We propose potential-based analyses for first-order algorithms applied to constrained and composite minimization problems. We first propose “idealized” frameworks for algorithms in the strongly and non-strongly convex cases and argue based on a potential that methods following the framework achieve the best possible rate. Then we show that the geometric descent (GD) algorithm by Bubeck … Read more
We consider a class of risk-averse submodular maximization problems (RASM) where the objective is the conditional value-at-risk (CVaR) of a random nondecreasing submodular function at a given risk level. We propose valid inequalities and an exact general method for solving RASM under the assumption that we have an efficient oracle that computes the CVaR of … Read more
We study the equivalence among a nonconvex QOP, its CPP and DNN relaxations under the assumption that the aggregated and correlative sparsity of the data matrices of the CPP relaxation is represented by a block-clique graph $G$. By exploiting the correlative sparsity, we decompose the CPP relaxation problem into a clique-tree structured family of smaller … Read more
Political districting in the United States is a decennial process of redrawing the boundaries of congressional and state legislative districts. The notion of fairness in political districting has been an important topic of subjective debate, with district maps having consequences to multiple stakeholders. Even though districting as an optimization problem has been well-studied, existing models … Read more
One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over … Read more
We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical strength of cutting planes. In particular, we show that its value lies … Read more
Mixed-integer supply chain models typically are very large but are also very sparse and can be decomposed into loosely coupled blocks. In this paper, we use general-purpose techniques to obtain a block decomposition of supply chain instances and apply a tailored penalty alternating direction method, which exploits the structural properties of the decomposed instances. We … Read more
In this paper, we propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. We use the general framework of Bregman distance functions to compute the proximal maps. Our method not only allows using two different extrapolation points to evaluate gradients and adding the inertial force, but also takes advantage … Read more
We consider spot-market trading of electricity including storage operators as additional agents besides producers and consumers. Storages allow for shifting produced electricity from one time period to a later one. Due to this, multiple market equilibria may occur even if classical uniqueness assumptions for the case without storages are satisfied. For models containing storage operators, … Read more
Bilevel problems are highly challenging optimization problems that appear in many applications of energy market design, critical infrastructure defense, transportation, pricing, etc. Often, these bilevel models are equipped with integer decisions, which makes the problems even harder to solve. Typically, in such a setting in mathematical optimization one develops primal heuristics in order to obtain … Read more