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On equating the difference between optimal and marginal values of general convex programs

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Abstract

Unlike elementary finite linear programming, the optimal program value of a convex optimization problem is generally different from the vector product of the marginal price vector and the resource right-hand side vector. In this paper, a duality approach is developed, based on objective function parametrizations, to characterize this difference under rather general circumstances.

The approach generalizes the concept of Kuhn-Tucker vectors of a convex program. It is shown that nonstandard polynomial Kuhn-Tucker vectors exist for any convex program having finite value. Two examples illustrate the procedure.

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References

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Author information

Authors and Affiliations

  1. Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania

    K. O. Kortanek (Professor)

  2. Department of Industrial Engineering and Operations Research, Virginia Polytechnic Institute and State University, Blacksburg, Virginia

    A. L. Soyster (Professor)

Authors
  1. K. O. Kortanek
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  2. A. L. Soyster
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Additional information

Communicated by A. V. Fiacco

An earlier version of this paper was presented at the International Symposium on Extremal Methods and Systems Analysis on the Occasion of Professor A. Charnes' Sixtieth Birthday, Austin, Texas, 1977. Partial support of the research of the first author was provided by NSF Grants Nos. ENG-76-05191 and ENG-78-25488. The authors gratefully acknowledge Professor F. J. Gould, University of Chicago, for bringing the valuable Balinski-Baumol reference (Ref. 1) to their attention. They also gratefully acknowledge criticisms of a referee reminding them of the sophistication which a convex analysis approach can bring to bear on the main problem treated in this paper. This paper is dedicated to Professor Charnes.

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Kortanek, K.O., Soyster, A.L. On equating the difference between optimal and marginal values of general convex programs. J Optim Theory Appl 33, 57–68 (1981). https://doi.org/10.1007/BF00935176

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