Volume 3

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1. From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

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.

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2. Minimal angle spread in the probability simplex with respect to the uniform distribution

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.

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3. Second-order conditions for non-uniformly convex integrands: quadratic growth in $L^1$

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.

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4. Analysis of the implicit Euler time-discretization of passive linear descriptor complementarity systems

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.

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