This work introduces an inexact cubic regularization method with adaptive reuse of Hessian approximations to solve general non-convex optimization problems. In the proposed approach, the gradient is computed inexactly and updated at every iteration, whereas the Hessian approximation is updated at a specific iteration and then reused for $m$ subsequent iterations (a lazy strategy), where … Read more
We propose and analyze an inexact proximal point method for minimizing locally Lipschitz functions on Euclidean spaces with a piecewise star-convex structure. More precisely, the space is covered by finitely many closed convex sets, and on each set the objective function satisfies a star-convex inequality with respect to the minimizers of its restriction. This class … Read more
We develop a trust-region method for efficiently minimizing the sum of a smooth function, a nonsmooth convex function, and the composition of a finite-valued support function with a smooth function. Optimization problems with this structure arise in numerous applications including risk-averse stochastic programming and subproblems for nonsmooth penalty nonlinear programming methods. Our method permits the … Read more
An algorithm is proposed, analyzed, and tested for minimizing locally Lipschitz objective functions that may be nonconvex and/or nonsmooth. The algorithm, which is built upon the gradient-sampling methodology, is designed specifically for cases when objective function and generalized gradient values might be subject to bounded uncontrollable errors. Similarly to state-of-the-art guarantees for noisy smooth optimization … Read more
In this paper, we propose a descent method for composite optimization problems with linear operators. Specifically, we first design a structure-exploiting preconditioner tailored to the linear operator so that the resulting preconditioned proximal subproblem admits a closed-form solution through its dual formulation. However, such a structure-driven preconditioner may be poorly aligned with the local curvature … Read more
We study generalized variational inequalities with a three-level hierarchical structure. This setting extends nested GVI models beyond the bilevel case, for which $\mathcal{O}(\delta^{-4})$ complexity bounds are known for any prescribed positive tolerance $\delta$, to a fully three-level hierarchical structure. We analyze a projected averaged subgradient method combined with a Tikhonov-like regularization scheme. Under compactness, maximal … Read more
We describe inexact proximal Newton-like methods for solving degenerate regularized optimization problems and for the broader problem of finding a zero of a generalized equation that is the sum of a continuous map and a maximal monotone operator. Superlinear convergence for both the distance to the solution set and a certain measure of first-order optimality … Read more
We generalize tilt stability, a fundamental concept in perturbation analysis of optimization problems in Euclidean spaces, to the setting of Riemannian manifolds. We prove the equivalence of the following conditions: Riemannian tilt stability, Riemannian variational strong convexity, Riemannian uniform quadratic growth, local strong monotonicity of Riemannian subdifferential, strong metric regularity of Riemannian subdifferential, and positive … Read more
This paper proposes an improved quasi-Newton penalty decomposition algorithm for the minimization of continuously differentiable functions, possibly nonconvex, over sparse symmetric sets. The method solves a sequence of penalty subproblems approximately via a two-block decomposition scheme: the first subproblem admits a closed-form solution without sparsity constraints, while the second subproblem is handled through an efficient … Read more
We analyze the problem of identifying large cliques in graphs that are affected by adversarial uncertainty. More specifically, we consider a new formulation, namely the adversarial maximum clique problem, which extends the classical maximum-clique problem to graphs with edges strategically perturbed by an adversary. The proposed mathematical model is thus formulated as a two-player zero-sum … Read more