We revisit the Ravine method of Gelfand and Tsetlin from a dynamical system perspective, study its convergence properties, and highlight its similarities and differences with the Nesterov accelerated gradient method. The two methods are closely related. ...

Convex representatives are proposed for the value function of an infinite-dimensional constrained nonconvex variational problem. All the involved variables in this problem take their values in (possibly of infinite dimension, not necessarily separable or ...

We correct an error in the proof of Theorem 3.1 in [E. Casas and M. Mateos, Critical cones for sufficient second order conditions in PDE constrained optimization, SIAM J. Optim., 30 (2020), pp. 585--603]. With this correction, all results in that paper ...

Conjugate gradient methods are important first-order optimization algorithms both in Euclidean spaces and on Riemannian manifolds. However, while various types of conjugate gradient methods have been studied in Euclidean spaces, there are relatively fewer ...

We present a model-based derivative-free method for optimization subject to general convex constraints, which we assume are unrelaxable and accessed only through a projection operator that is cheap to evaluate. We prove global convergence and a worst-case ...

We consider the consensus interdiction problem (CIP), in which the goal is to maximize the convergence time of consensus averaging dynamics subject to removing a limited number of network edges. We first show that CIP can be cast as an effective ...

Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set $\mathbf{X} \subseteq \mathbb{R}^n$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-...

Newton differentiability is an important concept for analyzing generalized Newton methods for nonsmooth equations. In this work, for a convex function defined on an infinite-dimensional space, we discuss the relation between Newton and Bouligand ...

In this paper we describe a systematic procedure to analyze the convergence of degenerate preconditioned proximal point algorithms. We establish weak convergence results under mild assumptions that can be easily employed in the context of splitting methods ...

Ben-Tal and Teboulle [Management Sci., 32 (1986), pp. 1445--1466] introduce the concept of optimized certainty equivalent (OCE) of an uncertain outcome as the maximum present value of a combination of the cash to be taken out from the uncertain income at ...

We study multistage distributionally robust optimization (DRO) to hedge against ambiguity in quantifying the underlying uncertainty of a problem. Recognizing that not all the realizations and scenario paths might have an “effect” on the optimal value, we ...

In this paper, we compute the tightest possible bounds on the probability that the optimal value of a combinatorial optimization problem in maximization form with a random objective exceeds a given number, assuming only knowledge of the marginal ...

We provide a proximal average with respect to a 1-coercive Legendre function. In the sense of Bregman distance, the Bregman envelope of the proximal average is a convex combination of Bregman envelopes of individual functions. The Bregman proximal mapping of ...

In this paper, we propose two new solution schemes to solve the stochastic strongly monotone variational inequality (VI) problems: the stochastic extra-point solution scheme and the stochastic extra-momentum solution scheme. The first one is a general scheme ...

In this paper, we study a solution approach for set optimization problems with respect to the lower set less relation. This approach can serve as a base for numerically solving set optimization problems by using established solvers from multiobjective ...

The conjugate gradient method (CG) has long been the workhorse for inner-iterations of second-order algorithms for large-scale nonconvex optimization. Prominent examples include line-search based algorithms, e.g., Newton-CG, and those based on a trust-...

Random projection techniques based on the Johnson--Lindenstrauss lemma are used for randomly aggregating the constraints or variables of optimization problems while approximately preserving their optimal values, which leads to smaller-scale optimization ...

We consider the problem of minimizing a continuously differentiable function $f$ of $m$ linear forms in $n$ variables on the Euclidean unit sphere. We show that this problem is equivalent to minimizing the same function of related $m$ linear forms (but now ...

We focus on convex semi-infinite programs with an infinite number of quadratically parametrized constraints. In our setting, the lower-level problem, i.e., the problem of finding the constraint that is the most violated by a given point, is not necessarily ...

Recent work has shown that stochastically perturbed gradient methods can efficiently escape strict saddle points of smooth functions. We extend this body of work to nonsmooth optimization, by analyzing an inexact analogue of a stochastically perturbed ...