Advertisement

Convex minimization under Lipschitz constraints

  • P. T. Thach
Contributed Papers

Abstract

We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf i (x)≤0,i=1,...,m. By transforming a system of Lipschitz constraintsf i (x)≤0,i=l,...,m, into a single constraints of the formh(x)-∥x2≤0, withh(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.

Key Words

Global optimization convex minimization Lipschitz constraints reverse convex constraints 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Tuy, H.,Concave Programming under Linear Constraints, Doklady Akademii Nauk SSSR, Vol. 159, pp. 32–35, 1964.Google Scholar
  2. 2.
    Falk, J. E., andHoffman, K. L.,A Successive Underestimation Method for Concave Minimization Problems, Mathematics of Operations Research, Vol. 1, pp. 251–259, 1976.Google Scholar
  3. 3.
    Thoai, N. V., andTuy, H.,Convergent Algorithms for Minimizing a Concave Function, Mathematics of Operations Research, Vol. 5, pp. 556–566, 1980.Google Scholar
  4. 4.
    Horst, R.,An Algorithm for Nonconvex Programming Problem, Mathematical Programming, Vol. 10, pp. 312–321, 1976.Google Scholar
  5. 5.
    Hoffman, K. L.,A Method for Globally Minimizing Concave Functions over Convex Sets, Mathematical Programming, Vol. 20, pp. 22–32, 1981.Google Scholar
  6. 6.
    Tuy, H.,On Outer Approximation Methods for Solving Concave Minimization Problems, Acta Mathematica Vietnamica, Vol. 8, pp. 3–34, 1983.Google Scholar
  7. 7.
    Thieu, T. V., Tam, B. T., andBan, V. T.,An Outer Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set, Acta Mathematica Vietnamica, Vol. 8, pp. 21–40, 1983.Google Scholar
  8. 8.
    Tuy, H., Thieu, T. V., andThai, N. Q.,A Conical Algorithm for Globally Minimizing a Concave Function over a Closed Convex Set, Mathematics of Operations Research, Vol. 10, pp. 489–514, 1985.Google Scholar
  9. 9.
    Hillestad, R. J., andJacobsen, S. E.,Reverse Convex Programming, Applied Mathematics and Optimization, Vol. 6, pp. 63–78, 1980.Google Scholar
  10. 10.
    Hillestad, R. J., andJacobsen, S. E.,Linear Programs with an Additional Reverse Convex Constraint, Applied Mathematics and Optimization, Vol. 6, pp. 257–269, 1980.Google Scholar
  11. 11.
    Tuy, H.,Convex Program with an Additional Reverse Convex Constraint, Journal of Optimization Theory and Applications, Vol. 52, pp. 463–486, 1987.Google Scholar
  12. 12.
    Tuy, H., andThuong, N. V.,Minimizing a Convex Function over the Complement of a Convex Set, Methods of Operations Research, Vol. 49, pp. 85–99, 1984.Google Scholar
  13. 13.
    Thuong, N. V., andTuy, H.,A Finite Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint, Nondifferentiable Optimization: Motivations and Applications, Edited by V. F. Demyanov and D. Pallaschke, Springer-Verlag, Berlin, Germany, pp. 291–304, 1984.Google Scholar
  14. 14.
    Muu, L. D.,A Convergent Algorithm for Solving Linear Programs with an Additional Reverse Convex Constraint, Kybernetika, Vol. 21, pp. 428–435, 1985.Google Scholar
  15. 15.
    Thach, P. T.,Convex Programs with Several Additional Reverse Convex Constraints, Acta Mathematica Vietnamica, Vol. 10, pp. 35–57, 1985.Google Scholar
  16. 16.
    Thoai, N. V.,A Modified Version of Tuy's Method for Solving DC Programming Problems, Core Discussion Paper No. 8528, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1985.Google Scholar
  17. 17.
    Tuy, H.,A General Deterministic Approach to Global Optimization via DC Programming, Mathematics Studies, Vol. 129, pp. 273–304, 1986.Google Scholar
  18. 18.
    Tuy, H.,Global Minimization of a Difference of Two Convex Functions, Mathematical Programming Study, Vol. 30, pp. 150–182, 1987.Google Scholar
  19. 19.
    Thach, P. T., andTuy, H.,Global Optimization under Lipschitzian Constraints, Japan Journal of Applied Mathematics, Vol. 4, pp. 205–217, 1987.Google Scholar
  20. 20.
    Thach, P. T.,The Design Centering Problem as a DC Program, Mathematical Programming, Vol. 41, pp. 229–248, 1988.Google Scholar
  21. 21.
    Horst, R., Vries, D., andThoai, N. V.,On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization, Operations Research Letters, Vol. 7, pp. 85–90, 1988.Google Scholar
  22. 22.
    Geoffrion, A. M.,Generalized Benders Decomposition, Journal of Optimization Theory and Applications, Vol. 10, pp. 237–260, 1972.Google Scholar
  23. 23.
    Thach, P. T.,Concave Minimization under Nonconvex Constraints with Special Structure, Essays on Nonlinear Analysis and Optimization Problems, Institute of Mathematics, Hanoi, Vietnam, pp. 121–139, 1987.Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • P. T. Thach
    • 1
  1. 1.Institute of Mathematics, Bo HoHanoiVietnam

Personalised recommendations