The third volume of the Journal of Nonsmooth Analysis and Optimization (2022)
Gerd Wachsmuth.
We consider a generalized equation governed by a strongly monotone and
Lipschitz single-valued mapping and a maximally monotone set-valued mapping in
a Hilbert space. We are interested in the sensitivity of solutions w.r.t.
perturbations of both mappings. We demonstrate that the directional
differentiability of the solution map can be verified by using the directional
differentiability of the single-valued operator and of the resolvent of the
set-valued mapping. The result is applied to quasi-generalized equations in
which we have an additional dependence of the solution within the set-valued
part of the equation.
Section:
Original research articles
Heinz H. Bauschke ; Peter A. V. DiBerardino.
We compute the minimal angle spread with respect to the uniform distribution
in the probability simplex. The resulting optimization problem is analytically
solved. The formula provided shows that the minimal angle spread approaches
zero as the dimension tends to infinity. We also discuss an application in
cognitive science.
Section:
Original research articles
Daniel Wachsmuth ; Gerd Wachsmuth.
We study no-gap second-order optimality conditions for a non-uniformly convex
and non-smooth integral functional. The integral functional is extended to the
space of measures. The obtained second-order derivatives contain integrals on
lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which
is an integral functional on the space of continuous functions. Applications to
non-smooth optimal control problems are given.
Section:
Original research articles
Bernard Brogliato ; Alexandre Rocca.
This article is largely concerned with the time-discretization of descriptor-variable systems coupled to with complementarity constraints. They are named descriptor-variable linear complementarity systems (DVLCS). More speci cally passive DVLCS with minimal state space representation are studied. The Euler implicit discretization of DVLCS is analysed: the one-step non-smooth problem (OSNSP), that is a generalized equation, is shown to be well-posed under some conditions. Then the convergence of the discretized solutions is studied. Several examples illustrate the applicability and the limitations of the developments.
Section:
Original research articles