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New iterative methods for solving linear equations are presented that are easy to use, generalize good existing methods, and appear to be faster. The new algorithms mix two kinds of linear recurrence formulas. Older methods have either high…

Numerical Analysis · Mathematics 2012-03-13 Joseph F. Grcar

We consider strongly convex optimization problems with affine-type restrictions. We build dual problem and solve dual problem by Fast Gradient Method. We use primal-dual structure of this method to construct the solution of the primal…

Optimization and Control · Mathematics 2017-06-23 Anton Anikin , Alexander Gasnikov , Pavel Dvurechensky , Alexander Turin , Alexey Chernov

We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our…

Combinatorics · Mathematics 2025-01-31 Samuel Fiorini , Gwenaël Joret , Stefan Weltge , Yelena Yuditsky

We develop a first-order accelerated algorithm for a class of constrained bilinear saddle-point problems with applications to network systems. The algorithm is a modified time-varying primal-dual version of an accelerated mirror-descent…

Optimization and Control · Mathematics 2024-10-04 Weijian Li , Xianlin Zeng , Lacra Pavel

Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates…

Optimization and Control · Mathematics 2024-03-29 Richard Cole , Christoph Hertrich , Yixin Tao , László A. Végh

In this paper, we propose a new decomposition approach named the proximal primal dual algorithm (Prox-PDA) for smooth nonconvex linearly constrained optimization problems. The proposed approach is primal-dual based, where the primal step…

Optimization and Control · Mathematics 2016-04-05 Mingyi Hong

We consider a class of optimization problems that involve determining the maximum value that a function in a particular class can attain subject to a collection of difference constraints. We show that a particular linear programming…

Data Structures and Algorithms · Computer Science 2022-11-16 Sungjin Im , Benjamin Moseley , Hung Q. Ngo , Kirk Pruhs , Alireza Samadian

Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…

Symbolic Computation · Computer Science 2019-05-14 Seung Gyu Hyun , Vincent Neiger , Éric Schost

We propose a scalable, efficient and statistically motivated computational framework for Graphical Lasso (Friedman et al., 2007b) - a covariance regularization framework that has received significant attention in the statistics community…

Machine Learning · Statistics 2011-10-26 Rahul Mazumder , Deepak K. Agarwal

In this paper we propose a class of randomized primal-dual methods to contend with large-scale saddle point problems defined by a convex-concave function $\mathcal{L}(\mathbf{x},y)\triangleq\sum_{i=1}^m f_i(x_i)+\Phi(\mathbf{x},y)-h(y)$. We…

Optimization and Control · Mathematics 2023-03-17 E. Yazdandoost Hamedani , A. Jalilzadeh , N. S. Aybat

We consider a class of linear programs on graphs with total variation regularization and a budgetary constraint. For these programs, we give a characterization of basic solutions in terms of rooted spanning forests with orientation on the…

Optimization and Control · Mathematics 2026-05-20 Dominic Yang

Primal-dual splitting schemes are a class of powerful algorithms that solve complicated monotone inclusions and convex optimization problems that are built from many simpler pieces. They decompose problems that are built from sums, linear…

Optimization and Control · Mathematics 2015-07-31 Damek Davis

Sum of squares (SOS) optimization is a powerful technique for solving problems where the positivity of a polynomials must be enforced. The common approach to solve an SOS problem is by relaxation to a Semidefinite Program (SDP). The main…

Optimization and Control · Mathematics 2024-10-29 Daniel Keren , Margarita Osadchy , Roi Poranne

We consider empirical risk minimization of linear predictors with convex loss functions. Such problems can be reformulated as convex-concave saddle point problems, and thus are well suitable for primal-dual first-order algorithms. However,…

Optimization and Control · Mathematics 2017-03-09 Jialei Wang , Lin Xiao

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what…

Optimization and Control · Mathematics 2014-05-23 Niao He , Anatoli Juditsky , Arkadi Nemirovski

For the solution of discrete ill-posed problems, in this paper a novel preconditioned iterative method based on the Arnoldi algorithm for matrix functions is presented. The method is also extended to work in connection with Tikhonov…

Numerical Analysis · Mathematics 2011-11-18 Paolo Novati , Michela Redivo-Zaglia , Maria Rosaria Russo

This work presents a universal accelerated first-order primal-dual method for affinely constrained convex optimization problems. It can handle both Lipschitz and H\"{o}lder gradients but does not need to know the smoothness level of the…

Optimization and Control · Mathematics 2022-11-09 Hao Luo

Optimization methods are at the core of many problems in signal/image processing, computer vision, and machine learning. For a long time, it has been recognized that looking at the dual of an optimization problem may drastically simplify…

Numerical Analysis · Computer Science 2014-12-04 Nikos Komodakis , Jean-Christophe Pesquet

This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…

Numerical Analysis · Mathematics 2024-06-11 Dongping Li , Xue Wang , Xiuying Zhang

Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…