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Convex programs with an additional reverse convex constraint

Contributed Papers

Abstract

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)≥0, whereD is a closed convex subset ofR n andf,g are convex finite functionsR n . Under suitable stability hypotheses, it is shown that a feasible point\(\bar x\) is optimal if and only if 0=max{g(x):xD,f(x)≤f(\(\bar x\))}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.

Key Words

Reverse convex constraints convex maximization concave minimization outer approximation methods 

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References

  1. 1.
    Rosen, J. B.,Iterative Solution of Nonlinear Optimal Control Problems, SIAM Journal on Control, Vol. 4, pp. 223–244, 1966.Google Scholar
  2. 2.
    Avriel, M., andWilliams, A. C.,Complementary Geometric Programming, SIAM Journal on Applied Mathematics, Vol. 19, pp. 125–141, 1970.Google Scholar
  3. 3.
    Meyer, R.,The Validity of a Family of Optimization Methods, SIAM Journal on Control, Vol. 8, pp. 41–54, 1970.Google Scholar
  4. 4.
    Ueing, U.,A Combinatorial Method to Compute a Global Solution of Certain Nonconvex Optimization Problems, Numerical Methods for Nonlinear Optimization, Edited by F. A. Lootsma, Academic Press, New York, New York, pp. 223–230, 1972.Google Scholar
  5. 5.
    Bansal, P. P., andJacobsen, S. E.,Characterization of Local Solutions for a Class of Nonconvex Programs, Journal of Optimization Theory and Applications, Vol. 15, pp. 549–564, 1975.Google Scholar
  6. 6.
    Hillestad, R. J., andJacobsen, S. E.,Reverse Convex Programming, Applied Mathematics and Optimization, Vol. 6, pp. 63–78, 1980.Google Scholar
  7. 7.
    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
  8. 8.
    Tuy, H.,Global Minimization of a Concave Function Subject to Mixed Linear and Reverse Convex Constraints, Proceedings of the IFIP Working Conference on Recent Advances in System Modeling and Optimization, Hanoi, Vietnam, 1983.Google Scholar
  9. 9.
    Thuong, Ng. V., andTuy, H. A Finite Algorithm For Solving Linear Programs with an Additional Reverse Convex Constraint, Proceedings of the IIASA Workshop on Nondifferentiable Optimization, Sopron, Hungary, 1984.Google Scholar
  10. 10.
    Zaleesky, A. B.,Nonconvexity of Feasible Domains and Optimization of Management Decisions (in Russian), Ekonomika i Matematitcheskie Metody, Vol. 16, pp. 1069–1081, 1980.Google Scholar
  11. 11.
    Singer, I.,Minimization of Continuous Convex Functionals on Complements of Convex Sets, Mathematische Operationsforschung und Statistik, Series Optimization, Vol. 11, pp. 221–234, 1980.Google Scholar
  12. 12.
    Tuy, H., andThuong, Ng. V.,Minimizing a Convex Function over the Complement of a Convex Set, Proceedings of the 9th Symposium on Operations Research, Osnabrück, Germany, 1984.Google Scholar
  13. 13.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  14. 14.
    Polyakova, L. N.,Necessary Conditions for an Extremum of Quasidifferentiable Functions (in Russian), Vestnik Leningrad University, Mathematics, Vol. 13, pp. 241–247, 1981.Google Scholar
  15. 15.
    Toland, J. F.,On Subdifferential Calculus and Duality in Nonconvex Optimization, Analyse Nonconvexe, Bulletin de la Société Mathématique de France, Mémoire 60, pp. 177–183, 1979.Google Scholar
  16. 16.
    Hoffman, K. L.,A Method for Globally Minimizing Concave Functions over Convex Sets, Mathematical Programming, Vol. 20, pp. 22–32, 1981.Google Scholar
  17. 17.
    Horst, R.,An Algorithm for Nonconvex Programming Problems, Mathematical Programming, Vol. 10, pp. 312–321, 1976.Google Scholar
  18. 18.
    Thieu, T. V., Tam, B. T., andBan, V. T.,An Outer Approximation Method for Globally Minimizing a Concave Function over a Compact Convex Set, Proceedings of the IFIP Working Conference on Recent Advances in System Modeling and Optimization, Hanoi, Vietnam, 1983.Google Scholar
  19. 19.
    Tuy, H., Thieu, T. V. andThai, Ng. Q.,A Conical Algorithm for Globally Minimizing a Concave Function over a Closed Convex Set, Mathematics of Operations Research, Vol. 10, pp. 498–514, 1985.Google Scholar
  20. 20.
    Tuy, H.,On Outer Approximation Methods for Solving Concave Minimization Problems, Acta Mathematica Vietnamica, Vol. 8, pp. 3–34, 1983.Google Scholar
  21. 21.
    Tuy, H.,Concave Programming under Linear Constraints, Soviet Mathematics, Vol. 5, pp. 1437–1440, 1964.Google Scholar
  22. 22.
    Tuy, H., andThai, Ng. Q.,Minimizing a Concave Function over a Compact Convex Set, Acta Mathematica Vietnamica, Vol. 8, pp. 13–20, 1983.Google Scholar
  23. 23.
    Tuy, H. Global Minimization of a Difference of Two Convex Functions, Selected Topics in Operations Research and Mathematical Economics, Edited by H. Pallaschlke, Springer-Verlag, Berlin, Germany, pp. 98–118, 1984.Google Scholar
  24. 24.
    Falk, J. E., andHoffman, K. R.,A Successive Underestimation Method for Concave Minimization Problems, Mathematics of Operations Research, Vol. 1, pp. 251–259, 1976.Google Scholar

Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • H. Tuy
    • 1
  1. 1.Institute of MathematicsHanoiVietnam

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