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In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to…

In this paper, we generalize the chance optimization problems and introduce constrained volume optimization where enables us to obtain convex formulation for challenging problems in systems and control. We show that many different problems…

We provide a simple proof of the radial symmetry of any nonnegative minimizer for a general class of quasi-linear minimization problems

We prove partial regularity for minimizers of vectorial integrals of the Calculus of Variations, with general growth condition, imposing quasiconvexity assumptions only in an asymptotic sense.

We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of…

We prove topological regularity results for isoperimetric sets in PI spaces having a suitable deformation property, which prescribes a control on the increment of the perimeter of sets under perturbations with balls. More precisely, we…

Inspired by a planar partitioning problem involving multiple improper chambers, this article investigates using classical techniques what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint…

This article deals with the variable coefficient thin obstacle problem in $n+1$ dimensions. We address the regular free boundary regularity, the behavior of the solution close to the free boundary and the optimal regularity of the solution…

In this paper, we establish an $\varepsilon$-regularity theorem for minimizers of an Alt-Phillips type functional subject to constraint maps. We prove that under sufficiently small energy, the minimizers exhibit regularity, and hence…

Common computational problems, such as parameter estimation in dynamic models and PDE constrained optimization, require data fitting over a set of auxiliary parameters subject to physical constraints over an underlying state. Naive…

We establish lower semicontinuity results for perimeter functionals with measure data on $\mathbb{R}^n$ and deduce the existence of minimizers to these functionals with Dirichlet boundary conditions, obstacles, or volume-constraints. In…

As a starting point of our research, we show that, for a fixed order $\gamma\geq 1$, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to…

In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is…

For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…

In this paper, we consider the asymptotical regularization with convex constraints for nonlinear ill-posed problems. The method allows to use non-smooth penalty terms, including the L1-like and the total variation-like penalty functionals,…

We establish a regularity result for optimal sets of the isoperimetric problem with double density under mild ($\alpha$-)H\"older regularity assumptions on the density functions. Our main Theorem improves some previous results and allows to…

This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincare inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with…

PDE-constrained optimal control problems require regularisation to ensure well-posedness, introducing small perturbations that make the solutions challenging to approximate accurately. We propose a finite element approach that couples both…

In this paper we present a general convex optimization approach for solving high-dimensional multiple response tensor regression problems under low-dimensional structural assumptions. We consider using convex and weakly decomposable…

Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…