Jove
Visualize
Contact Us

Related Experiment Videos

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
Summary
This summary is machine-generated.

Related Concept Videos

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Dualizing the Poisson summation formula.

Proceedings of the National Academy of Sciences of the United States of America·1991
Same author

The Markoff-Duffin-Schaeffer inequalities abstracted.

Proceedings of the National Academy of Sciences of the United States of America·1985
Same author

Rubel's universal differential equation.

Proceedings of the National Academy of Sciences of the United States of America·1981
Same author

Bounds for the rth characteristic frequency of a beaded string or of an electrical filter.

Proceedings of the National Academy of Sciences of the United States of America·1980
Same author

Convex programs having some linear constraints.

Proceedings of the National Academy of Sciences of the United States of America·1977
Same author

Hilbert transforms in yukawan potential theory.

Proceedings of the National Academy of Sciences of the United States of America·1972
Same journal

Chemotactic self-organization captures the dynamics of mammalian hair follicle patterning.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Tomographic imaging of superconducting order using particle-hole interference.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Inhibitory potential of autologous neutralizing antibodies sets quantitative limits on the rebound-competent HIV-1 reservoir.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Inferring epidemiological parameters under an infectious phylogeography model with visitor dynamics.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Analytical modeling for suction cup designs for skin-interfaced wearable devices.

Proceedings of the National Academy of Sciences of the United States of America·2026
Same journal

Improving cell-free metabolism through direct integration of artificial respiratory chains.

Proceedings of the National Academy of Sciences of the United States of America·2026
See all related articles
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

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.