First-Order Conditions for Set-Constrained Optimization

Steven M. Rovnyak (), Edwin K. P. Chong and James Rovnyak
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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

Mathematics, 2023, vol. 11, issue 20, 1-14

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.

Keywords: constrained optimization; local minimizer; necessary condition; sufficient condition; KKT theorem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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