Distributionally Fair Stochastic Optimization using Wasserstein Distance

A traditional stochastic program under a finite population typically seeks to optimize efficiency by maximizing the expected profits or minimizing the expected costs, subject to a set of constraints. However, implementing such optimization-based decisions can have varying impacts on individuals, and when assessed using the individuals’ utility functions, these impacts may differ substantially across demographic … Read more

Refined TSSOS

The moment-sum of squares hierarchy by Lasserre has become an established technique for solving polynomial optimization problems. It provides a monotonically increasing series of tight bounds, but has well-known scalability limitations. For structured optimization problems, the term-sparsity SOS (TSSOS) approach scales much better due to block-diagonal matrices, obtained by completing the connected components of adjacency … Read more

A widespread belief about county splits in political districting plans is wrong

Consider the task of dividing a state into k contiguous political districts whose populations must not differ by more than one person, following current practice for congressional districting in the USA. A widely held belief among districting experts is that this task requires at least k-1 county splits. This statement has appeared in expert testimony, … Read more

Facets of the knapsack polytope from non-minimal covers

We propose two new classes of valid inequalities (VIs) for the binary knapsack polytope, based on non-minimal covers. We also show that these VIs can be obtained through neither sequential nor simultaneous lifting of well-known cover inequalities. We further provide conditions under which they are facet-defining. The usefulness of these VIs is demonstrated using computational … Read more

Branch-and-Bound versus Lift-and-Project Relaxations in Combinatorial Optimization

In this paper, we consider a theoretical framework for comparing branch-and-bound with classical lift-and-project hierarchies. We simplify our analysis of streamlining the definition of branch-and-bound. We introduce “skewed $k$-trees” which give a hierarchy of relaxations that is incomparable to that of Sherali-Adams, and we show that it is much better for some instances. We also … Read more

An Integer Programming Approach To Subspace Clustering With Missing Data

In the Subspace Clustering with Missing Data (SCMD) problem, we are given a collection of n partially observed d-dimensional vectors. The data points are assumed to be concentrated near a union of low-dimensional subspaces. The goal of SCMD is to cluster the vectors according to their subspace membership and recover the underlying basis, which can … Read more

A Bilevel Optimization Approach for a Class of Combinatorial Problems with Disruptions and Probing

\(\) We consider linear combinatorial optimization problems under uncertain disruptions that increase the cost coefficients of the objective function. A decision-maker, or planner, can invest resources to probe the components (i.e., the coefficients) in order to learn their disruption status. In the proposed probing optimization problem, the planner, knowing just the disruptions’ probabilities, selects which … Read more

Improving Conflict Analysis in MIP Solvers by Pseudo-Boolean Reasoning

Conflict analysis has been successfully generalized from Boolean satisfiability (SAT) solving to mixed integer programming (MIP) solvers, but although MIP solvers operate with general linear inequalities, the conflict analysis in MIP has been limited to reasoning with the more restricted class of clausal constraint. This is in contrast to how conflict analysis is performed in … Read more

Delay-Resistant Robust Vehicle Routing with Heterogeneous Time Windows

We consider a robust variant of the vehicle routing problem with heterogeneous time windows (RVRP-HTW) with a focus on delay-resistant solutions. Here, customers have different availability time windows for every vehicle and must be provided with a preferably tight appointment window for the planned service. Different vehicles are a possibility to model different days on … Read more

A Short Proof of Tight Bounds on the Smallest Support Size of Integer Solutions to Linear Equations

\(\) In this note, we study the size of the support of solutions to linear Diophantine equations $Ax=b, ~x\in\Z^n$ where $A\in\Z^{m\times n}, b\in\Z^n$. We give an asymptotically tight upper bound on the smallest support size, parameterized by $\|A\|_\infty$ and $m$, and taken as a worst case over all $b$ such that the above system has … Read more