Numerical solution for optimization over the efficient set by d.c. optimization algorithms. (English) Zbl 0871.90074

Summary: We outline a d.c. optimization scheme and use it for (locally) maximizing a concave, a convex or a quadratic function \(f\) over the efficient set of a multiple objective convex program. We also propose a decomposition method for globally solving problem with \(f\) concave. Numerical experiences are discussed.


90C29 Multi-objective and goal programming
Full Text: DOI


[1] An, L. T.H., Analyse numérique des algorithmes de l’Optimisation d.c. Approches locales et globales. Code et simulations numériques en grande dimension. Applications, Thèse de Doctorat de l’Université de Rouen (1994), France
[2] Benson, H. P., Optimization over the efficient set, J. Math. Anal. Appl., 98, 562-580 (1984) · Zbl 0534.90077
[3] Benson, H. P., An all-linear programming relaxation algorithm for optimizing over the efficient set, J. Global Optim., 1, 83-104 (1991) · Zbl 0739.90056
[4] Benson, H. P., A finite, nonadjacent extreme-point search algorithm for optimization over the efficient set, J. Optim. Theory Appl., 73, 47-64 (1992) · Zbl 0794.90048
[6] Durier, R., On locally polyhedral convex functions, (Trends in Mathematical Mathematics (1988), Birkhäuser Verlag), 55-66
[7] Hiriart-Urruty, J. B., From convex optimization to nonconvex optimization. Part I: Necessary and sufficient conditions for global optimality, (Nonsmooth Optimization and Related Topics. Nonsmooth Optimization and Related Topics, Ettore Majorana International Sciences, Ser. 43 (1988), Plenum Press: Plenum Press Oxford) · Zbl 0735.90056
[8] Horst, R.; Tuy, H., Global optimization (Deterministic approaches) (1993), Springer: Springer Berlin
[9] Mahey, P.; Tao, P. D., Partial regularization of the sum of two maximal monotone operators, Math. Modell. Numer. Anal., 27, 375-395 (1993) · Zbl 0778.65042
[10] Mahey, P.; Tao, P. D., Proximal decomposition of the graph of maximal monotone operator, SIAM J. Optim., 5, 454-468 (1995) · Zbl 0834.90105
[11] Muu, Le. D., (A method for optimization of a linear function over the efficient set, Vol. 15 (1991), Institute of Mathematics: Institute of Mathematics Hanoi), preprint
[12] Tao, P. D., Contribution à la théorie de normes and ses applications à l’analyse numérique, (Thèse de Doctorat d’Etat Es Science (1981), Université Joseph Fourier: Université Joseph Fourier Grenoble)
[13] Tao, P. D., Convergence of subgradient method for computing the bound norm of matrices, Linear Alg. Appl., 62, 163-182 (1984) · Zbl 0563.65029
[14] Tao, P. D., Algorithmes de calcul du maximum d’une forme quadratique sur la boule unité de la norme du maximum, Numer. Math., 45, 377-440 (1985)
[15] Tao, P. D., Algorithms for solving a class of non convex optimization problems. Methods of subgradients, (Hiriart Urruty, J. B., Fermat Days 85. Mathematics for Optimization (1986), Elsevier: Elsevier North-Holland)
[16] Tao, P. D., Iterative behaviour, Fixed point of a class of monotone operators. Application to non symmetric threshold function, Discrete Math., 70, 85-105 (1988) · Zbl 0646.65050
[17] Tao, P. D.; El Bernoussi, S., Duality in d.c. (difference of convex functions) optimization. Subgradient methods, (Trends in Mathematical Optimization, International Series of Numer Math., Vol. 84 (1988), Birkhauser), 276-294
[20] Philip, J., Algorithms for the vector maximization problem, Math. Programming, 2, 207-229 (1972) · Zbl 0288.90052
[21] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0229.90020
[22] Toland, J. F., On subdifferential calculus and duality in nonconvex optimization, Bull. Soc. Math. France, Mémoire, 60, 177-183 (1979) · Zbl 0417.90088
[23] Tuy, H., Effect of subdivision strategy on convergence and efficiency of some global optimization algorithms, J. Global Optim., 1, 23-33 (1991) · Zbl 0744.90083
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.