This paper provides a comprehensive study of the nonmonotone forward–backward splitting (FBS) method for solving a class of nonsmooth composite problems in Hilbert spaces. The objective function is the sum of a Fréchet differentiable (not necessarily convex) function and a proper lower...

Spatial branch-and-bound algorithms for global minimization of non-convex problems require both lower and upper bounding procedures that finally converge to a globally optimal value in order to ensure termination of these methods. Whereas convergence of lower bounds is commonly guaranteed for...

Since shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations, we show how shape optimization techniques can also be applied to an interface identification problem constrained by a nonlocal Dirichlet problem. Here, we...

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance ε0, we show how to compute in finite time an...

We develop a semismooth Newton framework for the numerical solution of fixed-point equations that are posed in Banach spaces. The framework is motivated by applications in the field of obstacle-type quasi-variational inequalities and implicit obstacle problems. It is discussed in a general...

Total variation regularization has proven to be a valuable tool in the context of optimal control of differential equations. This is particularly attributed to the observation that TV-penalties often favor piecewise constant minimizers with well-behaved jumpsets. On the downside, their intricate...

We provide quantitative results on a seminal Tseng-type primal-dual splitting algorithm for solving monotone inclusions due to Combettes and Pesquet which involves a mixture of sums, linear compositions and parallel sums of set-valued and Lipschitzian operators. For that, we first give...

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Data-driven approaches for inverse problems aim for learning model and regularization parameters from observed data samples, and investigate their generalization properties when...

We introduce a framework for shape optimization in the class of simple polytopes in arbitrary dimensions. We use a parametrization via hyperplanes and provide conditions for stability of simple polytopes under small perturbations of the hyperplanes. Next, we construct a mapping based on...

To determine the optimal set of hyperparameters of a Gaussian process based on a large number of training data, both a linear system and a trace estimation problem must be solved. In this paper, we focus on establishing numerical methods for the case where the covariance matrix is given as the...