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We study the computation of competitive equilibrium for Fisher markets with n agents and m divisible chores. Prior work showed that competitive equilibria correspond to the nonzero KKT points of the Nash welfare minimization program, which is a non-...

Nonlinearly constrained nonconvex and nonsmooth optimization models play an increasingly important role in machine learning, statistics, and data analytics. In this paper, based on the augmented Lagrangian function, we introduce a flexible first-order ...

First-order methods (FOMs) have recently been applied and analyzed for solving problems with complicated functional constraints. Existing works show that FOMs for functional constrained problems have lower-order convergence rates than those for ...

The conditions of relative smoothness and relative strong convexity were recently introduced for the analysis of Bregman gradient methods for convex optimization. We introduce a generalized left-preconditioning method for gradient descent and show that ...

In this paper, we propose a primal-dual algorithm with a novel momentum term using the partial gradients of the coupling function that can be viewed as a generalization of the method proposed by Chambolle and Pock in [Math. Program., 159 (2016), pp. 253--287]...

The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant $L$. However, in many settings the differentiable ...

We consider the extragradient method to minimize the sum of two functions, the first one being smooth and the second being convex. Under the Kurdyka---źojasiewicz assumption, we prove that the sequence produced by the extragradient method converges to a ...

Nesterov's momentum trick is famously known for accelerating gradient descent, and has been proven useful in building fast iterative algorithms. However, in the stochastic setting, counterexamples exist and prevent Nesterov's momentum from providing ...

We study the round complexity of minimizing the average of convex functions under a new setting of distributed optimization where each machine can receive two subsets of functions. The first subset is from a random partition and the second subset is ...

Motivated by big data applications, first-order methods have been extremely popular in recent years. However, naive gradient methods generally converge slowly. Hence, much effort has been made to accelerate various first-order methods. This paper ...

A conservative invariant domain preserving arbitrary Lagrangian Eulerian method for solving nonlinear hyperbolic systems is introduced. The method is explicit in time, works with continuous finite elements, and is first-order accurate in space. One original ...

We propose a numerical method for solving general hyperbolic systems in any space dimension using forward Euler time stepping and continuous finite elements on nonuniform grids. The properties of the method are based on the introduction of an artificial ...

A small improvement in the structure of a material could potentially lower manufacturing costs. Thus, the free material design can be formulated as an optimization problem. However, due to its large scale, second-order methods cannot solve the free material ...

We introduce a flexible optimization framework for nuclear norm minimization of matrices with linear structure, including Hankel, Toeplitz, and moment structures and catalog applications from diverse fields under this framework. We discuss various first-...