Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time

Abstract

The -subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be performed in logarithmic worst-case time. Furthermore, a new algorithm to compute the entire graph of the -subdifferential in linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined function with little restrictions. This is a pre-print of an article published in Set-Valued and Variational Analysis (SVAA). The final authenticated version is available online at: http://dx.

New global optimality conditions for nonsmooth DC optimization problems

“…However, the same statement is true for most of the general global optimality conditions. In particular, it is true for the well-known global optimality condition in terms of ε-subdifferentials [28,29,30] due to the fact that ε-subdifferentials can be efficiently computed only in few particular cases (see, e.g., [41]). Let us note that in the case when the function f is piecewise affine, there always exists a global codifferential of the function f such that both sets df (x) and df (x) are convex polytops [27].…”