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This paper presents an algorithmic study and complexity analysis for solving distributionally robust multistage convex optimization (DR-MCO). We generalize the usual consecutive dual dynamic programming (DDP) algorithm to DR-MCO and propose…
In this paper, we study $\Delta$- convergence of iterations for a sequence of strongly quasi-nonexpansive mappings as well as the strong convergence of the Halpern type regularization of them in Hadamard spaces. Then, we give some their…
We develop a system-theoretic framework for the structured analysis of distributed optimization algorithms with decomposable cost functions. We model such algorithms as a network of interacting dynamical systems and derive tests for…
We discuss recent positive experiences applying convex feasibility algorithms of Douglas--Rachford type to highly combinatorial and far from convex problems.
Recently, several authors have shown local and global convergence rate results for Douglas-Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization…
We aim to provide a general framework of for computational photography that recovers the real scene from imperfect images, via the Deep Nonparametric Convexified Filtering (DNCF). It is consists of a nonparametric deep network to resemble…
We establish a region of convergence for the proto-typical non-convex Douglas-Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration [2] was only able to establish local…
Many iterative methods for solving optimization or feasibility problems have been invented, and often convergence of the iterates to some solution is proven. Under favourable conditions, one might have additional bounds on the distance of…
Although the performance of popular optimization algorithms such as Douglas-Rachford splitting (DRS) and the ADMM is satisfactory in small and well-scaled problems, ill conditioning and problem size pose a severe obstacle to their reliable…
Clustering high-dimensional datasets is hard because interpoint distances become less informative in high-dimensional spaces. We present a clustering algorithm that performs nonlinear dimensionality reduction and clustering jointly. The…
Interacting random matrix systems are fundamental to modern theoretical physics and data science, yet a unified framework for their analysis has been lacking. This work introduces such a universal framework, built upon two novel concepts:…
Proximal operators are now ubiquitous in non-smooth optimization. Since their introduction in the seminal work of Moreau, many papers have shown their effectiveness on a wide variety of problems, culminating in their use to construct…
Decentralized optimization is well studied for smooth unconstrained problems. However, constrained problems or problems with composite terms are an open direction for research. We study structured (or composite) optimization problems, where…
We provide two weakly convergent algorithms for finding a zero of the sum of a maximally monotone operator, a cocoercive operator, and the normal cone to a closed vector subspace of a real Hilbert space. The methods exploit the intrinsic…
We expand upon previous work that examined behavior of the iterated Douglas-Rachford method for a line and a circle by considering two generalizations: that of a line and an ellipse and that of a line together with a $p$-sphere. With…
Generalized inverses play a fundamental role in numerical linear algebra, particularly when matrices are rectangular, singular, or rank deficient. Even when the input matrix is sparse, generalized inverses such as the M-P pseudoinverse are…
The fact that deep neural networks are susceptible to crafted perturbations severely impacts the use of deep learning in certain domains of application. Among many developed defense models against such attacks, adversarial training emerges…
We provide a simple analysis of the Douglas-Rachford splitting algorithm in the context of $\ell^1$ minimization with linear constraints, and quantify the asymptotic linear convergence rate in terms of principal angles between relevant…
Large optimization problems with hard constraints arise in many settings, yet classical solvers are often prohibitively slow, motivating the use of deep networks as cheap "approximate solvers." Unfortunately, naive deep learning approaches…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…