Abstract
Characterizations of global optimality are given for general difference convex (DC) optimization problems involving convex inequality constraints. These results are obtained in terms of ε-subdifferentials of the objective and constraint functions and do not require any regularity condition. An extension of Farkas' lemma is obtained for inequality systems involving convex functions and is used to establish necessary and sufficient optimality conditions. As applications, optimality conditions are also given for weakly convex programming problems, convex maximization problems and for fractional programming problems.
Similar content being viewed by others
References
I.M. Bomze and G. Danninger (1993), A global optimization algorithm for concave quadratic programming problems, SIAM Journal on Optimization 3, 826–842.
B.D. Craven (1978), Mathematical Programming and Control Theory, Chapman and Hall, London, England.
R. Ellala (1992), Fractional programming: duality and applications, Preprint, Universite Paul Sabatier, France, To appear in Journal of Global Optimization.
B.M. Glover, V. Jeyakumar, and W. Oettli (1994), A Farkas lemma for difference sublinear systems and quasidifferentiable programming, Mathematical Programming, Series A 63, 109–125.
J. Gwinner (1987), Results of Farkas type, Numerical Functional Analysis and Optimization 9, 471–520.
Chung-wei Ha (1979), On systems of convex inequalities, Journal of Mathematical Analysis and Applications 68, 25–34.
J.-B. Hiriart-Urruty (1995), Conditions for global optimality, in Handbook of Global Optimization, Eds. R. Horst and P.M. Pardalos, Kluwer Academic Publishers, The Netherlands, 1–26.
J.-B. Hiriart-Urruty (1989), From convex to nonconvex minimization: Necessary and sufficient conditions for global optimality, in Nonsmooth Optimization and Related Topics, 219–240, Plenum, New York.
{au{gnJ.-B.} {fnHiriart-Urruty}} (1982), ε-subdifferential calculus, in Convex Analysis and Optimization, Eds. J.P. Aubin and R. Vinter, Pitman, 43–92.
J.-B. Hiriart-Urruty and C. Lemarechal (1993), Convex Analysis and Minimization Algorithms, Volumes I and II, Springer-Verlag, Berlin Heidelberg.
R.B. Holmes (1975), Geometric Functional Analysis and its Applications, Springer, New York.
R. Horst and P.M. Pardalos (1995), Handbook of Global Optimization, Kluwer Academic Publishers, The Netherlands.
R. Horst and H. Tuy (1990), Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin.
V. Jeyakumar (1994), Asymptotic dual conditions characterizing optimality for infinite convex programs, Applied Mathematics Research Report AMR94/30, University of New South Wales. (Submitted for publication in Journal of Optimization Theory and Applications).
V. Jeyakumar and B.M. Glover (1993), A new Farkas' lemma and global convex maximization, Applied Mathematics Letters 6(5), 39–43.
V. Jeyakumar and B.M. Glover (1995), Nonlinear extensions of Farkas' lemma with applications to global optimization and least squares, Applied Mathematics Research Report No. AMP93/1. To appear in Mathematics of Operations Research, 1995.
V. Jeyakumar and H. Wolkowicz (1990), Zero duality gaps in infinite dimensional programming, Journal of Optimization Theory and Applications 67(1), 87–108.
V. Jeyakumar and H. Wolkowicz (1992), Generalizations of Slater's constraint qualifications for infinite convex programs, Mathematical Programming, series B 57, 85–101.
P.M. Pardalos (1994), On the passage from local to global in optimization, Mathematical Programming: State of the art, Proceedings of the 15th International Symposium of Mathematical Programming, Edited by J.R. Birge and K.G. Murty, University of Michigan.
P.M. Pardalos and J.B. Rosen (1987), Constrained Global Optimization: Algorithms and Applications, Lecture Notes in Computer Science, 268, Springer-Verlag, Berlin.
R.T. Rockafellar (1974), Conjugate Duality and Optimization, SIAM Regional Conference Series in Applied Mathematics.
A. Rubinov, B.M. Glover, and V. Jeyakumar (1994), A general approach to dual characterizations of the solvability of inequality systems with applications, Applied mathematics Research Report No. AMR94/22, University of New South Wales, to appear in J. Convex Analysis Applications, 1995.
H. Tuy (1995), D.C. Optimization: theory, methods and algorithms, in Handbook of Global Optimization, edited by R. Horst and P.M. Pardalos, Kluwer Academic Publishers, 149–216.
J.P. Vial (1983), Strong and weak convexity of sets and functions, Mathematics of Operations Research 3, 231–259.
M. Volle (1994), Calculus rules for global approximate minima and applications to approximate subdifferential calculus, Journal of Global Optimization 5(2), 131–158.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jeyakumar, V., Glover, B.M. Characterizing global optimality for DC optimization problems under convex inequality constraints. J Glob Optim 8, 171–187 (1996). https://doi.org/10.1007/BF00138691
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00138691