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Primal and dual optimality criteria in convex programming

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Summary

This paper considers the problem of minimizing a convex differentiable function subject to convex differentiable constraints. Necessary and sufficient conditions (not requiring any constraints qualification) for a point to be an optimal solution are given in terms of a parametric linear program. Dual characterization theorems are then derived, which generalizes the classical results ofKuhn-Tucker andJohn.

Zusammenfassung

Es wird das Problem betrachtet, eine konvexe differenzierbare Funktion unter konvexen differenzierbaren Nebenbedingungen zu minimieren. Unter Verwendung eines parametrischen linearen Optimierungsproblems werden notwendige und hinreichende Bedingungen für Optimalität eines Punktes angegeben, die keine constraints qualifications benötigen. Sodann werden duale Charakterisierungstheoreme hergeleitet, welche die klassischen Resultate vonKuhn-Tucker undJohn verallgemeinern.

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References

  • Bazaraa, M.S.: A Theorem of the Alternative with Application to Convex Programming: Optimality, Duality and Stability. J. Math. Anal. Applic.41 (3), 1973.

  • Ben-Israel, A., A. Charnes, andK.O. Kortanek: Duality and Asymptotic Solvability over Cones. Bull. Amer. Math. Soc.75, 1969, 318–324.

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  • Ben-Tal, A., A. Ben-Israel, andS. Zlobec: Characterization of Optimality in Convex Programming without a Constraint Qualification. J. Optimiz. Th. and Appl.20, (4), 1976.

  • Duffin, R.J.: Infinite Programs, in Linear Inequalities and Related Systems. Ed. byH. W. Kuhn andA. W. Tucker. Annals of Math. Studies No. 38. Princeton, N.J., 1956, 157–170.

  • John, F.: Extremum Problems with Inequalities as Subsidiary Conditions. In Studies and Essays Courant Anniversary Volume. Ed. byK.O. Friedrichs, I.E. Neugebauer andJ.J. Stoker, N.Y., 1948, 187–204.

  • Kuhn, H. W., andA. W. Tucker: Nonlinear Programming. Proc. Second Berkeley Symp. on Math. Statistics and Probability. Ed. byJ. Neyman, Berkeley 1951.

  • Rockafellar, R. T.: Some Convex Programs whose Duals are Linearly Constrained. In Nonlinear Programming Ed. byJ.B. Rosen, O.L. Mangasarian andK. Ritter, N.Y. 1970.

  • —: Ordinary Convex Program without a Duality Gap. J. Optimiz. Th. and Appl.7 (3), 1971, 143–148.

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Authors and Affiliations

  1. Department of Computer Science, TECHNION, Haifa, Israel

    A. Ben-Tal

  2. Center for cybernetic studies, The University of Texas at Austin, 78712, Austin, Texas, USA

    A. Charnes (Director)

Authors
  1. A. Ben-Tal
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  2. A. Charnes
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Additional information

This research was partly supported by Project No. NR 047-021, ONR Contract N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Ben-Tal, A., Charnes, A. Primal and dual optimality criteria in convex programming. Zeitschrift für Operations Research 21, 197–209 (1977). https://doi.org/10.1007/BF01965717

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  • DOI: https://doi.org/10.1007/BF01965717

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