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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 introduces two decomposition-based methods for two-block mixed-integer linear programs (MILPs), which aim to take advantage of separable structures of the original problem by solving a sequence of lower-dimensional MILPs. The…

We develop two new variants of alternating direction methods of multipliers (ADMM) and two parallel primal-dual decomposition algorithms to solve a wide range class of constrained convex optimization problems. Our approach relies on a novel…

Also known as likelihood-free methods, approximate Bayesian computational (ABC) methods have appeared in the past ten years as the most satisfactory approach to untractable likelihood problems, first in genetics then in a broader spectrum…

We present an acceleration method for sequences of large-scale linear systems, such as the ones arising from the numerical solution of time-dependent partial differential equations coupled with algebraic constraints. We discuss different…

Large-scale constrained optimization is pivotal in modern scientific, engineering, and industrial computation, often involving complex systems with numerous variables and constraints. This paper provides a unified and comprehensive…

Stochastic gradient methods are among the most widely used algorithms for large-scale optimization and machine learning. A key technique for improving the statistical efficiency and stability of these methods is the use of averaging schemes…

Recently, inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications. After the discretization, many of inverse problems are reduced to linear systems.…

The approximation of tensors has important applications in various disciplines, but it remains an extremely challenging task. It is well known that tensors of higher order can fail to have best low-rank approximations, but with an important…

In the past two decades, some major efforts have been made to reduce exact (e.g. integer, rational, polynomial) linear algebra problems to matrix multiplication in order to provide algorithms with optimal asymptotic complexity. To provide…

The Barzilai-Borwein (BB) method is an effective gradient descent algorithm for solving unconstrained optimization problems. Based on the observation of two classical BB step sizes, by constructing an interpolated least squares model, we…

Tabled evaluation is a recognized and powerful technique that overcomes some limitations of traditional Prolog systems in dealing with recursion and redundant sub-computations. We can distinguish two main categories of tabling mechanisms:…

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…

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered, the starting point being the classical Bjorck-Pereyra algorithms for…

In this paper, we propose an acceleration framework for a class of iterative methods using the Reduced Order Method (ROM). Assuming that the underlying iterative scheme generates a rich basis for the solution space, we construct the next…

Lagrangian-based methods are classical methods for solving convex optimization problems with equality constraints. We present novel prediction-correction frameworks for such methods and their variants, which can achieve $O(1/k)$ non-ergodic…

In the following article we consider approximate Bayesian computation (ABC) for certain classes of time series models. In particular, we focus upon scenarios where the likelihoods of the observations and parameter are intractable, by which…

The Alternating Direction Method of Multipliers (ADMM) has been studied for years. The traditional ADMM algorithm needs to compute, at each iteration, an (empirical) expected loss function on all training examples, resulting in a…

This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization…

In this paper, we introduce principal asymmetric least squares (PALS) as a unified framework for linear and nonlinear sufficient dimension reduction. Classical methods such as sliced inverse regression (Li, 1991) and principal support…