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The use of convex relaxations has lately gained considerable interest in Power Systems. These relaxations play a major role in providing global optimality guarantees for non-convex optimization problems. For the Optimal Power Flow (OPF)…
A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a…
A strongly polynomial algorithm is given for the generalized flow maximization problem. It uses a new variant of the scaling technique, called continuous scaling. The main measure of progress is that within a strongly polynomial number of…
In this paper we provide new randomized algorithms with improved runtimes for solving linear programs with two-sided constraints. In the special case of the minimum cost flow problem on $n$-vertex $m$-edge graphs with integer…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
In this paper we apply a reduced basis framework for the computation of flow bifurcation (and stability) problems in fluid dynamics. The proposed method aims at reducing the complexity and the computational time required for the…
This paper presents a tensor-recovery method to solve probabilistic power flow problems. Our approach generates a high-dimensional and sparse generalized polynomial-chaos expansion that provides useful statistical information. The result…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict…
In this paper, we develop a theory of new classes of discrete convex functions, called L-extendable functions and alternating L-convex functions, defined on the product of trees. We establish basic properties for optimization: a…
We consider a class of learning problems that involve a structured sparsity-inducing norm defined as the sum of $\ell_\infty$-norms over groups of variables. Whereas a lot of effort has been put in developing fast optimization methods when…
The paper deals with the problem of finding sparse solutions to systems of polynomial equations possibly perturbed by noise. In particular, we show how these solutions can be recovered from group-sparse solutions of a derived system of…
In this paper we propose a general methodology for solving a broad class of continuous, multifacility location problems, in any dimension and with $\ell_\tau$-norms proposing two different methodologies: 1) by a new second order cone mixed…
We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original…
This paper studies the fundamental problem of how to reroute $k$ unsplittable flows of a certain demand in a capacitated network from their current paths to their respective new paths, in a congestion-free manner and fast. This scheduling…
Semidefinite Programming (SDP) provides tight lower bounds for Optimal Power Flow problems. However, solving large-scale SDP problems requires exploiting sparsity. In this paper, we experiment several clique decomposition algorithms that…
The Moment/Sum-of-squares hierarchy provides a way to compute the global minimizers of polynomial optimization problems (POP), at the cost of solving a sequence of increasingly large semidefinite programs (SDPs). We consider large-scale…
This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide…
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
The problem of minimizing a polynomial over a set of polynomial inequalities is an NP-hard non-convex problem. Thanks to powerful results from real algebraic geometry, one can convert this problem into a nested sequence of…