## Exact Algorithms for Arc and Node Routing Problems

Routing problems stand among the hardest combinatorial problems to find high quality bounds or to prove new optimal solutions. In this thesis, we tackle the Capacitated Arc Routing Problem (CARP) and the Generalized Vehicle Routing Problem (GVRP). For both problems, there are a set of customers spread over a given graph, where each customer has … Read more

## Completely Positive Reformulations for Polynomial Optimization

Polynomial optimization encompasses a very rich class of problems in which both the objective and constraints can be written in terms of polynomials on the decision variables. There is a well stablished body of research on quadratic polynomial optimization problems based on reformulations of the original problem as a conic program over the cone of … Read more

## A Lagrangian-DNN Relaxation: a Fast Method for Computing Tight Lower Bounds for a Class of Quadratic Optimization Problems

We propose an efficient computational method for linearly constrained quadratic optimization problems (QOPs) with complementarity constraints based on their Lagrangian and doubly nonnegative (DNN) relaxation and first-order algorithms. The simplified Lagrangian-CPP relaxation of such QOPs proposed by Arima, Kim, and Kojima in 2012 takes one of the simplest forms, an unconstrained conic linear optimization problem … Read more

## Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing. This problem is referred to as near-separable NMF and requires that there exists a cone spanned by a small subset of … Read more

## Mathematical Programming: Turing completeness and applications to software analysis

Mathematical Programming is Turing complete, and can be used as a general-purpose declarative language. We present a new constructive proof of this fact, and showcase its usefulness by discussing an application to finding the hardest input of any given program running on a Minsky Register Machine. We also discuss an application of Mathematical Programming to … Read more

## Accelerated Gradient Methods for Nonconvex Nonlinear and Stochastic Programming

In this paper, we generalize the well-known Nesterov’s accelerated gradient (AG) method, originally designed for convex smooth optimization, to solve nonconvex and possibly stochastic optimization problems. We demonstrate that by properly specifying the stepsize policy, the AG method exhibits the best known rate of convergence for solving general nonconvex smooth optimization problems by using first-order … Read more

## Conic Geometric Programming

We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as linear programs (LPs) and semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints … Read more

## Optimal scenario set partitioning for multistage stochastic programming with the progressive hedging algorithm

In this paper, we propose a new approach to reduce the total running time (RT) of the progressive hedging algorithm (PHA) for solving multistage stochastic programs (MSPs) defined on a scenario tree. Instead of using the conventional scenario decomposition scheme, we apply a multi-scenario decomposition scheme and partition the scenario set in order to minimize … Read more

## Dynamic Linear Programming Games with Risk-Averse Players

Motivated by situations in which independent agents, or players, wish to cooperate in some uncertain endeavor over time, we study dynamic linear programming games, which generalize classical linear production games to multi-period settings under uncertainty. We specifically consider that players may have risk-averse attitudes towards uncertainty, and model this risk aversion using coherent conditional risk … Read more

## Time Consistency Versus Law Invariance in Multistage Stochastic Optimization with Coherent Risk Measures: Multilevel Optimization Modeling and Computational Complexity

Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which un-certainly is resolved at more than one stage, however, use of coherent risk measures within a standard single-level optimization framework presents the modeler with an uncomfortable choice between two desirable model properties, time consistency … Read more