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Assessment of the degree of boundedness/stability of multidimensional nonlinear systems with time-dependent and nonperiodic coefficients is an important problem in various applied areas which has no adequate resolution yet. Most of the…
In this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region. By exploiting an interior point technique, we present first- and…
We establish the consistency of classical scaling under a broad class of noise models, encompassing many commonly studied cases in literature. Our approach requires only finite fourth moments of the noise, significantly weakening standard…
Gradient based optimization algorithms deployed in Machine Learning (ML) applications are often analyzed and compared by their convergence rates or regret bounds. While these rates and bounds convey valuable information they don't always…
Algorithmic stability is a central concept in statistics and learning theory that measures how sensitive an algorithm's output is to small changes in the training data. Stability plays a crucial role in understanding generalization,…
We provide sufficient conditions for instability of the subgradient method with constant step size around a local minimum of a locally Lipschitz semi-algebraic function. They are satisfied by several spurious local minima arising in robust…
We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O.…
Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in…
This paper conducts sensitivity analysis of random constraint and variational systems related to stochastic optimization and variational inequalities. We establish efficient conditions for well-posedness, in the sense of robust Lipschitzian…
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if…
We consider optimization problems with a disjunctive structure of the constraints. Prominent examples of such problems are mathematical programs with equilibrium constraints or vanishing constraints. Based on the concepts of directional…
We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays…
We present stability conditions for deterministic time-varying nonlinear discrete-time systems whose inputs aim to minimize an infinite-horizon time-dependent cost. Global asymptotic and exponential stability properties for general…
This paper provides a set of sensitivity analysis and activity identification results for a class of convex functions with a strong geometric structure, that we coined "mirror-stratifiable". These functions are such that there is a…
The paper introduces and characterizes new notions of Lipschitzian and H\"olderian full stability of solutions to general parametric variational systems described via partial subdifferential and normal cone mappings acting in Hilbert…
Bilevel optimization and bilevel minimax optimization have recently emerged as unifying frameworks for a range of machine-learning tasks, including hyperparameter optimization and reinforcement learning. The existing literature focuses on…
In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool,…
In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not…
A new result on stability of an optimal nonlinear filter with respect to small perturbations on every step is established.
The dynamical stability of the iterates during training plays a key role in determining the minima obtained by optimization algorithms. For example, stable solutions of gradient descent (GD) correspond to flat minima, which have been…