We consider trust-region methods for solving optimization problems where the objective is the sum of a smooth, nonconvex function and a nonsmooth, convex regularizer. We extend the global convergence theory of such methods to include worst-case complexity bounds in the case of unbounded model Hessian growth, and introduce a new, simple nonsmooth trust-region subproblem solver … Read more
This paper deals with the Constrained Riemannian Optimization (CRO) problem, which involves minimizing a function subject to equality and inequality constraints on Riemannian manifolds. The study aims to advance optimization theory in the Riemannian setting by presenting and analyzing a penalty-type method for solving CRO problems. The proposed approach is based on techniques that involve … Read more
In this article, we present several examples of special nonlinear conjugate gradient directions for nonlinear (non-convex) multi-objective optimization. These directions provide a descent direction for the objectives, independent of the line-search. This way, we can provide an algorithm with simple, Armijo-like backtracking and prove convergence to first-order critical points. In contrast to other popular conjugate … Read more
This work considers the nonconvex, nonsmooth problem of minimizing a composite objective of the form $f(g(x))+h(x)$ where the inner mapping $g$ is a smooth finite summation or expectation amenable to variance reduction. In such settings, prox-linear methods can enjoy variance-reduced speed-ups despite the existence of nonsmoothness. We provide a unified convergence theory applicable to a … Read more
Explainable artificial intelligence is one of the most important trends in modern machine-learning research. The idea is to explain the outcome of a model by presenting a certain change in the input of the model so that the outcome changes significantly. In this paper, we study this question for linear optimization problems as an automated … Read more
In this paper we consider an unconstrained stochastic optimization problem where the objective function exhibits a high order of smoothness. In particular, we propose a stochastic first-order method (SFOM) with multi-extrapolated momentum, in which multiple extrapolations are performed in each iteration, followed by a momentum step based on these extrapolations. We show that our proposed … Read more
We consider model-based derivative-free optimization (DFO) for large-scale problems, based on iterative minimization in random subspaces. We provide the first worst-case complexity bound for such methods for convergence to approximate second-order critical points, and show that these bounds have significantly improved dimension dependence compared to standard full-space methods, provided low accuracy solutions are desired and/or … Read more
It is well known that the performance of the classical Markowitz model for portfolio optimization is extremely sensitive to estimation errors on the expected asset returns. Robust optimization mitigates this issue. We focus on ellipsoidal uncertainty sets around a point estimate of the expected asset returns. An important issue is the choice of the parameters … Read more
In this paper, we propose a multilevel stochastic framework for the solution of nonconvex unconstrained optimization problems. The proposed approach uses random regularized first-order models that exploit an available hierarchical description of the problem, being either in the classical variable space or in the function space, meaning that different levels of accuracy for the objective … Read more
This paper introduces an optimization technique for scenario modeling in uncertain business situations, termed the Gradient-Driven Solution Based on Indifference Analysis (GIA). GIA evolves the conventional methods of scenario planning by applying a reverse-strategy approach, where future financial goals are specified, and the path to attain these targets are engineered backward. It adopts economic concepts … Read more