Related papers: Geometric characterizations of Lipschitz stability…
Abstracting neural networks with constraints they impose on their inputs and outputs can be very useful in the analysis of neural network classifiers and to derive optimization-based algorithms for certification of stability and robustness…
This paper introduces and investigates a regularity condition in the asymptotic sense for optimization problems whose objective functions are polynomial. Under this regularity condition, the normalization argument in asymptotic analysis…
We consider the inverse problem of determining the Lam\'{e} parameters and the density of a three-dimensional elastic body from the local time-harmonic Dirichlet-to-Neumann map. We prove uniqueness and Lipschitz stability of this inverse…
In this paper, we consider the inverse problem of recovering an isotropic elastic tensor from the Neumann-to-Dirichlet map. To this end, we prove a Lipschitz stability estimate for Lam\'e parameters with certain regularity assumptions. In…
Optimization of frame structures is formulated as a~non-convex optimization problem, which is currently solved to local optimality. In this contribution, we investigate four optimization approaches: (i) general non-linear optimization, (ii)…
We study stability of solutions of the Cauchy problem on the line for the Camassa-Holm equation $u_t-u_{xxt}+3uu_x-2u_xu_{xx}-uu_{xxx}=0$ with initial data $u_0$. In particular, we derive a new Lipschitz metric $d_\D$ with the property that…
Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations.…
In Hilbert space setting we prove local lipchitzness of projections onto parametric polyhedral sets represented as solutions to systems of inequalities and equations with parameters appearing both in left-hand-sides and right-hand-sides of…
New approaches to the study of stability of solutions of Set Differential Equations (SDEs) based on convex geometry and the theory of mixed volumes were proposed. The stability of the forms of program solutions of linear SDEs with a stable…
We study the iteration complexity of Lipschitz convex optimization problems satisfying a general error bound. We show that for this class of problems, subgradient descent with either Polyak stepsizes or decaying stepsizes achieves minimax…
We show that convex-concave Lipschitz stochastic saddle point problems (also known as stochastic minimax optimization) can be solved under the constraint of $(\epsilon,\delta)$-differential privacy with \emph{strong (primal-dual) gap} rate…
Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function so that along the iterations the objective function decreases. Such a simple principle allows to solve a large…
Quasi-variational inequalities (QVIs) of obstacle type in many cases have multiple solutions that can be ordered. We study a multitude of properties of the operator mapping the source term to the minimal or maximal solution of such QVIs. We…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
In this paper, we study the problem of optimizing the stability of positive semi-Markov jump linear systems. We specifically consider the problem of tuning the coefficients of the system matrices for maximizing the exponential decay rate of…
Composite optimization offers a powerful modeling tool for a variety of applications and is often numerically solved by means of proximal gradient methods. In this paper, we consider fully nonconvex composite problems under only local…
We discuss the problem of deciding when a metrisable topological group $G$ has a canonically defined local Lipschitz geometry. This naturally leads to the concept of minimal metrics on $G$, that we characterise intrinsically in terms of a…
In this paper, we study local convergence of high-order Tensor Methods for solving convex optimization problems with composite objective. We justify local superlinear convergence under the assumption of uniform convexity of the smooth…
Modern data science applications demand solving large-scale optimization problems. The prevalent approaches are first-order methods, valued for their scalability. These methods are implemented to tackle highly irregular problems where…
We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove…