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Existence of Global Minima for Constrained Optimization

Author

Listed:
  • A. E. Ozdaglar

    (Massachusetts Institute of Technology)

  • P. Tseng

    (University of Washington)

Abstract

We present a unified approach to establishing the existence of global minima of a (non)convex constrained optimization problem. Our results unify and generalize previous existence results for convex and nonconvex programs, including the Frank-Wolfe theorem, and for (quasi) convex quadratically constrained quadratic programs and convex polynomial programs. For example, instead of requiring the objective/constraint functions to be constant along certain recession directions, we only require them to linearly recede along these directions. Instead of requiring the objective/constraint functions to be convex polynomials, we only require the objective function to be a (quasi)convex polynomial over a polyhedral set and the constraint functions to be convex polynomials or the composition of coercive functions with linear mappings.

Suggested Citation

  • A. E. Ozdaglar & P. Tseng, 2006. "Existence of Global Minima for Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 128(3), pages 523-546, March.
    • Handle: RePEc:spr:joptap:v:128:y:2006:i:3:d:10.1007_s10957-006-9039-2
    DOI: 10.1007/s10957-006-9039-2
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    References listed on IDEAS

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    1. Alfred Auslender, 1996. "Noncoercive Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 21(4), pages 769-782, November.
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    Cited by:

    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    2. Werner Schachinger & Immanuel Bomze, 2009. "A Conic Duality Frank--Wolfe-Type Theorem via Exact Penalization in Quadratic Optimization," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 83-91, February.
    3. Fabián Flores-Bazán & Filip Thiele, 2022. "On the Lower Semicontinuity of the Value Function and Existence of Solutions in Quasiconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 195(2), pages 390-417, November.

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