Local Convergence Analysis of an Inexact Trust-Region Method for Nonsmooth Optimization

In [R. J. Baraldi and D. P. Kouri, Mathematical Programming, (2022), pp. 1–40], we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex function and a nonsmooth convex function in Hilbert space—a class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. This algorithm has demonstrated … Read more

Efficient Proximal Subproblem Solvers for a Nonsmooth Trust-Region Method

In [R. J. Baraldi and D. P. Kouri, Mathematical Programming, (2022), pp. 1-40], we introduced an inexact trust-region algorithm for minimizing the sum of a smooth nonconvex and nonsmooth convex function. The principle expense of this method is in computing a trial iterate that satisfies the so-called fraction of Cauchy decrease condition—a bound that ensures … Read more

Exact convergence rate of the last iterate in subgradient methods

\(\) We study the convergence of the last iterate in subgradient methods applied to the minimization of a nonsmooth convex function with bounded subgradients. We first introduce a proof technique that generalizes the standard analysis of subgradient methods. It is based on tracking the distance between the current iterate and a different reference point at … Read more

The complexity of first-order optimization methods from a metric perspective

A central tool for understanding first-order optimization algorithms is the Kurdyka-Lojasiewicz inequality. Standard approaches to such methods rely crucially on this inequality to leverage sufficient decrease conditions involving gradients or subgradients. However, the KL property fundamentally concerns not subgradients but rather “slope”, a purely metric notion. By highlighting this view, and avoiding any use of … Read more

On a Frank-Wolfe Approach for Abs-smooth Functions

We propose an algorithm which appears to be the first bridge between the fields of conditional gradient methods and abs-smooth optimization. Our problem setting is motivated by various applications that lead to nonsmoothness, such as $\ell_1$ regularization, phase retrieval problems, or ReLU activation in machine learning. To handle the nonsmoothness in our problem, we propose … Read more

On the Optimization Landscape of Burer-Monteiro Factorization: When do Global Solutions Correspond to Ground Truth?

In low-rank matrix recovery, the goal is to recover a low-rank matrix, given a limited number of linear and possibly noisy measurements. Low-rank matrix recovery is typically solved via a nonconvex method called Burer-Monteiro factorization (BM). If the rank of the ground truth is known, BM is free of sub-optimal local solutions, and its true solutions … Read more

Minimizing the difference of convex and weakly convex functions via bundle method

We consider optimization problems with objective and constraint being the difference of convex and weakly convex functions. This framework covers a vast family of nonsmooth and nonconvex optimization problems, particularly those involving certain classes of composite and nonconvex value functions. We investigate several stationary conditions and extend the proximal bundle algorithm of [van Ackooij et … Read more

A Levenberg-Marquardt Method for Nonsmooth Regularized Least Squares

\(\) We develop a Levenberg-Marquardt method for minimizing the sum of a smooth nonlinear least-squares term \(f(x) = \frac{1}{2} \|F(x)\|_2^2\) and a nonsmooth term \(h\). Both \(f\) and \(h\) may be nonconvex. Steps are computed by minimizing the sum of a regularized linear least-squares model and a model of \(h\) using a first-order method such … Read more

The cost of nonconvexity in deterministic nonsmooth optimization

\(\) We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a black-box algorithm of Zhang et al. that approximates an $\epsilon$-stationary point of any directionally differentiable Lipschitz objective using $O(\epsilon^{-4})$ calls … Read more

The projective exact penalty method for general constrained optimization

A new projective exact penalty function method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing its value at the projection of an infeasible point on the feasible set with the distance to the projection. The … Read more