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Training and fine-tuning large language models (LLMs) come with challenges related to memory and computational requirements due to the increasing size of the model weights and the optimizer states. Various techniques have been developed to…
Approximate linear programming (ALP) is an efficient approach to solving large factored Markov decision processes (MDPs). The main idea of the method is to approximate the optimal value function by a set of basis functions and optimize…
In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel…
In this paper, we consider nonconvex optimization problems with nonsmooth nonconvex objective function and nonlinear equality constraints. We assume that both the objective function and the functional constraints can be separated into 2…
This work proposes a general learned proximal alternating minimization algorithm, LPAM, for solving learnable two-block nonsmooth and nonconvex optimization problems. We tackle the nonsmoothness by an appropriate smoothing technique with…
To construct a parallel approach for solving optimization problems with orthogonality constraints is usually regarded as an extremely difficult mission, due to the low scalability of the orthonormalization procedure. However, such demand is…
This work proposes a rapid algorithm, BM-Global, for nuclear-norm-regularized convex and low-rank matrix optimization problems. BM-Global efficiently decreases the objective value via low-cost steps leveraging the nonconvex but smooth…
We consider constructing a surface from a given set of point cloud data. We explore two fast algorithms to minimize the weighted minimum surface energy in [Zhao, Osher, Merriman and Kang, Comp.Vision and Image Under., 80(3):295-319, 2000].…
We study the computational complexity certification of inexact gradient augmented Lagrangian methods for solving convex optimization problems with complicated constraints. We solve the augmented Lagrangian dual problem that arises from the…
The Alternating Direction Method of Multipliers (ADMM) has now days gained tremendous attentions for solving large-scale machine learning and signal processing problems due to the relative simplicity. However, the two-block structure of the…
Alternating gradient-descent-ascent (AltGDA) is an optimization algorithm that has been widely used for model training in various machine learning applications, which aims to solve a nonconvex minimax optimization problem. However, the…
Euler's elastica model has been extensively studied and applied to image processing tasks. However, due to the high nonlinearity and nonconvexity of the involved curvature term, conventional algorithms suffer from slow convergence and high…
We present a primal-dual majorization-minimization method for solving large-scale linear programs. A smooth barrier augmented Lagrangian (SBAL) function with strict convexity for the dual linear program is derived. The…
In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a…
We study a generalized nonconvex Burer-Monteiro formulation for low-rank minimization problems. We use recent results on non-Euclidean first order methods to provide efficient and scalable algorithms. Our approach uses geometries induced by…
Low-rank plus diagonal (LRPD) decompositions provide a powerful structural model for large covariance matrices, simultaneously capturing global shared factors and localized corrections that arise in covariance estimation, factor analysis,…
We study connections between Dykstra's algorithm for projecting onto an intersection of convex sets, the augmented Lagrangian method of multipliers or ADMM, and block coordinate descent. We prove that coordinate descent for a regularized…
This paper provides an overview of the historical progression of distributed optimization techniques, tracing their development from early duality-based methods pioneered by Dantzig, Wolfe, and Benders in the 1960s to the emergence of the…
This paper presents a new approach and algorithm for solving a class of constrained Bi-Level Optimization (BLO) problems in which the lower-level problem involves constraints coupling both upper-level and lower-level variables. Such…
In this paper, we develop a variant of the well-known Gauss-Newton (GN) method to solve a class of nonconvex optimization problems involving low-rank matrix variables. As opposed to the standard GN method, our algorithm allows one to handle…