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Modified regularity conditions for nonlinear programming problems over mixed cone domains

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Abstract

In earlier results by Sposito and David, Kuhn—Tucker duality was established over nondegenerate cone domains (not necessarily polyhedral) without differentiability under a certain natural modification of the Slater condition, in addition to the convexity of a certain auxiliary set. This note extends Kuhn—Tucker duality to optimization problems with both nondegenerate and degenerate cone domains. Moreover, under a different condition than presented in earlier results by the author, this note develops Kuhn—Tucker duality for a certain class of nonlinear problems with linear constraints and an arbitrary objective function.

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References

  1. R.A. Abrams and A. Ben-Israel, “A duality theorem for complex quadratic programming”,Journal of Optimization Theory and Applications 4 (1969) 244–252.

    Google Scholar 

  2. K. Arrow, L. Hurwicz and H. Uzawa, “Constraint qualifications in maximization problems”,Naval Research Logistics Quarterly 8 (1961) 175–191.

    Google Scholar 

  3. C.R. Bector, “Duality in fractional and indefinite programming”,Zeitschrift für Angewandte Mathematik und Mechanik 48 (1968).

  4. A. Ben-Israel, “Linear equations and inequalities or finite dimensional, real or complex, vector spaces: a unified theory”,Journal of Mathematical Analysis and Applications 27 (1969) 367–389.

    Google Scholar 

  5. A. Ben-Israel, A. Charnes and K.O. Kortanek, “Duality and asymptotic solvability over cones”,Bulletin of the American Mathematical Society 75 (1969) 318–324.

    Google Scholar 

  6. A. Berman and A. Ben-Israel, “Linear inequalities, mathematical programming and matrix theory”,Mathematical Programming 1 (1971) 291–300.

    Google Scholar 

  7. M. Canon, C. Cullum and E. Polak, “Constrained minimization problems in finite-dimensional spaces”,SIAM Journal on Control 4 (1966) 528–547.

    Google Scholar 

  8. A. Charnes and W.W. Cooper, “Programming with linear fractional functionals”,Naval Research Logistics Quarterly 9 (1962) 181–186.

    Google Scholar 

  9. R.A. Cottle, “Theorem of Fritz John in mathematical programming”, RAND Memorandum RM-3858-PR, Santa Monica, Calif. (1963).

  10. R.J. Duffin, “Infinite programs”, in:Linear inequalities and related systems Eds. H.W. Kuhn and A.W. Tucker (Princeton University Press, Princeton, N.J., 1965) pp. 157–170.

    Google Scholar 

  11. Ky. Fan, “A generalization of the Alaoglu—Bourbaki theorem and its applications”,Mathematische Zeitschrift 88 (1965) 48–60.

    Google Scholar 

  12. J. Farkas, “Über die Theorie der einfachen Ungleichungen”,Journal für die Reine und Angewandte Mathematik 124 (1902) 1–27.

    Google Scholar 

  13. F.J. Gould and J.W. Tolle, “A necessary and sufficient qualification for constrained optimization”,SIAM Journal on Applied Mathematics 20 (1971) 164–172.

    Google Scholar 

  14. F.J. Gould and J.W. Tolle, “Geometry of optimality conditions and constraint qualifications”,Mathematical Programming 2 (1972) 1–18.

    Google Scholar 

  15. S. Karlin, Mathematical methods and theory of games, programming and economics (Addison—Wesley, Reading, Mass., 1959).

    Google Scholar 

  16. K.S. Kretschmer, “Programmes in paired spaces”,Canadian Journal of Mathematics 13 (1961) 221–238.

    Google Scholar 

  17. H.W. Kuhn and A.W. Tucker, “Nonlinear programming”, in:Second Berkeley symposium proceedings on mathematical statistics and probability (University of California Press, Berkeley, Calif., 1950) pp. 481–492.

    Google Scholar 

  18. H. Kunzi and W. Krelle,Nonlinear programming (Blaisdell, Waltham, Mass., 1966).

    Google Scholar 

  19. O.L. Mangasarian,Nonlinear programming (McGraw—Hill, New York, 1969).

    Google Scholar 

  20. O.L. Mangasarian and S. Fromovitz, “The Fritz John necessary optimality conditions in the presence of equality and inequality constraints”,Journal of Mathematical Analysis and Applications 17 (1967) 37–47.

    Google Scholar 

  21. M. Slater, “Lagrange multipliers revisited: a contribution to nonlinear programming”, Rand Corporation Rept. RM-676, Santa Monica, Calif., (1951).

  22. V.A. Sposito, “A note on dual quadratic programs over arbitrary cone domains”, to appear.

  23. V.A. Sposito and H.T. David, “Saddlepoint optimality criteria of nonlinear programming problems over cones without differentiability”,SIAM Journal on Applied Mathematics 20 (1971) 698–708.

    Google Scholar 

  24. V.A. Sposito and H.T. David, “A note on Farkas lemmas over cone domains”,SIAM Journal on Applied Mathematics 22 (1972) 356–358.

    Google Scholar 

  25. K. Swarup, “Some aspects of linear fractional programs”,Zeitschrift für Angewandte Mathematik und Mechanik 47 (1968).

  26. K. Ritter, “Optimization theory in linear spaces”,Mathematische Annalen 184 (1970) 133–154.

    Google Scholar 

  27. R.T. Rockafellar,Convex analysis (Princeton University Press, Princeton, N.J., 1970).

    Google Scholar 

  28. R.M. van Slyke and R.J. Wets, “A duality theory for abstract mathematical programs with applications to optimal control theory”,Journal of Mathematical Analysis and Applications 22 (1968) 679–706.

    Google Scholar 

  29. P.P. Varaiya, “Nonlinear programming in Banach space”,SIAM Journal on Applied Mathematics 15 (1967) 284–293.

    Google Scholar 

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Authors and Affiliations

  1. Iowa State University, Ames, Iowa, USA

    V. A. Sposito

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  1. V. A. Sposito
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Sposito, V.A. Modified regularity conditions for nonlinear programming problems over mixed cone domains. Mathematical Programming 6, 167–179 (1974). https://doi.org/10.1007/BF01580234

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  • DOI: https://doi.org/10.1007/BF01580234

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