The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions

Sixuan Bai (), Minghua Li (), Chengwu Lu (), Daoli Zhu () and Sien Deng ()
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Sixuan Bai: Chongqing Jiaotong University
Minghua Li: Chongqing University of Arts and Sciences
Chengwu Lu: Chongqing University of Arts and Sciences
Daoli Zhu: Antai College of Economics and Management and Sino-US Global Logistics Institute
Sien Deng: Northern Illinois University

Journal of Optimization Theory and Applications, 2022, vol. 194, issue 1, No 9, 220-245

Abstract: Abstract We start by establishing the equivalence of three types of error bounds: weak sharp minima, level-set subdifferential error bounds and Łojasiewicz (for short Ł) inequalities for weakly convex functions with exponent $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] and approximately convex functions. Then we apply these equivalence results to a class of nonconvex optimization problems, whose objective functions are the sum of a convex function and a composite function with a locally Lipschitz function and a smooth vector-valued function. Finally, applying a characterization for lower-order regularization problems, we show that the level-set subdifferential error bound with exponent 1 and the Ł inequality with exponent $$\frac{1}{2}$$ 1 2 hold at a local minimum point.

Keywords: Weak sharp minima; Level-set subdifferential error bounds; Łojasiewicz inequalities; Lower-order regularization problems; 65K10; 90C26; 90C31 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-022-02016-z

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