On Optimal Universal First-Order Methods for Minimizing Heterogeneous Sums

This work considers minimizing a convex sum of functions, each with potentially different structure ranging from nonsmooth to smooth, Lipschitz to non-Lipschitz. Nesterov’s universal fast gradient method provides an optimal black-box first-order method for minimizing a single function that takes advantage of any continuity structure present without requiring prior knowledge. In this paper, we show … Read more

An Improved Unconstrained Approach for Bilevel Optimization

In this paper, we focus on the nonconvex-strongly-convex bilevel optimization problem (BLO). In this BLO, the objective function of the upper-level problem is nonconvex and possibly nonsmooth, and the lower-level problem is smooth and strongly convex with respect to the underlying variable $y$. We show that the feasible region of BLO is a Riemannian manifold. … Read more

An adaptive superfast inexact proximal augmented Lagrangian method for smooth nonconvex composite optimization problems

This work presents an adaptive superfast proximal augmented Lagrangian (AS-PAL) method for solving linearly-constrained smooth nonconvex composite optimization problems. At each iteration, AS-PAL inexactly solves a possibly nonconvex proximal augmented Lagrangian subproblem with prox stepsize chosen aggressively large so as to speed up its termination. An adaptive ACG variant of FISTA, namely R-FISTA, is then … Read more

Convergence to a second-order critical point of composite nonsmooth problems by a trust region method

An algorithm for finding a first-order and second-order critical point of composite nonsmooth problems is proposed in this paper. For smooth problems, algorithms for searching such a point usually utilize the so called negative-curvature directions. In this paper, the method recently proposed for nonlinear semidefinite problems by the current author is extended for solving general … Read more

Multi-fidelity robust controller design with gradient sampling

Robust controllers that stabilize dynamical systems even under disturbances and noise are often formulated as solutions of nonsmooth, nonconvex optimization problems. While methods such as gradient sampling can handle the nonconvexity and nonsmoothness, the costs of evaluating the objective function may be substantial, making robust control challenging for dynamical systems with high-dimensional state spaces. In … Read more

A Proximal Gradient Method for Multi-objective Optimization Problems Using Bregman Functions

In this paper, a globally convergent proximal gradient method is developed for convex multi-objective optimization problems using Bregman distance. The proposed method is free from any kind of a priori chosen parameters or ordering information of objective functions. At every iteration of the proposed method, a subproblem is solved to find a descent direction. This … Read more

Integral Global Optimality Conditions and an Algorithm for Multiobjective Problems

In this work, we propose integral global optimality conditions for multiobjective problems not necessarily differentiable. The integral characterization, already known for single objective problems, are extended to multiobjective problems by weighted sum and Chebyshev weighted scalarizations. Using this last scalarization, we propose an algorithm for obtaining an approximation of the weak Pareto front whose effectiveness … Read more

Identifiability, the KL property in metric spaces, and subgradient curves

Identifiability, and the closely related idea of partial smoothness, unify classical active set methods and more general notions of solution structure. Diverse optimization algorithms generate iterates in discrete time that are eventually confined to identifiable sets. We present two fresh perspectives on identifiability. The first distills the notion to a simple metric property, applicable not … Read more

Escaping Spurious Local Minima of Low-Rank Matrix Factorization Through Convex Lifting

This work proposes a rapid global solver for nonconvex low-rank matrix factorization (MF) problems that we name MF-Global. Through convex lifting steps, our method efficiently escapes saddle points and spurious local minima ubiquitous in noisy real-world data, and is guaranteed to always converge to the global optima. Moreover, the proposed approach adaptively adjusts the rank … Read more

Error Bound and Isocost Imply Linear Convergence of DCA-based Algorithms to D-stationarity

We consider a class of structured nonsmooth difference-of-convex minimization, which can be written as the difference of two convex functions possibly nonsmooth with the second one in the format of the maximum of a finite convex smooth functions. We propose two extrapolation proximal difference-of-convex based algorithms for potential acceleration to converge to a weak/standard d-stationary … Read more