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Polyextremal principles and separably-infinite programs

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Abstract

As a direct extension of Charnes' characterization of two-person zero-sum constrained games by linear programming, we show how a general class of saddle value problems can be reduced to a pair of uniextremal dual separably-infinite programs. These programs have an infinite number of variables and an infinite number of constraints, but only a finite number of variables appear in an infinite number of constraints and only a finite number of constraints have an infinite number of variables. The conditions under which the characterization holds are among the more general ones appearing in the literature sufficient to guarantee the existence of a saddle point of a concave-convex function.

The key construction involves augmenting a given player's original set of variables by generalized finite sequences determined by the other player's constraint set and objective function. A duality theory is developed which includes complementarity conditions, thereby making contact with the numerical treatment of semi-infinite programming.

Zusammenfassung

Als eine direkte Erweiterung von Charnes' Charakterisierung von Zweipersonen-Nullsummenspielen durch lineare Programme wird gezeigt, daß eine allgemeine Klasse von Sattelpunktproblemen auf ein Paar dualer separabel-infiniter Programme zurückgeführt werden kann. Diese Programme haben unendlich viele Variablen und unendlich viele Nebenbedingungen, wobei nur endlich viele Variablen in unendlich vielen Restriktionen vorkommen und nur endlich viele Nebenbedingungen unendlich viele Variablen enthalten. Es wird eine Dualitätstheorie entwickelt, die Komplementaritätsbedingungen einschließt, wobei auf die numerische Behandlung semi-infiniter Optimierungsprobleme Bezug genommen wird.

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Authors and Affiliations

  1. The University of Texas at Austin, Texas

    A. Charnes

  2. Pacific Gas and Electric, San Francisco, California

    P. R. Gribik

  3. Carnegie-Mellon University, Pittsburgh, Pennsylvania

    K. O. Kortanek

Authors
  1. A. Charnes
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  2. P. R. Gribik
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  3. K. O. Kortanek
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Additional information

This research was partially supported by Project NR047-021, ONR Contract N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas, and by the National Science Foundation Grant NSF ENG-7825488 with Carnegie-Mellon University. Reproduction in whole or in part is permitted for any purpose of the United States Government.

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Charnes, A., Gribik, P.R. & Kortanek, K.O. Polyextremal principles and separably-infinite programs. Zeitschrift für Operations Research 24, 211–234 (1980). https://doi.org/10.1007/BF01919901

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