On convergence of solutions to nonlocal optimal control problems with quasi-minimization constraints

Projected Subgradient Methods for a Class of Nonsmooth and Nonconvex Optimization Problems

We investigate the optimization problem of minimizing a nonsmooth function that satisfies a nonsmooth version of the descent lemma over a nonempty and closed but not necessarily convex set. The objective function belongs to the class of upper-$\mathcal{C}^2$ functions, whereas the constraints may promote a sparse or low-rank structure. We propose a projected subgradient method with two different globalization strategies: (a) a nonmonotone linesearch and, under additional assumptions, (b) an auto-conditioned method, where the stepsize is given by a formula depending on data from past iterations. We show that both methods converge to solutions that satisfy a stronger stationarity concept than one would expect from the subdifferential sum-rule, which is particularly important since the optimization problems of interest are inherently nonconvex. Finally, we present promising numerical results when applying the algorithm to an MPEC-style problem as well as the matrix optimization problems MAXCUT and Robust PCA.

A Local-Linearly Convergent Algorithm for Nonconvex Equality-Constrained Optimization

For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.

On Constrained Riesz Minimum Energy Problems

The constrained equilibrium problem for the logarithmic potential was introduced by Rakhmanov (1996) as he realized that the asymptotic distribution of the zeros of discrete orthogonal polynomials in a compact interval could be described in terms of the equilibrium measure of this interval in a class of measures subject to a certain constraint. Other authors such as Dragnev and Saff (1997), and Kuijlaars and Van Assche (1999), extended this approach to more general settings. These problems have proven to be useful for describing asymptotic distributions in different settings. In the current paper, we consider constrained equilibrium problems in the Riesz setting, that is, for $s$-Riesz potentials in the hyperplane $\mathbb{R}^d,$ with $d\geq 1$ and $\max (0,d-2)<s<d$. Along with some general results, an illustrative example consisting of the solution of a constrained equilibrium problem in the unit ball is studied in detail. This is the main part of the paper. Finally, a number of numerical experiments show how the constrained equilibrium measure, the solution of the problem, may be discretized using the so-called'constrained Leja points'introduced by Coroian and Dragnev (2001).

A single loop method for quadratic minmax optimization

This work considers a quadratic minmax problem with coupled inner constraints and proposes a method to compute a class of stationary points and shows in particular that the method is polynomial in the special case where the inner feasible set of the authors' constrained minmax problem is independent from outer variables.

New inexact adaptive proximal gradient algorithms for nonconvex composite optimization problems

A geometric approach to nonlocal 2-Hessian equations

We study a nonlocal 2-Hessian equation, given by an infimum of linear deformations of the fractional Laplacian, that is, $\inf_{A\in {A}_2}\Delta^s (u\circ A)(A^{-1}x)$. We characterize the class of coefficient matrices ${A}_2$, which determines the behavior of the operator, and provide a detailed geometric description of the possible degeneracies. Our main theorem shows that the nonlocal 2-Hessian equation remains uniformly elliptic for strictly positive right-hand sides, which leads to regularity estimates. The results hold under weaker hypotheses than those previously considered in the literature, and for the full range $s\in(0,1)$. In particular, no convexity on the solutions is required. Our hypotheses can be interpreted as nonlocal counterparts of the local notions of semiconcavity and 2-convexity. Moreover, all the results are stable as $s\to1$, recovering the local case. The geometric methods developed here are new, even in the local setting, and may be relevant to a broader class of nonlocal fully nonlinear equations and curvature-type problems.