Journal of Convex Analysis 30 (2023), No. 3, 793--834
Copyright Heldermann Verlag 2023
The Minimization of Piecewise Functions: Pseudo Stationarity
Ying Cui
Dept. of Industrial and Systems Engineering, University of Minnesota, Minneapolis, U.S.A.
yingcui@umn.edu
Junyi Liu
Dept. of Industrial Engineering, Tsinghua University, Beijing, China
junyiliu@mail.tsinghua.edu.cn
Jong-Shi Pang
The D.-J.-Epstein Dept. of Industrial and Systems Engineering, University of Southern California, Los Angeles, U.S.A.
jongship@usc.edu
There are many significant applied contexts that require the solution of discontinuous optimization problems in finite dimensions. Yet these problems are very difficult, both computationally and analytically. With the functions being discontinuous and a minimizer (local or global) of the problems, even if it exists, being impossible to verifiably compute, a foremost question is what kind of "stationary solutions" one can expect to obtain; these solutions provide promising candidates for minimizers; i.e., their defining conditions are necessary for optimality. Motivated by recent results on sparse optimization, we introduce in this paper such a kind of solution, termed "pseudo B- (for Bouligand) stationary solution", for a broad class of discontinuous optimization problems with objective and constraint defined by indicator functions of the positive real axis composite with functions that are possibly nonsmooth. We present two approaches for computing such a solution. One approach is based on lifting the problem to a higher dimension via the epigraphical formulation of the indicator functions; this requires the addition of some auxiliary variables. The other approach is based on certain continuous (albeit not necessarily differentiable) piecewise approximations of the indicator functions and the convergence to a pseudo B-stationary solution of the original problem is established. The conditions for convergence are discussed and illustrated by an example.
Keywords: Piecewise functions, stationarity, epigraphical formulation, nonifier approximation.
MSC: 90C26, 90C59.
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