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A survey of recent developments in multiobjective optimization

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Abstract

Multiobjective Optimization (MO) has many applications in such fields as the Internet, finance, biomedicine, management science, game theory and engineering. However, solving MO problems is not an easy task. Searching for all Pareto optimal solutions is expensive and a time consuming process because there are usually exponentially large (or infinite) Pareto optimal solutions. Even for simple problems determining whether a point belongs to the Pareto set is \(\mathcal{NP}\) -hard. In this paper, we discuss recent developments in MO. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon Pareto optimal solution, and heuristics.

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References

  • Aghezzaf, B., & Hachimi, M. (2000). Generalized invexity and duality in multiobjective programming problems. Journal of Global Optimization, 18, 91–101.

    Article  Google Scholar 

  • Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: theory, algorithms, and applications. Jersey: New Prentice–Hall.

    Google Scholar 

  • Ansoff, H. I. (1968). Corporate strategy. Harmandsworth: Penguin.

    Google Scholar 

  • Bell, D. E., & Raiffa, H. (1988). Risky choice revisited. In D. E. Bell, H. Raiffa & A. Tversky (Eds.), Decision making: descriptive, normative and prescriptive interactions (pp. 99–112). Cambridge: Cambridge University Press.

    Google Scholar 

  • Bhaskar, K. (1979). A multiple objective approach to capital budgeting. Accounting and Business Research, 9, 25–46.

    Google Scholar 

  • Bitran, G. R. (1977). Linear multiple objective programs with zero-one variables. Mathematical Programming, 13, 121–139.

    Article  Google Scholar 

  • Bitran, G. R. (1979). Theory and algorithms for linear multiple objective programs with zero-one variables. Mathematical Programming, 17, 362–390.

    Article  Google Scholar 

  • Bitran, G. R. (1981). Duality for nonlinear multi-criteria optimization problems. Journal of Optimization Theory and Applications, 35, 367–401.

    Article  Google Scholar 

  • Bouri, A., Martel, J. M., & Chabchoub, H. (2002). A multi-criterion approach for selecting attractive portfolio. Journal of Multi-Criteria Decision Analysis, 11, 269–277.

    Article  Google Scholar 

  • Brumbaugh-Smith, J., & Shier, D. (1989). An empirical investigation of some bicriterion-shortest path algorithms. European Journal of Operational Research, 43, 216–224.

    Article  Google Scholar 

  • Camerini, P.M., Galbiati, G., & Maffioli, F. (1984). The complexity of multi-constrained spanning tree problems. In Theory of algorithms (pp. 53–101). Colloquium Pecs 1984.

  • Chalmet, L. G., Lemonidis, L., & Elzinga, D. J. (1986). An algorithm for the bi-criterion integer programming problem. European Journal of Operations Research, 25, 292–300.

    Article  Google Scholar 

  • Chankong, V., & Haimes, Y. Y. (1983). Multiobjective decision making theory and methodology. New York: Elsevier Science.

    Google Scholar 

  • Cloquette, J. F., Gerard, M., & Hadhri, M. (1995). An empirical analysis of Belgian daily returns using GARCH models. Cahiers Economiques de Bruxelles, 418, 513–535.

    Google Scholar 

  • Corley, H. W. (1985). Efficient spanning trees. Journal of Optimization Theory and Applications, 45, 481–485.

    Article  Google Scholar 

  • Craven, B. D. (1981). Duality for the generalized convex fractional programs. In S. Schiable & W. T. Ziemba (Eds.), Generalized Concavity in Optimization and Economics (pp. 473–489). New York: Academic.

    Google Scholar 

  • Das, L. N., & Nanda, S. (1997). Symmetric dual multiobjective programming. European Journal of Operational Research, 97, 167–171.

    Article  Google Scholar 

  • Dijkstra, E. W. (1959). A note on two problems in connexion with graphs. Numerical Mathematics, 1, 262–271.

    Article  Google Scholar 

  • Eben-Chaime, M. (1996). Parametric solution for linear bicriteria knapsack models. Management Science, 42, 1565–1575.

    Google Scholar 

  • Egudo, R. (1989). Efficiency and generalized convex duality for multiobjective programs. Journal of Mathematical Analysis and Applications, 138, 84–94.

    Article  Google Scholar 

  • Ehrgott, M., & Gandibleux, X. (2000). A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spektrum, 22, 425–460.

    Google Scholar 

  • Etzioni, O., Hanks, S., Jiang, T., Karp, R. M., Madari, O., & Waarts, O. (1996). Efficient information gathering on the Internet. In Proceedings of the 37th IEEE symposium on foundations of computer science (pp. 234–243).

  • Gandibleux, X., & Freville, A. (2000). Tabu search based procedure for solving the 0–1 multiobjective knapsack problem: the two objective case. Journal of Heuristics, 6, 361–383.

    Article  Google Scholar 

  • Garfinkel, R. S., & Nemhauser, G. L. (1972). Integer programming. New York: Wiley.

    Google Scholar 

  • Geoffrion, A. M. (1968). Proper efficiency and the theory of vector maximization. Journal of Mathematical Analysis and Applications, 22, 618–630.

    Article  Google Scholar 

  • Hachimi, M., & Aghezzaf, B. (2004). Sufficiency and duality in differentiable multiobjective programming involving generalized type I functions. Journal of Mathematical Analysis and Applications, 296, 382–392.

    Article  Google Scholar 

  • Haimes, Y. Y., Lasdon, L. S., & Wismer, D. A. (1971). On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Transactions on Systems, Man and Cybernetics, 1, 296–297.

    Google Scholar 

  • Hamacher, H. W., & Ruhe, G. (1994). On spanning tree problems with multipleobjectives. Annals of Operations Research, 52, 209–230.

    Article  Google Scholar 

  • Hanson, M. A. (1961). A duality theorem in nonlinear programming with nonlinear constraints. Australian Journal of Statistics, 3, 67–71.

    Article  Google Scholar 

  • Hanson, M. A. (1981). On sufficiency of the Kuhn-Tucker conditions. Journal of Mathematical Analysis and Applications, 80, 545–550.

    Article  Google Scholar 

  • Henig, M. I. (1985). The shortest path problem with two objective functions. European Journal of Operational Research, 25, 281–291.

    Article  Google Scholar 

  • Hillermeier, C. (2001). Nonlinear multiobjective optimization: a generalized homotopy approach. Boston: Birkhauser Verlag.

    Google Scholar 

  • Huarng, F., Pulat, P. S., & Shih, L. (1996). A computational comparison of some bicriterion shortest path algorithms. Journal of the Chinese Institution of Industrial Engineers, 13, 121–125.

    Google Scholar 

  • Hurson, C., & Zopounidis, C. (1995). On the use of multi-criteria decision aid methods to portfolio selection. Journal of Euro-Asian Management, 1, 69–94.

    Google Scholar 

  • Jacquillat, B. (1972). Les modèles d’évaluation et de sélection des valeurs mobilières: panorama des recherches américaines. Analyse Financière, 11, 68–88.

    Google Scholar 

  • Jahn, J. (2004). Vector optimization: theory, applications and extensions. Berlin: Springer.

    Google Scholar 

  • Jeyakumar, V. (1985). Strong and weak invexity in mathematical programming. Methods of Operations Research, 55, 109–125.

    Google Scholar 

  • Jeyakumar, V., & Mond, B. (1992). On generalized convex mathematical programming. Journal of the Australian Mathematical Society, Series B, 34, 43–53.

    Google Scholar 

  • Ibaraki, T. (1987). Enumerative approaches to combinatorial optimization, part ii. Annals of Operations Research, 11, 343–602.

    Article  Google Scholar 

  • Khan, Z., & Hanson, M. A. (1997). On ratio invexity in mathematical programming. Journal of Mathematical Analysis and Applications, 205, 330–336.

    Article  Google Scholar 

  • Khoury, N., Marte, J. M., & Veilleux, M. (1993). Methode multicritere de selection de portefeuilles indiciels interantionaux. Acualite Economique, 69, 171–190.

    Google Scholar 

  • Klamroth, K., & Wiecek, M. (2000a). Dynamic programming approaches to the multiple criteria knapsack problem. Naval Research Logistics, 47, 57–76.

    Article  Google Scholar 

  • Klamroth, K., & Wiecek, M. (2000b). Time-dependent capital budgeting with multiple criteria. In Y. Y. Haimes & R. E. Steuer (Eds.), Lecture notes in economics and mathematical systems: Vol. 487. Research and practice in multiple criteria decision making (pp. 421–432). Berlin: Springer.

    Google Scholar 

  • Klamroth, K., Tind, J., & Zust, S. (2004). Integer programming duality in multiple objective programming. Journal of Global Optimization, 29, 1–18.

    Article  Google Scholar 

  • Klein, D., & Hannan, E. (1982). An algorithm for the multiple objective integer linear programming problem. European Journal of Operations Research, 93, 378–385.

    Article  Google Scholar 

  • Kostreva, M. M., & Wiecek, M. M. (1993). Time dependency in multiple objective dynamic programming. Journal of Mathematical Analysis and Applications, 173, 289–307.

    Article  Google Scholar 

  • Kruskal, J. B. (1956). On the shortest spanning tree of a graph and the traveling salesman problem. In Proceedings of American mathematical society 7 (pp. 48–50).

  • Kuhn, H. W., & Tucker, A. W. (1951). Nonlinear programming. In J. Neyman (Ed.), Proceedings of the second Berkeley symposium on mathematical statistics and probability (pp. 481–492). Los Angeles: University of California Press.

    Google Scholar 

  • Lee, S. M., & Lerro, A. J. (1974). Capital budgeting for multiple objectives. Management Science, 36, 1106–1119.

    Google Scholar 

  • Liang, Z. A., Huang, H. X., & Pardalos, P. M. (2001). Optimality conditions and duality for a class of nonlinear fractional programming problems. Journal of Optimization Theory and Applications, 110, 611–619.

    Article  Google Scholar 

  • Liang, Z. A., Huang, H. X., & Pardalos, P. M. (2003). Efficiency conditions and duality for a class of multiobjective fractional programming problems. Journal of Global Optimization, 27, 447–471.

    Article  Google Scholar 

  • Luc, D. T. (1984). On duality theory in multiobjective programming. Journal of Optimization Theory and Applications, 43, 557–582.

    Article  Google Scholar 

  • Luc, D. T., & Schaible, S. (1997). Efficiency and generalized concavity. Journal of Optimization Theory and Applications, 94, 147–153.

    Article  Google Scholar 

  • Maeda, T. (1994). Constraint qualifications in multiobjective optimization problems: differentiable case. Journal of Optimization Theory and Applications, 80, 483–500.

    Article  Google Scholar 

  • Markowitz, H. M. (1952). Portfolio selection. Journal of Finance, 7, 77–91.

    Article  Google Scholar 

  • Martello, S., & Toth, P. (1990). Knapsack problems: algorithms and computer implementations. New York: Wiley.

    Google Scholar 

  • Martello, S., Pisinger, D., & Toth, P. (1997). New trends in exact algorithms for the 0–1 knapsack problem. In J. Barceló (Ed.), Proceedings of EURO/IMFORMS-97 (pp. 151–160), Barcelona

  • Marusciac, I. (1982). On Fritz John type optimality criterion in multiobjective optimization. L’Analyse Numérique et la Theorie de l’Approximation, 11, 109–114.

    Google Scholar 

  • Miettinen, K. M. (1999). Nonlinear multiobjective optimization. Boston: Kluwer Academic.

    Google Scholar 

  • Mond, B., & Weir, T. (1981). Generalized concavity and duality. In S. Schaible & W. T. Ziemba (Eds.), Generalized convexity in optimization and economics (pp. 263–280). New York: Academic.

    Google Scholar 

  • Nakayama, H. (1985). Duality theory in vector optimization: an overview, decision making with multiple objectives. In Y.Y. Haimes & V. Chankong (Eds.), Lecture Notes in Economics and Mathematical Systems (Vol. 337, pp. 109–125). Berlin: Springer.

  • Papadimitriou, C. H., & Yannakakis, M. (2000). On the approximability of trade-offs and optimal access of web sources, In Proceedings of the 41st annual symposium on foundations of computer science (pp. 86–92).

  • Pardalos, P. M., Siskos, Y., & Zopounidis, C. (Eds.). (1995). Advances in multicriteria analysis. Netherlands: Kluwer Academic.

    Google Scholar 

  • Pareto, V. (1964). Course d’economie politique. Genève: Libraire Drotz. The first edition in 1986.

    Google Scholar 

  • Preda, V. (1992). On efficiency and duality for multiobjective programs. Journal of Mathematical Analysis and Applications, 166, 365–377.

    Article  Google Scholar 

  • Prim, R. C. (1957). Shortest connection networks and some generations. Bell System Technical Journal, 36, 1389–1401.

    Google Scholar 

  • Ramos, R. M., Aloso, S., Sicilia, J., & González, C. (1998). The problem of the optimal biobjective spanning tree. European Journal of Operational Research, 111, 617–628.

    Article  Google Scholar 

  • Reddy, L. V., & Mukherjee, R. N. (1999). Some results on mathematical programming with generalized ratio invexity. Journal of Mathematical Analysis and Applications, 240, 299–310.

    Article  Google Scholar 

  • Rosenblatt, M. J., & Sinuany-Stern, Z. (1989). Generating the discrete efficient frontier to the capital budgeting problem. Operations Research, 37, 38–394.

    Google Scholar 

  • Ruíz-Canales, P., & Rufián-Lizana, A. (1995). A characterization of weakly efficient points. Mathematical Programming, 68, 205–212.

    Google Scholar 

  • Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of multiobjective optimization. Orlando: Academic.

    Google Scholar 

  • Schweigert, D. (1990). Linear extensions and vector-valued spanning trees. Methods of Operations Research, 60, 219–222.

    Google Scholar 

  • Serafini, P. (1986). Some considerations about computational complexity for multi objective combinatorial problems. In J. Jahn & W. Krabs (Eds.), Lecture notes in economics and mathematical systems: Vol. 294. Recent advances and historical development of vector optimization (pp. 222–231). Berlin: Springer.

    Google Scholar 

  • Singh, C., & Hanson, M. A. (1991). Multiobjective fractional programming duality theory. Naval Research Logistics, 38, 925–933.

    Google Scholar 

  • Singh, C., Bhatia, D., & Rueda, N. (1996). Duality in nonlinear multiobjective programming using augmented Lagrangian functions. Journal of Optimization Theory and Applications, 88, 659–670.

    Article  Google Scholar 

  • Skriver, A. J. V., & Andersen, K. A. (2000). A label correcting approach for solving bicriterion shortest path problems. Computers and Operations Research, 27, 507–524.

    Article  Google Scholar 

  • Sniedovich, M. (1988). A multi-objective routing problem revisited. Engineering Optimization, 13, 99–108.

    Google Scholar 

  • Spronk, J., & Hallerbach, W. G. (1997). Financial modelling: where to go? With an illustration for portfolio management. European Journal of Operational Research, 99, 113–127.

    Article  Google Scholar 

  • Steuer, R. E. (1986). Multiple criteria optimization: theory, computation and application. New York: Wiley.

    Google Scholar 

  • Steuer, R. E., & Na, P. (2003). Multiple criteria decision making combined with finance: a categorized bibliography. European Journal of Operational Research, 150, 496–515.

    Article  Google Scholar 

  • Tanina, T., & Sawaragi, Y. (1979). Duality theory in multiobjective programming. Journal of Optimization Theory and Applications, 27, 509–529.

    Article  Google Scholar 

  • Thanassoulis, E. (1985). Selecting a suitable solution method for a multiobjective programming capital budgeting problem. Journal of Business Finance and Accounting, 12, 453–471.

    Article  Google Scholar 

  • Ulungu, E. L., & Teghem, J. (1994). Application of the two phases method to solve the bi-objective knapsack problem. Technical report, Dept. of Mathematics & Operational Research, Faculté Polytechnique de Mons, Mons, Belgium, 1994.

  • Ulungu, E. L., & Teghem, J. (1997). Solving multiobjective knapsack problems by a branch and bound procedure. In J. N. Climaco (Ed.), Multicriteria analysis (pp. 269–278). New-York: Springer.

    Google Scholar 

  • Vial, J. P. (1983). Strong and weak convexity set and functions. Mathematics of Operations Research, 8, 231–259.

    Google Scholar 

  • Villarreal, B., & Karwan, M. H. (1981). Multicriteria integer programming: a hybrid dynamic programming recursive approach. Mathematical Programming, 21, 204–223.

    Article  Google Scholar 

  • Visée, M., Teghem, J., Pirlot, M., & Ulungu, E. L. (1996). Two-phases method and branch and bound procedures to solve the bi-objective knapsack problem. Technical report, Department of Mathematics & Operational Research, Faculté Polytechnique de Mons, Belgium, 1996.

  • Wang, S. (1991). Second-order necessary and sufficient conditions in multiobjective programming. Numerical functional Analysis and Applications, 12, 237–252.

    Google Scholar 

  • Warburton, A. (1987). Approximation of a Pareto optima in multiple-objective shortest-path problems. Operations Research, 35, 70–79.

    Article  Google Scholar 

  • Weingartner, H. M. (1963). Mathematical programming and the analysis of capital budgeting problems. New Jersey: Prentice–Hall.

    Google Scholar 

  • Weir, T. (1990). A note on invex functions and duality in generalized fractional programming. Research report, Department of Mathematics, The University of New South Wales, ACT 2600, Australia.

  • Weir, T., & Mond, B. (1988). Symmetric and self duality in multiobjective programming. Asia-Pacific Journal of Operational Research, 5, 124–133.

    Google Scholar 

  • Zadeh, L. (1963). Optimality and non-scalar-valued performance criteria. IEEE Transactions on Automatic Control, 8, 59–60.

    Article  Google Scholar 

  • Zhou, G., & Gen, M. (1999). Genetic algorithm approach on multi-criteria minimum spanning tree problem. European Journal of Operational Research, 114, 141–152.

    Article  Google Scholar 

  • Zopounidis, C. (1999). Multicriteria decision aid in financial management. European Journal of Operational Research, 119, 404–415.

    Article  Google Scholar 

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  1. Dept. of Industrial and Systems Engineering, University of Florida, 303 Weil Hall, Gainesville, FL, 32611, USA

    Altannar Chinchuluun & Panos M. Pardalos

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  1. Altannar Chinchuluun
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Chinchuluun, A., Pardalos, P.M. A survey of recent developments in multiobjective optimization. Ann Oper Res 154, 29–50 (2007). https://doi.org/10.1007/s10479-007-0186-0

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