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DC Programming: Overview

Authors:

N. V. ThoaiAuthors Info & Claims

Journal of Optimization Theory and Applications, Volume 103, Issue 1

Pages 1 - 43

Published: 01 October 1999 Publication History

Abstract

Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

References

[1]

Tikhonov, A. N., On a Reciprocity Principle, Soviet Mathematics Doklady, Vol. 22, pp. 100---103, 1980.

[2]

Horst, R., Pardalos, P. M., and Thoai, N. V., Introduction to Global Optimization, Kluwer Academic Publishers, Dordrecht, Holland, 1995.

Digital Library

[3]

Hartman, P., On Functions Representable as a Difference of Convex Functions, Pacific Journal of Mathematics, Vol. 9, pp. 707---713, 1959.

[4]

Tuy, H., DC Optimization: Theory, Methods, and Algorithms, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 149---216, 1995.

[5]

Luo, Z. Q., Pang, J. S., and Ralph, D., Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, England, 1996.

[6]

Fukushima, M., Luo, Z. Q., and Pang, J. S., A Globally Convergent Sequential Quadratic Programming Algorithm for Mathematical Programs with Linear Complementarity Constraints, Computational Optimization and Applications, Vol. 10, pp. 5---34, 1998.

Digital Library

[7]

Vidigal, L., and Director, S., A Design Centering Algorithm for Nonconvex Regions of Acceptability, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, Vol. 1, pp. 13---24, 1982.

Digital Library

[8]

Strodiot, J. J., Nguyen, V. H., and Thoai, N. V., On an Optimum Shape Design Problem, Report 85/5, Department of Mathematics, University of Namur, Namur, Belgium, 1985.

[9]

Thach, P. T., The Design Centering Program as a DC Programming Problem, Mathematical Programming, Vol. 41, pp. 229---248, 1988.

Digital Library

[10]

Nguyen, V. H., and Strodiot, J. J., Computing a Global Optimal Solution to a Design Centering Problem, Mathematical Programming, Vol. 53, pp. 111---123, 1992.

Digital Library

[11]

Idrissi, H., Loridan, P., and Michelot, C., Approximation of Solutions for Location Problems, Journal of Optimization Theory and Applications, Vol. 56, pp. 127---143, 1988.

Digital Library

[12]

Chen, P. C., Hansen, P., Jaumard, B., and Tuy, H., Weber's Problem with Attraction and Repulsion, Journal of Regional Science, Vol. 32, pp. 467---486, 1992.

[13]

Chen, P. C., Hansen, P., Jaumard, B., and Tuy, H., Solution of the Multifacility Weber and Conditional Weber Problems by DC Programming, Cahier du GERAD G-92-35, Ecole Polytechnique, Montréal, Québec, Canada, 1992.

[14]

Goldberg, M., The Packing of Equal Circles in a Square, Mathematical Programming, Vol. 43, pp. 24---30, 1970.

[15]

Peikert, R., WÜrtz, D., Monagan, M., and De Groot, C., Packing Circles in a Square: A Review and New Results, Proceedings of the 15th IFIP Conference, Zürich, Switzerland, pp. 45---54, 1991.

[16]

Maranas, C. D., Floudas, C. A., and Pardalos, P. M., New Results in the Packing of Equal Circles in a Square, Discrete Mathematics, Vol. 142, pp. 287---293, 1995.

Digital Library

[17]

Philip, J., Algorithms for the Vector Maximization Problem, Mathematical Programming, Vol. 2, pp. 207---229, 1972.

[18]

Dauer, J. P., and Fosnaugh, T. A., Optimization over the Efficient Set, Journal of Global Optimization, Vol. 7, pp. 261---277, 1995.

[19]

FÜlÖp, J., A Cutting Plane Algorithm for Linear Optimization over the Efficient Set, Generalized Convexity, Edited by S. Komlösi, T. Rapcsàk, and S. Schaible, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, Vol. 405, pp. 374---385, 1994.

[20]

Le-Thi, H. A., Pham, D. T., and Muu, L. D., Numerical Solution for Optimization over the Efficient Set by DC Optimization Algorithms, Operations Research Letters, Vol. 19, pp. 117---128, 1996.

Digital Library

[21]

Muu, L. D., and Luc, L. T., On Equivalence between Convex Maximization and Optimization over the Efficient Set, Vietnam Journal of Mathematics, Vol. 24, pp. 439---444, 1996.

[22]

Benson, H. P., and Lee, D., Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem, Journal of Optimization Theory and Applications, Vol. 88, pp. 77---105, 1996.

Digital Library

[23]

Horst, R., and Thoai, N. V., Maximizing a Concave Function over the Efficient or Weakly-Efficient Set, European Journal of Operational Research, Vol. 27, pp. 239---252, 1999.

[24]

Rockafellar, R. T., Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.

[25]

Thoai, N. V., On Tikhonov's Reciprocity Principle and Optimality Conditions in DC Optimization, Journal of Mathematical Analysis and Applications, Vol. 25, pp. 673---678, 1998.

[26]

Hiriart-Urruty, J. B., Generalized Differentiability, Duality, and Optimization for Problems Dealing with Differences of Convex Functions, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 256, pp. 37---69, 1985.

[27]

Hiriart-Urruty, J. B., From Convex Optimization to Nonconvex Optimization, Part 1: Necessary and Sufficient Conditions for Global Optimality, Nonsmooth Optimization and Related Topics, Edited by F. Clarke, Plenum Press, New York, New York, 1989.

[28]

Hiriart-Urruty, J. B., Conditions for Global Optimality, Handbook of Global Optimization, Edited by R. Horst and P. Pardalos, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1---26, 1995.

[29]

DÜr, M., Horst, R., and Locatelli, M., Necessary and Sufficient Global Optimality Conditions for Convex Maximization Revisited, Journal of Mathematical Analysis and Applications, Vol. 27, pp. 637---649, 1998.

[30]

Singer, I., A Fenchel-Rockafellar Type Duality Theorem for Maximization, Bulletin of the Australian Mathematical Society, Vol. 20, pp. 193---198, 1979.

[31]

Singer, I., Minimization of Continuous Convex Functionals on Complements of Convex Sets, Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 11, pp. 221---234, 1980.

[32]

Singer, I., A General Theory of Dual Optimization Problems, Journal of Mathematical Analysis and Applications, Vol. 116, pp. 77---130, 1986.

[33]

Singer, I., Some Further Duality Theorem for Optimization Problems with Reverse Convex Constraint Sets, Journal of Mathematical Analysis and Applications, Vol. 171, pp. 205---219, 1992.

[34]

Toland, J. F., Duality in Nonconvex Optimization, Journal of Mathematical Analysis and Applications, Vol. 66, pp. 399---415, 1978.

[35]

Strekalovski, A. C., On Problems of Global Extremum in Nonconvex Extremal Problems, Izvestya Vuzov, Series Matematika, Vol. 8, pp. 74---80, 1990 (in Russian).

[36]

Strekalovski, A. C., On Conditions for Global Extremum in a Nonconvex Minimization Problem, Izvestya Vuzov, Series Matematika, Vol. 2, pp. 94---96, 1991 (in Russian).

[37]

Ellaia, R., and Hiriart-Urruty, J. B., The Conjugate of the Difference of Convex Functions, Journal of Optimization Theory and Applications, Vol. 49, pp. 493---498, 1986.

Digital Library

[38]

Pshenichnyi, B. N., Leçons sur les Jeux Différentiels, Contrôle Optimal et Jeux Différentiels, Cahiers de l'IRIA, No. 4, 1971.

[39]

Volle, M., Concave Duality Applications to Problems Dealing with Difference of Functions, Mathematical Programming, Vol. 41, pp. 261---278, 1988.

[40]

Thuong, N. V., and Tuy, H., A Finite Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint, Nondifferentiable Optimization, Edited by V. Demyanov and H. Pallaschke, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 225, pp. 291---302, 1984.

[41]

Horst, R., Deterministic Global Optimization with Partition Sets Whose Feasibility Is Not Known: Application to Concave Minimization, Reverse Convex Constraints, DC Programming, and Lipschitz Optimization, Journal of Optimization Theory and Applications, Vol. 58, pp. 11---37, 1988.

Digital Library

[42]

Hillestad, R. J., Optimization Problems Subject to a Budget Constraint with Economies of Scale, Operations Research, Vol. 23, pp. 707---713, 1975.

[43]

Hillestad, R. J., and Jacobsen, S. E., Reverse Convex Programming, Applied Mathematics and Optimization, Vol. 6, pp. 63---78, 1980.

[44]

Hillestad, R. J., and Jacobsen, S. E., Linear Programs with an Additional Reverse Convex Constraint, Applied Mathematics and Optimization, Vol. 6, pp. 257---269, 1980.

[45]

Muu, L. D., A Convergent Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint, Kibernetika, Vol. 21, pp. 428---435, 1985.

[46]

Tuy, H., Convex Programs with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications, Vol. 52, pp. 463---486, 1987.

[47]

Horst, R., Phong, T. Q., and Thoai, N. V., On Solving General Reverse Convex Programming Problems by a Sequence of Linear Programs and Line Searches, Annals of Operations Research, Vol. 25, pp. 1---18, 1990.

Digital Library

[48]

Horst, R., Phong, T. Q., Thoai, N. V., and De Vries, J., On Solving a DC Programming Problem by a Sequence of Linear Programs, Journal of Global Optimization, Vol. 1, p. 183---203, 1991.

[49]

Horst, R., and Thoai, N. V., Constraint Decomposition Algorithms in Global Optimization, Journal of Global Optimization, Vol. 5, pp. 333---348, 1994.

[50]

FÜlÖp, J., A Finite Cutting Plane Method for Solving Linear Programs with an Additional Reverse Convex Constraint, European Journal of Operations Research, Vol. 44, pp. 395---409, 1990.

[51]

FÜlÖp, J., An Outer Approximation Method for Solving Canonical DC Problems, Operations Research 91, Edited by P. Gritzmann, R. Horst, and E. Sachs, Physica Verlag, Heidelberg, Germany, pp. 95---98, 1992.

[52]

Thoai, N. V., On Canonical DC Programs and Applications, Essays on Nonlinear Analysis and Optimization Problems, Institute of Mathematics, Hanoi, Vietnam, pp. 88---100, 1987.

[53]

Thoai, N. V., and Tuy, H., Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556---566, 1980.

[54]

Horst, R., and Tuy, H., Global Optimization (Deterministic Approaches), 3rd Edition, Springer Verlag, Berlin, Germany, 1996.

[55]

Horst, R., Thoai, N. V., and Benson, H. P., Concave Minimization via Conical Partitions and Polyhedral Outer Approximation, Mathematical Programming, Vol. 50, pp. 259---274, 1991.

Digital Library

[56]

Thoai, N. V., Employment of Conical Algorithm and Outer Approximation Method in DC Programming, Vietnam Journal of Mathematics, Vol. 22, pp. 71---85, 1994.

[57]

Tuy, H., Concave Programming under Linear Constraints, Soviet Mathematics, Vol. 5, pp. 1437---1440, 1964.

[58]

Horst, R., Thoai, N. V., and Tuy, H., Outer Approximation by Polyhedral Convex Sets, Operations Research Spektrum, Vol. 9, pp. 153---159, 1987.

[59]

Horst, R., Thoai, N. V., and De Vries, J., On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Operations Research Letters, Vol. 7, pp. 85---90, 1988.

[60]

Chen, P. C., Hansen, P., and Jaumard, B., On-Line and Off-Line Vertex Enumeration by Adjacency Lists, Operations Research Letters, Vol. 10, pp. 403---409, 1991.

Digital Library

[61]

Falk, J. E., and Hoffman, K. L., A Successive Underestimation Method for Concave Minimization Problems, Mathematics of Operations Research, Vol. 1, pp. 251---259, 1976.

Digital Library

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DC Programming: Overview

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Published In

cover image Journal of Optimization Theory and Applications

Journal of Optimization Theory and Applications Volume 103, Issue 1

October 1999

250 pages

ISSN:0022-3239

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Copyright © Copyright © 1999 Plenum Publishing Corporation.

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Plenum Press

United States

Publication History

Published: 01 October 1999

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https://dl.acm.org/doi/10.5555/3702676.3702870
Powell WLyu HSalakhutdinov RKolter ZHeller KWeller AOliver NScarlett JBerkenkamp F(2024)Stochastic optimization with arbitrary recurrent data samplingProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693735(41000-41038)Online publication date: 21-Jul-2024
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Abbaszadehpeivasti Hde Klerk EZamani M(2024)On the Rate of Convergence of the Difference-of-Convex Algorithm (DCA)Journal of Optimization Theory and Applications10.1007/s10957-023-02199-z202:1(475-496)Online publication date: 1-Jul-2024
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