–92.

    Google Scholar 

  • Sayin, S. (2000), Optimizing over the efficient set using a top-down search of faces. Operations Research 48, 65–72.

    Google Scholar 

  • Sawaragi, Y., Nakayama, H. and Tanino, T. (1985), Theory ofMultiobjective Optimization. Academic Press, Orland.

    Google Scholar 

  • Shi, J.M. and Yamamoto, Y. (1997), A global optimization method for minimum maximal flow problem. Acta Mathematica Vietnamica 22, 271–287.

    Google Scholar 

  • Steuer, R.E. (1985), Multiple Criteria Optimization: Theory, Computation and Application. Wiley, New York.

    Google Scholar 

  • Steuer, R.E. (1995), Manual for the ADBASE multiple objective linear programming package. Department of Management Science and Information Technology, University of Georgia.

    Google Scholar 

  • Stoer, J. and Witzgall, C. (1970), Convexity and Optimization in Finite Dimensions I. Springer-Verlag, Berlin.

    Google Scholar 

  • Thach, P.T. (1991), Quasiconjugates of functions, duality relationship between quasiconvex minimization under a reverse convex constraints and quasiconvex maximization under a cnvex constraint, and applications. Journal of Mathematical Analysis and Applications 159, 299–322.

    Google Scholar 

  • Thach, P.T., (1993), Global optimality criterion and a duality with a zero gap in nonconvex optimization. SIAM Journal on Mathematical Analysis 24, 1537–1556.

    Google Scholar 

  • Thach, P.T. (1994), A nonconvex duality with zero gap and applications. SIAM Journal on Optimization 4, 44–64.

    Google Scholar 

  • Thach, P.T., Konno, H. and Yokota, D. (1996), Dual approach to nminimization on the set of paretooptimal solutions. Journal of Optimization Theory and Applications 88, 689–707.

    Google Scholar 

  • Thieu, T.V., Tam, B.T. and Ban, V.T. (1983), An outer-approximation method for globally minimizing a concave function over a compact convex set. Acta Mathematica Vietnamica 8, 21–40.

    Google Scholar 

  • Thoai, N.V. (2000), A class of optimization problems over the efficient set of a multiple criteria nonlinear programming problem. European Journal of Operational Research 122, 58–68.

    Google Scholar 

  • Thoai, N.V. (2001), Conical algorithm in global optimization for optimizing over efficient sets Journal of Global Optimization, 18, 321–336.

    Google Scholar 

  • Tuy, H. (1998), Convex Analysis and Global Optimization. Kluwer, Dordrecht.

    Google Scholar 

  • White, D.J. (1982), Optimality and Efficiency. John Wiley & Sons, Chichester.

    Google Scholar 

  • White, D.J. (1996), The maximization of a function over the efficient set via a penalty function approach, European Journal of Operational Research 94, 143–153.

    Google Scholar 

  • Yamada, S., Tanino, T. and Inuiguchi, M. (2000), An inner approximation method for optimization over the weakly efficient set. Journal of Global Optimization 16, 197–217.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, 305-8573, Japan

    Yoshitsugu Yamamoto

Authors
  1. Yoshitsugu Yamamoto
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yamamoto, Y. Optimization over the efficient set: overview. Journal of Global Optimization 22, 285–317 (2002). https://doi.org/10.1023/A:1013875600711

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013875600711

Search

Navigation