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Clark's Theorem on linear programs holds for convex programs.

R J Duffin1

  • 1Department of Mathematics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213.

Proceedings of the National Academy of Sciences of the United States of America
|April 1, 1978
PubMed
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Clark's Theorem, originally for linear programs, extends to convex programs. If one program's feasible set is bounded, the dual convex program's feasible set is unbounded, with equal optimal values.

Area of Science:

  • Optimization Theory
  • Convex Analysis
  • Mathematical Programming

Background:

  • Linear programming duality establishes a relationship between minimization and maximization problems.
  • F. E. Clark's Theorem states that for linear programs, if one feasible set is bounded, the other is unbounded.
  • Convex programming involves minimizing convex functions under convex constraints.

Purpose of the Study:

  • To investigate the applicability of Clark's Theorem to convex programming.
  • To analyze the properties of dual convex programs and their feasible sets.
  • To determine if the duality gap holds for convex programs under specific conditions.

Main Methods:

  • Defining the dual of a convex program using the Lagrange function.
  • Identifying feasible Lagrange multipliers based on nonnegativity and dual objective function bounds.

Related Experiment Videos

  • Applying the principles of Clark's Theorem to the established dual convex program framework.

    • Main Results:

    • Clark's Theorem holds true for dual convex programs without modification.
    • The theorem implies that if the primal feasible set is bounded, the dual feasible set is unbounded.
    • The values of the primal and dual convex programs are proven to be equal.

    Conclusions:

    • Clark's Theorem is a robust result applicable to both linear and convex programming duality.
    • The duality gap is zero for convex programs where the conditions of Clark's Theorem are met.
    • This extends fundamental understanding of duality in mathematical optimization.