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Quasi-Newton methods form an important class of methods for solving nonlinear optimization problems. In such methods, first order information is used to approximate the second derivative. The aim is to mimic the fast convergence that can be…

A number of variable selection methods have been proposed involving nonconvex penalty functions. These methods, which include the smoothly clipped absolute deviation (SCAD) penalty and the minimax concave penalty (MCP), have been…

The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an…

The method of block coordinate gradient descent (BCD) has been a powerful method for large-scale optimization. This paper considers the BCD method that successively updates a series of blocks selected according to a Markov chain. This kind…

This monograph presents a class of algorithms called coordinate descent algorithms for mathematicians, statisticians, and engineers outside the field of optimization. This particular class of algorithms has recently gained popularity due to…

Recent literature has advocated the use of randomized methods for accelerating the solution of various matrix problems arising throughout data science and computational science. One popular strategy for leveraging randomization is to use it…

In this paper, we aim to study non-convex minimization problems via second-order (in-time) dynamics, including a non-vanishing viscous damping and a geometric Hessian-driven damping. Second-order systems that only rely on a viscous damping…

The curse of dimensionality presents a pervasive challenge in optimization problems, with exponential expansion of the search space rapidly causing traditional algorithms to become inefficient or infeasible. An adaptive sampling strategy is…

Symplectic numerical methods have become a widely-used choice for the accurate simulation of Hamiltonian systems in various fields, including celestial mechanics, molecular dynamics and robotics. Even though their characteristics are…

Stochastic optimization methods have been hugely successful in making large-scale optimization problems feasible when computing the full gradient is computationally prohibitive. Using the theory of modified equations for numerical…

Selecting the best hyperparameters for a particular optimization instance, such as the learning rate and momentum, is an important but nonconvex problem. As a result, iterative optimization methods such as hypergradient descent lack global…

This paper focusses on the optimal control problems governed by fourth-order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretisation method (HDM). The HDM is an abstract framework that…

We develop algorithms that find and track the optimal solution trajectory of time-varying convex optimization problems which consist of local and network-related objectives. The algorithms are derived from the prediction-correction…

We study online inference and asymptotic covariance estimation for the stochastic gradient descent (SGD) algorithm. While classical methods (such as plug-in and batch-means estimators) are available, they either require inaccessible…

We develop a new Riemannian descent algorithm that relies on momentum to improve over existing first-order methods for geodesically convex optimization. In contrast, accelerated convergence rates proved in prior work have only been shown to…

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…

We study the theoretical convergence properties of random-search methods when optimizing non-convex objective functions without having access to derivatives. We prove that standard random-search methods that do not rely on second-order…

The Douglas-Rachford (DR) method is a widely used method for finding a point in the intersection of two closed convex sets (feasibility problem). However, the method converges weakly and the associated rate of convergence is hard to analyze…

With the development of machine learning and Big Data, the concepts of linear and non-linear optimization techniques are becoming increasingly valuable for many quantitative disciplines. Problems of that nature are typically solved using…

Distributed optimization is widely used in large-scale and privacy-preserving machine learning, where each agent stores a local objective and communicates only with its neighbors in a connected network. We study decentralized second-order…