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First-Order Conditions for Set-Constrained Optimization

Author

Listed:
  • Steven M. Rovnyak

    (Department of Electrical and Computer Engineering, Indiana University-Purdue University, Indianapolis, IN 46202, USA)

  • Edwin K. P. Chong

    (Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523, USA)

  • James Rovnyak

    (Department of Mathematics, University of Virginia, Charlottesville, VA 22904, USA)

Abstract

A well-known first-order necessary condition for a point to be a local minimizer of a given function is the non-negativity of the dot product of the gradient and a vector in a feasible direction. This paper proposes a series of alternative first-order necessary conditions and corresponding first-order sufficient conditions that seem not to appear in standard texts. The conditions assume a nonzero gradient. The methods use extensions of the notions of gradient, differentiability, and twice differentiability. Examples, including one involving the Karush–Kuhn–Tucker (KKT) theorem, illustrate the scope of the conditions.

Suggested Citation

  • Steven M. Rovnyak & Edwin K. P. Chong & James Rovnyak, 2023. "First-Order Conditions for Set-Constrained Optimization," Mathematics, MDPI, vol. 11(20), pages 1-14, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4274-:d:1259148
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