Abstract
We consider a class of important problems in global optimization, namely concave minimization over a bounded polyhedral convex set with an additional reverse convex constraint. Using a related exact penalty property, we reformulate the class of mixed zero-one concave minimization programs, concave minimization programs over efficient sets, bilevel programs, and concave minimization programs with mixed linear complementarity constraints in the form of equivalent d.c. (difference of convex functions) programs. Solution methods based on d.c. optimization algorithms (DCA) and the combined DCA – branch and bound are investigated.
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References
Le An, T.H.: Contribution à l’optimisation non convexe et l’optimisation globale: Théorie, Algorithmes et Applications, Habilitation à Diriger des Recherches, Université de Rouen (Juin 1997)
Le An, T.H., Tao, P.D.: Solving a class of linearly constrained indefinite quadratic problems by D.c. algorithms. Journal of Global Optimization 11, 253–285 (1997)
Le An, T.H., Tao, P.D.: A continuous approach for large-scale linearly constrained quadratic zero-one programming. Optimization 50(1-2), 93–120 (2001)
Le An, T.H., Tao, P.D., Le Muu, D.: Numerical solution for Optimization over the efficient set by D.c. Optimization Algorithm. Operations Research Letters 19, 117–128 (1996)
Le An, T.H., Tao, P.D., Le Muu, D.: Exact penalty in d.c. programming. Vietnam Journal of Mathematics 27(2), 169–178 (1999)
Le An, T.H., Tao, P.D., Le Muu, D.: Simplicially constrained d.c. Optimization for optimizing over the Efficient and weakly efficient sets. Journal of Optimization Theory and Applications 117(3), 503–531 (2003)
Benson, H.P.: Optimization over the Efficient Set. Journal of Mathematical Analysis and Applications 98, 562–580 (1984)
Bolitineanu, S.: Minimization of a Quasi-concave Function over an Efficient Set. Mathematical Programming 61, 89–110 (1993)
Hiriat Urruty, J.B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Heidelberg (1993)
Horst, R., Phong, T.Q., Thoai, N.V., Vries, J.: On Solving a D.C. Programming Problem by a Sequence of Linear Programs. Journal of Global Optimization 1, 183–203 (1991)
Horst, R., Tuy, H.: Global optimization (Deterministic approaches). Springer, Berlin (1993)
Loridan, P., Morgan, J.: Approximation solutions for two-level optimization problems. Trends in Mathematical Optimization, International Series of Numer. Math., vol. 84, pp. 181–196. Birkhäuser, Basel (1988)
Tao, P.D.: Algorithms for solving a class of non convex optimization problems. Methods of subgradients. In: Hiriart Urruty, J.B. (ed.) Fermat days 85. Mathematics for Optimization, Elsevier Science Publishers B.V., North-Holland (1986)
Tao, P.D., Bernoussi, S.E.: Duality in d.c (difference of convex functions) optimization. Subgradient methods. Trends in Mathematical Optimization, International Series of Numer Math., vol. 84, pp. 277–293. Birkhauser, Basel (1988)
Tao, P.D., Le An, T.H.: D.c. optimization algorithms for solving the trust region subproblem. SIAM J. Optimization 8(2), 476–505 (1998)
Tao, P.D., Le An, T.H.: Convex analysis approach to d.c. programming. Theory, algorithms and applications (Dedicated to Professor Hoang Tuy on the occasion of his 70th birth day). Acta Mathematica Vietnamica 22(1), 289–355 (1997)
Philip, J.: Algorithms for the vector maximization problem. Mathematical Programming 2, 207–229 (1972)
Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)
Shimizu, K., Aiyoshi, E.: A new computation method for Stackelberg and minmax problem by use of a penalty method. IEEE Transactions on Automatic Control AC-26(2), 460–466 (1981)
Simaan, M., Cruz, J.: On the Stackelberg strategy in nonzero-sum games. Journal of Optimization Theory and Applications 11(5), 533–555 (1973)
Steuer, R.E.: Multiple Criteria Optimization: Theory, Computation and Application. John Willey and Sons, New York (1986)
Vavasis, S.A.: Nonlinear Optimization, Complexity Issues. Oxford University Press, Oxford (1991)
Thach, P.T.: D.c. sets, d.c. functions and nonlinear equations. Mathematical Programming 58, 415–428 (1993)
Tuy, H.: A general deterministic approach to global optimization via d.c. programming. In: Hiriart Urruty, J.B. (ed.) Fermat days 85, Mathematics for Optimization, pp. 273–303. Elsevier Science Publishers B.V., North-Holland (1986)
Tuy, H.: Global minimization of difference of two convex functions. Mathematical Programming Study 30, 159–182 (1987)
Tuy, H.: D.c. programming. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht (1995)
Tuy, H.: Convex Analysis and Global Optimization. Kluwer Academic Publishers, Boston (1998)
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An, L.T.H. (2003). D.C. Programming for Solving a Class of Global Optimization Problems via Reformulation by Exact Penalty. In: Bliek, C., Jermann, C., Neumaier, A. (eds) Global Optimization and Constraint Satisfaction. COCOS 2002. Lecture Notes in Computer Science, vol 2861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39901-8_7
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DOI: https://doi.org/10.1007/978-3-540-39901-8_7
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