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Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the…
There is a recent surge of interest in nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem in the purpose of efficiency and scalability. Compared with the original convex formulations,…
This paper defines a convertible nonconvex function(CN function for short) and a weak (strong) uniform (decomposable, exact) CN function, proves the optimization conditions for their global solutions and proposes algorithms for solving the…
We derive necessary conditions for locally optimal shapes of a design problem governed by a non-smooth PDE. The main particularity of the state system is the lack of differentiability of the nonlinearity. We work in the framework of the…
In a general Hilbert framework, we consider continuous gradient-like dynamical systems for constrained multiobjective optimization involving non-smooth convex objective functions. Our approach is in the line of a previous work where was…
A strategy is proposed for characterizing the worst-case performance of algorithms for solving nonconvex smooth optimization problems. Contemporary analyses characterize worst-case performance by providing, under certain assumptions on an…
Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis…
This work demonstrates the utility of gradients for the global optimization of certain differentiable functions with many suboptimal local minima. To this end, a principle for generating search directions from non-local quadratic…
In non-smooth stochastic optimization, we establish the non-convergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold $M$…
Sparsity-promoting terms are incorporated into the objective functions of optimal control problems in order to ensure that optimal controls vanish on large parts of the underlying domain. Typical candidates for those terms are integral…
Designing high-performance electric machines that maintain their efficiency and reliability under uncertain material and operating conditions is crucial for industrial applications. In this paper, we present a novel framework for robust…
We consider several classes of highly important semidefinite optimization problems that involve both a convex objective function (smooth or nonsmooth) and additional linear or nonlinear smooth and convex constraints, which are ubiquitous in…
In this paper we study second-order optimality conditions for non-convex set-constrained optimization problems. For a convex set-constrained optimization problem, it is well-known that second-order optimality conditions involve the support…
This paper focus on the minimization of a possibly nonsmooth objective function over the Stiefel manifold. The existing approaches either lack efficiency or can only tackle prox-friendly objective functions. We propose a constraint…
The use of min-max optimization in adversarial training of deep neural network classifiers and training of generative adversarial networks has motivated the study of nonconvex-nonconcave optimization objectives, which frequently arise in…
This paper deals with $\varepsilon$-efficient and $\varepsilon$-properly efficient points with respect to a co-radiant set in vector optimization problems. In the first part of the paper, we establish a new nonlinear separation theorem for…
The paper concerns the investigation of nonconvex and nondifferentiable integral functionals on general Banach spaces, which may not be reflexive and/or separable. Considering two major subdifferentials of variational analysis, we derive…
This book is devoted to finite-dimensional problems of non-convex non-smooth optimization and numerical methods for their solution. The problem of nonconvexity is studied in the book on two main models of nonconvex dependencies: these are…
Non-convex optimization problems have multiple local optimal solutions. Non-convex optimization problems are commonly found in numerous applications. One of the methods recently proposed to efficiently explore multiple local optimal…
The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the…