# On solving a D.C. programming problem by a sequence of linear programs

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## Abstract

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ℝ^{ n }. This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.

## Key words

Nonlinear programming global optimization d.c. programming branch and bound outer approximation prismatic partition## Preview

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## References

- Benson, H. P. (1991), Separable Concave Minimization via Partial Outer Approximation and Branch and Bound, forthcoming in
*Operations Research Letters*.Google Scholar - Benson, H. P. and Horst, R. (1991), A Branch and Bound-Outer Approximation Algorithm for Concave Minimization over a Convex Set, forthcoming in
*J. of Computers and Mathematics with Applications*.Google Scholar - Falk, J. E. and Hoffman, K. R. (1976), A Successive Underestimation Method for Concave Minimization Problems,
*Mathematics of Operations Research***1**, 271–280.Google Scholar - Giannessi, F., Jurina, L., and Maier, G. (1979), Optimal Excavation Profile for a Pipeline Freely Resting on the Sea Floor,
*Engineering Structures***1**, 81–91.Google Scholar - Heron, B. and Sermange, M. (1982), Nonconvex Methods for Computing Free Boundary Equilibria of Axially Symmetric Plasmas,
*Applied Mathematics and Optimization***8**, 351–382.Google Scholar - Hiriart-Urruty, J. B. (1985), Generalize Differentiability, Duality and Optimization for Problems Dealing with Differences of Convex Functions,
*Lecture Notes in Economics and Mathematical Systems***256**, Springer-Verlag, 36–69.Google Scholar - Hoffman, K. L. (1981), A Method for Globally Minmizing Concave Functions over Convex Sets,
*Mathematical Programming***20**, 22–32.Google Scholar - Horst, R. (1976), An Algorithm for Nonconvex Programming Problems,
*Mathematical Programming***10**, 312–321.Google Scholar - Horst, R. (1980), A Note on the Convergence of an Algorithm for Nonconvex Programming Problems,
*Mathematical Programming***19**, 237–238.Google Scholar - Horst, R. (1986), A General Class of Branch and Bound Methods in Global Optimization with Some New Approaches for Concave Minimization,
*J. of Optimization Theory and Applications***51**, 271–291.Google Scholar - Horst, R. (1988), Deterministic Global Optimization with Partition Sets Whose Feasibility is Not Known. Application to Concave Minimization, Reverse Convex Constraints, D.C.-Programming and Lipschitzian Optimization,
*J. of Optimization Theory and Application***58**, 11–37.Google Scholar - Horst, R. (1989), On Consistency of Bounding Operations in Deterministic Global Optimization,
*J. of Optimization Theory and Applications***61**, 143–146.Google Scholar - Horst, R. (1990), Deterministic Global Optimization: Recent Advances and New Fields of Application,
*Naval Research Logistics***37**, 433–471.Google Scholar - Horst, R., Phong, T. Q. and Thoai, N. V. (1990), On Solving General Reverse Convex Programming Problems by a Sequence of Linear Programs and Line Searches,
*Annals of Operations Research***25**, 1–18.Google Scholar - Horst, R. and Thoai, N. V. (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems,
*Computing***42**, 271–289.Google Scholar - Horst, R., Thoai, N. V., and Benson, H. P. (1991), Concave Minimization via Conical Partitions and Polyhedral Outer Approximation, forthcoming in
*Mathematical Programming*.Google Scholar - Horst, R., Thoai, N. V., and Tuy, H. (1987), Outer Approximation by Polyhedral Convex Sets,
*Operations Research Spektrum***9**(3), 153–159.Google Scholar - Horst, R., Thoai, N. V., and Tuy, H. (1989), On an Outer Approximation Concept in Global Optimization,
*Optimization***20**, 255–264.Google Scholar - Horst, R., Thoai, N.V., and deVries, J. (1987), 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 Tuy, H. (1987), On the Convergence of Global Methods in Multiextremal Optimization,
*J. of Optimization Theory and Applications***54**, 253–271.Google Scholar - Horst, R. and Tuy, H. (1990),
*Global Optimization: Deterministic Approaches*, Springer, Berlin.Google Scholar - Nguyen, V. H. and Strodiot, J. J. (1988), Computing a Global Solution to a Design Centering Problem,
*Technical Report***88/12**, Facultés Universitaires de Namur.Google Scholar - Pardalos, P. M., Glick, J. H., and Rosen, J. B. (1987), Global Minimization of Indefinite Quadratic Problems,
*Computing***39**, 281–291.Google Scholar - Pardalos, P. M. and Rosen, J. B. (1987), Constrained Global Optimization: Algorithms and Applications,
*Lecture Notes on Computer Science***268**, Springer XIV.Google Scholar - Polak, E. (1987), On the Mathematical Foundations of Nondifferentiable Optimization in Engineering Design,
*SIAM Review***29**, 21–89.Google Scholar - Polak, E. and Vincentelly, A. S. (1979), Theoretical and Computational Aspects of the Optimal Design Centering Tolerancing and Tuning Problem,
*IEEE Transactions Circuits and Systems***CAS**-**26**, 795–813.Google Scholar - Rockafellar, R. T. (1970),
*Convex Analysis*, Princeton University Press.Google Scholar - Thach, P. T. (1988), The Design Centering Problem as a D.C. Program,
*Mathematical Programming***41**, 229–248.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. (1988), A Modified Version of Tuy's Method for Solving D.C. Programming Problems,
*Optimization***19**, 665–674.Google Scholar - Toland, J. F. (1979), A Duality Principle for Nonconvex Optimization and the Calculus of Variations,
*Archive of Radional Mechanics and Analysis***71**, 41–61.Google Scholar - Tuy, H. (1983), On Outer Approximation Methods for Solving Concave Minimization Problems,
*Acta Mathematica Vietnamica***8**, 3–34.Google Scholar - Tuy, H. (1986), A General Deterministic Approach to Global Optimization via D.C. Programming, in Hiriart-Urruty, J. B. (ed.),
*Fermat Days*1985:*Mathematics for Optimization*, North Holland, Amsterdam, 137–162.Google Scholar - Tuy, H. (1987), Convex Programs with an Additional Reverse Convex Constraint,
*J. of Optimization Theory and Applications***52**, 463–485.Google Scholar - Tuy, H. (1987a), Global Minimization of a Difference of Two Convex Functions,
*Mathematical Programming Study***30**, 150–182.Google Scholar - Tuy (1991), Effect of the Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms,
*J. of Global Optimization***1**, 23–36.Google Scholar - Tuy, H. and Horst, R. (1988), Convergence and Restart in Branch and Bound Algorithms for Global Optimization. Application to Concave Minimization and DC-Optimization Problems,
*Mathematical Programming***41**, 161–183.Google Scholar - Tuy, H., Thieu, T. V., and Thai, N. Q. (1985), A Conical Algorithm for Globally Minimizing a Concave Function over a Closed Convex Set,
*Mathematics of Operations Research***10**, 498–514.Google Scholar - Tuy, H. and Thuong, N. V. (1988), On the Global Minimization of a Convex Function under General Nonconvex Constraints,
*Applied Mathematics and Optimization***18**, 119–142.Google Scholar - Vidigal, L. M. and Director, S. W. (1982), A Design Centering Algorithm for Nonconvex Regions of Acceptability,
*IEEE Transactions on Computer-Aided-Design of Integrated Circuits and Systems***CAD-I**, 13–24.Google Scholar

## Copyright information

© Kluwer Academic Publishers 1991