Golden Ratio Algorithms for Variational Inequalities

The paper presents a fully explicit algorithm for monotone variational inequalities. The method uses variable stepsizes that are computed using two previous iterates as an approximation of the local Lipschitz constant without running a linesearch. Thus, each iteration of the method requires only one evaluation of a monotone operator $F$ and a proximal mapping $g$. … Read more

Block Coordinate Descent Almost Surely Converges to a Stationary Point Satisfying the Second-order Necessary Condition

Given a non-convex twice continuously differentiable cost function with Lipschitz continuous gradient, we prove that all of the block coordinate gradient descent, block mirror descent and proximal block coordinate descent methods converge to stationary points satisfying the second-order necessary condition, almost surely with random initialization. All our results are ascribed to the center-stable manifold theorem … Read more

DSCOVR: Randomized Primal-Dual Block Coordinate Algorithms for Asynchronous Distributed Optimization

Machine learning with big data often involves large optimization models. For distributed optimization over a cluster of machines, frequent communication and synchronization of all model parameters (optimization variables) can be very costly. A promising solution is to use parameter servers to store different subsets of the model parameters, and update them asynchronously at different machines … Read more

Chambolle-Pock and Tseng’s methods: relationship and extension to the bilevel optimization

In the first part of the paper we focus on two problems: (a) regularized least squares and (b) nonsmooth minimization over an affine subspace. For these problems we establish the connection between the primal-dual method of Chambolle-Pock and Tseng’s proximal gradient method. For problem (a) it allows us to derive a nonergodic $O(1/k^2)$ convergence rate … Read more

Dynamic Data-Driven Estimation of Non-Parametric Choice Models

We study non-parametric estimation of choice models, which was introduced to alleviate unreasonable assumptions in traditional parametric models, and are prevalent in several application areas. Existing literature focuses only on the static observational setting where all of the observations are given upfront, and lacks algorithms that provide explicit convergence rate guarantees or an a priori … Read more

Second-order cone programming formulation for two player zero-sum game with chance constraints

We consider a two player finite strategic zero-sum game where each player has stochastic linear constraints. We formulate the stochastic constraints of each player as chance constraints. We show the existence of a saddle point equilibrium if the row vectors of the random matrices, defining the stochastic constraints of each player, are elliptically symmetric distributed … Read more

Fixing and extending some recent results on the ADMM algorithm

We first point out several flaws in the recent paper {\it [R. Shefi, M. Teboulle: Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization, SIAM J. Optim. 24, 269–297, 2014]} that proposes two ADMM-type algorithms for solving convex optimization problems involving compositions with linear operators and show … Read more

Preconditioning PDE-constrained optimization with L^1-sparsity and control constraints

PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L^1-terms as well as additional inclusion of box … Read more

A first-order primal-dual algorithm with linesearch

The paper proposes a linesearch for the primal-dual method. Each iteration of the linesearch requires to update only the dual (or primal) variable. For many problems, in particular for regularized least squares, the linesearch does not require any additional matrix-vector multiplications. We prove convergence of the proposed method under the standard assumptions. We also show … Read more

Exploiting Problem Structure in Optimization under Uncertainty via Online Convex Optimization

In this paper, we consider two paradigms that are developed to account for uncertainty in optimization models: robust optimization (RO) and joint estimation-optimization (JEO). We examine recent developments on efficient and scalable iterative first-order methods for these problems, and show that these iterative methods can be viewed through the lens of online convex optimization (OCO). … Read more