A Study of Piecewise Linear-Quadratic Programs

Ying Cui (), Tsung-Hui Chang (), Mingyi Hong () and Jong-Shi Pang ()
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Ying Cui: University of Minnesota
Tsung-Hui Chang: The Chinese University of Hong Kong
Mingyi Hong: University of Minnesota
Jong-Shi Pang: University of Southern California

Journal of Optimization Theory and Applications, 2020, vol. 186, issue 2, No 9, 523-553

Abstract: Abstract Motivated by a growing list of nontraditional statistical estimation problems of the piecewise kind, this paper provides a survey of known results supplemented with new results for the class of piecewise linear-quadratic programs. These are linearly constrained optimization problems with piecewise linear-quadratic objective functions. We first summarize some local properties of a piecewise linear-quadratic function in terms of their first- and second-order directional derivatives. We next extend some well-known necessary and sufficient second-order conditions for local optimality of a quadratic program to a piecewise linear-quadratic program and provide a dozen such equivalent conditions for strong, strict, and isolated local optimality, showing in particular that a piecewise linear-quadratic program has the same characterizations for local minimality as a standard quadratic program. As a consequence of one such condition, we show that the number of strong, strict, or isolated local minima of a piecewise linear-quadratic program is finite; this result supplements a recent result about the finite number of directional stationary objective values. We also consider a special class of unconstrained composite programs involving a non-differentiable norm function, for which we show that the task of verifying the second-order stationary condition can be converted to the problem of checking the copositivity of certain Schur complement on the nonnegative orthant.

Keywords: Piecewise linear-quadratic programming; Directional stationarity; Second-order local optimality theory; Second-order directional; Semi- and sub-derivatives; 90C20; 90C26; 68Q25 (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10957-020-01716-8

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