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Convex programs having some linear constraints.

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
|January 1, 1977
PubMed
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This study proves that the minimum of a convex function equals the maximum of its dual problem under specific constraints. This applies to optimization in normed spaces like Hilbert spaces.

Area of Science:

  • Optimization Theory
  • Convex Analysis
  • Functional Analysis

Background:

  • Minimizing convex functions with inequality constraints is a fundamental problem in optimization.
  • Duality theory, involving Lagrange multipliers, provides an alternative perspective for solving such problems.
  • The Slater constraint qualification is crucial for establishing the duality gap.

Purpose of the Study:

  • To investigate the relationship between a primal minimization problem and its dual maximization problem for convex functions in normed spaces.
  • To provide an elementary proof for the equality of the infimum and supremum under specific constraint conditions.
  • To explore the applicability of Uzawa, Stoer, and Witzgall constraint qualifications.

Main Methods:

  • Formulation of a convex minimization problem with inequality constraints.

Related Experiment Videos

  • Construction of the corresponding dual maximization problem using Lagrange multipliers.
  • Application of Slater constraint qualifications, including the Uzawa, Stoer, and Witzgall forms.
  • Development of a short, elementary proof to connect the primal and dual problem solutions.

    • Main Results:

    • Demonstration that the infimum of the primal problem is equal to the supremum of the dual problem.
    • Validation of the chosen Slater constraint qualifications for the given problem structure.
    • Establishment of strong duality under the specified conditions.

    Conclusions:

    • The duality gap is zero for the considered class of convex optimization problems in normed spaces.
    • The elementary proof provides a clear and accessible method for verifying strong duality.
    • The findings contribute to the theoretical understanding of constrained optimization and its dual formulations.