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Many problems in geometric optics or convex geometry can be recast as optimal transport problems: this includes the far-field reflector problem, Alexandrov's curvature prescription problem, etc. A popular way to solve these problems…
Optimization of convex functions subject to eigenvalue constraints is intriguing because of peculiar analytical properties of eigenvalues, and is of practical interest because of wide range of applications in fields such as structural…
Adam is a popular variant of stochastic gradient descent for finding a local minimizer of a function. In the constant stepsize regime, assuming that the objective function is differentiable and non-convex, we establish the convergence in…
We propose a trust-region type method for a class of nonsmooth nonconvex optimization problems where the objective function is a summation of a (probably nonconvex) smooth function and a (probably nonsmooth) convex function. The model…
We study the smooth structure of convex functions by generalizing a powerful concept so-called self-concordance introduced by Nesterov and Nemirovskii in the early 1990s to a broader class of convex functions, which we call generalized…
We provide improved complexity results for symmetric primal--dual interior-point algorithms in conic optimization. The results follow from new uniform bounds on a key complexity measure for primal--dual metrics at pairs of primal and dual…
This paper addresses the study of a new class of nonsmooth optimization problems, where the objective is represented as a difference of two generally nonconvex functions. We propose and develop a novel Newton-type algorithm to solving such…
Developing a contemporary optimal transport (OT) solver requires navigating trade-offs among several critical requirements: GPU parallelization, scalability to high-dimensional problems, theoretical convergence guarantees, empirical…
This work introduces a new cubic regularization method for nonconvex unconstrained multiobjective optimization problems. At each iteration of the method, a model associated with the cubic regularization of each component of the objective…
The convergence rate of stochastic gradient search is analyzed in this paper. Using arguments based on differential geometry and Lojasiewicz inequalities, tight bounds on the convergence rate of general stochastic gradient algorithms are…
We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the…
The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm…
This paper proposes and justifies two globally convergent Newton-type methods to solve unconstrained and constrained problems of nonsmooth optimization by using tools of variational analysis and generalized differentiation. Both methods are…
Several optimization schemes have been known for convex optimization problems. However, numerical algorithms for solving nonconvex optimization problems are still underdeveloped. A progress to go beyond convexity was made by considering the…
In this paper, we propose a Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The objective function of the problem under consideration is given by…
In this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region. By exploiting an interior point technique, we present first- and…
We consider minimization of a smooth nonconvex objective function using an iterative algorithm based on Newton's method and the linear conjugate gradient algorithm, with explicit detection and use of negative curvature directions for the…
The stochastic gradient descent has been widely used for solving composite optimization problems in big data analyses. Many algorithms and convergence properties have been developed. The composite functions were convex primarily and…
This paper proposes a novel CTA (Combine-Then-Adapt)-based decentralized algorithm for solving convex composite optimization problems over undirected and connected networks. The local loss function in these problems contains both smooth and…
We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that…