Randomized Roundings for a Mixed-Integer Elliptic Control System

\(\) We present randomized reconstruction approaches for optimal solutions to mixed-integer elliptic PDE control systems. Approximation properties and relations to sum-up rounding are derived using the cut norm. This enables us to dispose of space-filling curves required for sum-up rounding. Rates of almost sure convergence in the cut norm and the SUR norm in control … Read more

Finding Groups with Maximum Betweenness Centrality via Integer Programming with Random Path Sampling

One popular approach to access the importance/influence of a group of nodes in a network is based on the notion of centrality. For a given group, its group betweenness centrality is computed, first, by evaluating a ratio of shortest paths between each node pair in a network that are “covered” by at least one node … Read more

Multi-stage robust optimization problems: A sampled scenario tree based approach

In this paper, we consider multi-stage robust convex optimization problems of the minimax type. We assume that the total uncertainty set is the cartesian product of stagewise compact uncertainty sets and approximate the given problem by a sampled subproblem. Instead of looking for the worst case among the infinite and typically uncountable set of uncertain … 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

Coordinate descent algorithms

Coordinate descent algorithms solve optimization problems by successively performing approximate minimization along coordinate directions or coordinate hyperplanes. They have been used in applications for many years, and their popularity continues to grow because of their usefulness in data analysis, machine learning, and other areas of current interest. This paper describes the fundamentals of the coordinate … Read more

Accelerated Proximal Stochastic Dual Coordinate Ascent for Regularized Loss Minimization

We introduce a proximal version of the stochastic dual coordinate ascent method and show how to accelerate the method using an inner-outer iteration procedure. We analyze the runtime of the framework and obtain rates that improve state-of-the-art results for various key machine learning optimization problems including SVM, logistic regression, ridge regression, Lasso, and multiclass SVM. … Read more

A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints

In this paper we present a variant of random coordinate descent method for solving linearly constrained convex optimization problems with composite objective function. If the smooth part has Lipschitz continuous gradient, then the method terminates with an ϵ-optimal solution in O(N2/ϵ) iterations, where N is the number of blocks. For the class of problems with … Read more

L1 Minimization via Randomized First Order Algorithms

In this paper we propose randomized first-order algorithms for solving bilinear saddle points problems. Our developments are motivated by the need for sublinear time algorithms to solve large-scale parametric bilinear saddle point problems where cheap online assessment of solution quality is crucial. We present the theoretical efficiency estimates of our algorithms and discuss a number … Read more

A Randomized Cutting Plane Method with Probabilistic Geometric Convergence

We propose a randomized method for general convex optimization problems; namely, the minimization of a linear function over a convex body. The idea is to generate N random points inside the body, choose the best one and cut the part of the body defined by the linear constraint. We first analyze the convergence properties of … Read more

Approximation Algorithms for Indefinite Complex Quadratic Maximization Problems

In this paper we consider the following two types of complex quadratic maximization problems: (i) maximize $z^{\HH} Q z$, subject to $z_k^m=1$, $k=1,…,n$, where $Q$ is a Hermitian matrix with $\tr Q=0$ and $z\in \C^n$ is the decision vector; (ii) maximize $\re y^{\HH}Az$, subject to $y_k^m=1$, $k=1,…,p$, and $z_l^m=1$, $l=1,…,q$, where $A\in \C^{p\times q}$ and … Read more