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 (PLQ) objective functions. Starting from a study of the representation of such a function in terms of a family of elementary functions consisting of squared affine functions, squared plus-composite-affine functions, and affine functions themselves, we summarize some local properties of a PLQ function in terms of their first and second-order directional derivatives. We extend some well-known necessary and sufficient second-order conditions for local optimality of a quadratic program to a PLQ program and provide a dozen such equivalent conditions for strong, strict, and isolated local optimality, showing in particular that a PLQ 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 PLQ program is finite; this result supplements a recent result about the finite number of directional stationary objective values. Interestingly, these finiteness results can be uncovered by invoking a very powerful property of subanalytic functions; our proof is fairly elementary, however. We discuss applications of PLQ programs in some modern statistical estimation problems. These problems lead to a special class of unconstrained composite programs involving the non-differentiable $\ell_1$-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.

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分段线性二次规划研究

受越来越多的分段类型的非传统统计估计问题的启发，本文提供了对已知结果的调查，并补充了分段线性二次程序类的新结果。这些是具有分段线性二次 (PLQ) 目标函数的线性约束优化问题。从研究这样一个函数在由平方仿射函数、平方加复合仿射函数和仿射函数本身组成的基本函数族方面的表示开始，我们总结了 PLQ 函数的一些局部性质它们的一阶和二阶方向导数。我们将二次程序局部最优的一些众所周知的充分必要的二阶条件扩展到 PLQ 程序，并为强、严格和孤立的局部最优提供了十几个这样的等价条件，特别表明 PLQ 程序具有局部极小性的特征与标准二次规划相同。作为这样一个条件的结果，我们证明了 PLQ 程序的强、严格或孤立的局部最小值的数量是有限的；该结果补充了最近关于有限数量的定向平稳目标值的结果。有趣的是，这些有限性结果可以通过调用子分析函数的一个非常强大的特性来揭示；然而，我们的证明是相当初级的。我们讨论了 PLQ 程序在一些现代统计估计问题中的应用。