We present a mathematical and computational framework for the problem of learning a dynamical system from noisy observations of a few trajectories and subject to side information. Side information is any knowledge we might have about the dynamical system we would like to learn besides trajectory data. It is typically inferred from domain-specific knowledge or … Read more
This paper proposes and analyzes a proximal augmented Lagrangian (NL-IAPIAL) method for solving smooth nonconvex composite optimization problems with nonlinear K-convex constraints, i.e., the constraints are convex with respect to the order given by a closed convex cone K. Each NL-IAPIAL iteration consists of inexactly solving a proximal augmented Lagrangian subproblem by an accelerated composite … Read more
We consider the linearly constrained separable convex optimization problem whose objective function is separable w.r.t. $m$ blocks of variables. A bunch of methods have been proposed and well studied. Specifically, a modified strictly contractive Peaceman-Rachford splitting method (SC-PRCM) has been well studied in the literature for the special case of $m=3$. Based on the modified … Read more
Many key problems in machine learning and data science are routinely modeled as optimization problems and solved via optimization algorithms. With the increase of the volume of data and the size and complexity of the statistical models used to formulate these often ill-conditioned optimization tasks, there is a need for new efficient algorithms able to … Read more
We develop a new interior-point method (IPM) for symmetric-cone optimization, a common generalization of linear, second-order-cone, and semidefinite programming. In contrast to classical IPMs, we update iterates with a geodesic of the cone instead of the kernel of the linear constraints. This approach yields a primal-dual-symmetric, scale-invariant, and line-search-free algorithm that uses just half the … Read more
In this note, we propose a FISTA-type first order algorithm, VAR-FISTA, to solve a composite optimization problem. A distinctive feature of VAR-FISTA is its ability to exploit the convexity of the function in the problem, resulting in an improved iteration complexity when the function is convex compared to when it is nonconvex. The iteration complexity … Read more
This paper investigates the problem of defining an optimal long-term investment strategy, where the investor can exit the investment before maturity without severe loss. Our setting is a multi-period one, where the aim is tomake a plan for allocating all of wealth among the n assets within a time horizon of m periods. In addition, … Read more
This paper develops algorithms for decentralized machine learning over a network, where data are distributed, computation is localized, and communication is restricted between neighbors. A line of recent research in this area focuses on improving both computation and communication complexities. The methods SSDA and MSDA \cite{scaman2017optimal} have optimal communication complexity when the objective is smooth … Read more
This paper establishes the iteration-complexity of an inner accelerated inexact proximal augmented Lagrangian (IAPIAL) method for solving linearly constrained smooth nonconvex composite optimization problems which is based on the classical Lagrangian function and, most importantly, performs a full Lagrangian multiplier update, i.e., no shrinking factor is incorporated on it. More specifically, each IAPIAL iteration consists … Read more
We modify the total-variation-regularized image segmentation model proposed by Chan, Esedoglu and Nikolova [SIAM Journal on Applied Mathematics 66, 2006] by introducing local regularization that takes into account spatial image information. We propose some techniques for defining local regularization parameters, based on the cartoon-texture decomposition of the given image, on the mean and median filters, … Read more