The Exact Modulus of the Generalized Concave Kurdyka-Łojasiewicz Property

Xianfu Wang () and Ziyuan Wang ()
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Xianfu Wang: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada
Ziyuan Wang: Department of Mathematics, University of British Columbia, Kelowna, British Columbia V1V 1V7, Canada

Mathematics of Operations Research, 2022, vol. 47, issue 4, 2765-2783

Abstract: We introduce a generalized version of the concave Kurdyka-Łojasiewicz (KL) property by employing nonsmooth desingularizing functions. We also present the exact modulus of the generalized concave KL property, which provides an answer to the open question regarding the optimal concave desingularizing function. The exact modulus is designed to be the smallest among all possible concave desingularizing functions. Examples are given to illustrate this pleasant property. In turn, using the exact modulus, we provide the sharpest upper bound for the total length of iterates generated by the celebrated Bolte-Sabach-Teboulle proximal alternating linearized minimization algorithm.

Keywords: Primary: 49J52; 26D10; 90C26; secondary: 26A51; 26B25; generalized concave Kurdyka-Łojasiewicz property; Kurdyka-Łojasiewicz property; optimal concave desingularizing function; Bolte-Daniilidis-Ley-Mazet desingularizing function; proximal alternating linearized minimization; nonconvex optimization (search for similar items in EconPapers)
Date: 2022
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