Edge expansion of a graph: SDP-based computational strategies

Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two variants of exact algorithms using semidefinite programming (SDP) to compute this constant for any graph. The first variant uses the SDP relax- ation first to reduce the search space considerably. One … Read more

Similarity-based Decomposition Algorithm for Two-stage Stochastic Scheduling

This paper presents a novel decomposition method for two-stage stochastic mixed-integer optimization problems. The algorithm builds upon the idea of similarity between finite sample sets to measure how similar the first-stage decisions are among the uncertainty realization scenarios. Using such a Similarity Index, the non-anticipative constraints are removed from the problem formulation so that the … Read more

Asynchronous Iterations in Optimization: New Sequence Results and Sharper Algorithmic Guarantees

We introduce novel convergence results for asynchronous iterations that appear in the analysis of parallel and distributed optimization algorithms. The results are simple to apply and give explicit estimates for how the degree of asynchrony impacts the convergence rates of the iterates. Our results shorten, streamline and strengthen existing convergence proofs for several asynchronous optimization … Read more

Distributed Projections onto a Simplex

Projecting a vector onto a simplex is a well-studied problem that arises in a wide range of optimization problems. Numerous algorithms have been proposed for determining the projection; however, all but one of these algorithms are serial. We address this gap by developing a method that preprocesses the input vector by decomposing and distributing it … Read more

Parallel Dual Dynamic Integer Programming for Large-Scale Hydrothermal Unit-Commitment

Unit commitment has been at the center of power system operation for well over 50 years. Yet, this problem cannot be considered solved due to its size and complexity. Today, operators rely on off-the-shelf optimization solvers to tackle this challenging problem, and often resort to simplifications to make the problem more tractable and solvable in … Read more

Scalable Parallel Nonlinear Optimization with PyNumero and Parapint

We describe PyNumero, an open-source, object-oriented programming framework in Python that supports rapid development of performant parallel algorithms for structured nonlinear programming problems (NLP’s) using the Message Passing Interface (MPI). PyNumero provides three fundamental building blocks for developing NLP algorithms: a fast interface for calculating first and second derivatives with the AMPL Solver Library (ASL), … Read more

Parallel Strategies for Direct Multisearch

Direct Multisearch (DMS) is a Derivative-free Optimization class of algorithms suited for computing approximations to the complete Pareto front of a given Multiobjective Optimization problem. It has a well-supported convergence analysis and simple implementations present a good numerical performance, both in academic test sets and in real applications. Recently, this numerical performance was improved with … Read more

String-Averaging Methods for Best Approximation to Common Fixed Point Sets of Operators: The Finite and Infinite Cases

Abstract String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm at hand requires to employ the operators in a specific order. Sequential orderings are well-known and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by … Read more

Improving Column-Generation for Vehicle Routing Problems via Random Coloring and Parallelization

We consider a variant of the Vehicle Routing Problem (VRP) where each customer has a unit demand and the goal is to minimize the total cost of routing a fleet of capacitated vehicles from one or multiple depots to visit all customers. We propose two parallel algorithms to efficiently solve the column-generation based linear-programming relaxation … Read more

A Novel Cooperative Multi-search Benders Decomposition for Solving the Hydrothermal Unit-Commitment Problem

Renewable energy and modernization of power operation demand Independent System Operators (ISOs) to solve ever more complex and larger programming problems to securely and economically schedule power resources. A key step in the scheduling process is the unit commitment (UC). In a hydro-dominated system, this process also involves managing reservoirs and is called hydrothermal UC … Read more