Visualization of the $$\varepsilon $$ ε -subdifferential of piecewise linear–quadratic functions

Abstract

The final publication is available at Springer via http://dx.doi.org/10.1007/s10589-017-9892-yComputing explicitly the ε-subdifferential of a proper function amounts to computing the level set of a convex function namely the conjugate minus a linear function. The resulting theoretical algorithm is applied to the the class of (convex univariate) piecewise linear-quadratic functions for which existing numerical libraries allow practical computations. We visualize the results in a primal, dual, and subdifferential views through several numerical examples. We also provide a visualization of the Brøndsted-Rockafellar Theorem.

Proximal Mappings and Moreau Envelopes of Single-Variable Convex Piecewise Cubic Functions and Multivariable Gauge Functions

“…To the best of our knowledge, the closed forms of Moreau envelopes of many examples given here have not been realized until now. The piecewise cubic work extends the results for piecewise linear-quadratic functions found in [2,7,17], and the study of Moreau envelopes of gauge functions is new.…”

“…Convex piecewise functions in general, and their Moreau envelopes, are explored in [18,19] and similar works. Properties of piecewise linear-quadratic (PLQ) functions in particular, and their Moreau envelopes, are developed in [2,7,17] and others. The new theory of piecewise cubic functions found in this section will enable the expansion of such works to polynomials of one degree higher, and any result developed here reverts to the piecewise linear-quadratic case by setting the cubic coefficients to zero.…”

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

“…To make the distinction with the approximate subdifferential introduced in [Iof84], following [BHL16] we use the accepted terminology -subdifferential, although historically [BR65] used the term "approximate subgradients". When = 0, ∂ 0 f (x) reduces to the convex subdifferential that we will denote ∂f (x).…”

“…will play a critical role in the computation due to the following fact. While [BHL16] uses the CCA numerical library [Luc16] to compute f * explicitly, we will avoid such computation by relying on the natural parametrization of f * previously exploited in [GL11,HUL07]. A function f : R → R ∪ {+∞} is piecewise linear-quadratic (PLQ) if dom(f ) can be represented as the union of finitely many closed intervals on each of which f is linear or quadratic.…”

“…multifunctions such as the -subdifferential. Bajaj et al [BHL16] proposed an algorithm to compute the -subdifferential of a univariate convex PLQ function as the level set of its conjugate minus an affine function, and used the CCA numerical library to carry out that computation in linear time.…”