# 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): minimize*f*(*x*), subject to*x*∈*D*,*g*(*x*)≥0, where*D* is a closed convex subset of*R*^{ n } and*f*,*g* are convex finite functions*R*^{ n }. Under suitable stability hypotheses, it is shown that a feasible point\(\bar x\) is optimal if and only if 0=max{*g*(*x*):*x*∈*D*,*f*(*x*)≤*f*(\(\bar x\))}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblems*Q*_{ k },*k*=1, 2, ..., each of which consists in maximizing the convex function*g*(*x*) over some polyhedron*S*_{ 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## Preview

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## References

- 1.Rosen, J. B.,
*Iterative Solution of Nonlinear Optimal Control Problems*, SIAM Journal on Control, Vol. 4, pp. 223–244, 1966.Google Scholar - 2.Avriel, M., andWilliams, A. C.,
*Complementary Geometric Programming*, SIAM Journal on Applied Mathematics, Vol. 19, pp. 125–141, 1970.Google Scholar - 3.Meyer, R.,
*The Validity of a Family of Optimization Methods*, SIAM Journal on Control, Vol. 8, pp. 41–54, 1970.Google Scholar - 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.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.Hillestad, R. J., andJacobsen, S. E.,
*Reverse Convex Programming*, Applied Mathematics and Optimization, Vol. 6, pp. 63–78, 1980.Google Scholar - 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.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.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.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.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.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.Rockafellar, R. T.,
*Convex Analysis*, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar - 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.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.Hoffman, K. L.,
*A Method for Globally Minimizing Concave Functions over Convex Sets*, Mathematical Programming, Vol. 20, pp. 22–32, 1981.Google Scholar - 17.Horst, R.,
*An Algorithm for Nonconvex Programming Problems*, Mathematical Programming, Vol. 10, pp. 312–321, 1976.Google Scholar - 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.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.Tuy, H.,
*On Outer Approximation Methods for Solving Concave Minimization Problems*, Acta Mathematica Vietnamica, Vol. 8, pp. 3–34, 1983.Google Scholar - 21.Tuy, H.,
*Concave Programming under Linear Constraints*, Soviet Mathematics, Vol. 5, pp. 1437–1440, 1964.Google Scholar - 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.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.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

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© Plenum Publishing Corporation 1987