The Ramsey number R(m,n) is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size m or a red clique of size n. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that … Read more
For decades, researchers have complained that contiguity constraints make problems like districting much harder than other combinatorial optimization problems. This has prompted the use of integer programming models with inexact contiguity constraints that may be fast in practice but are invalid in the sense that they cut off or “overlook” some contiguous solutions. Examples include … Read more
This paper proposes a four-stage decision-support framework for structuring and reducing large multi-objective solution sets into compact and interpretable collections of representative alternatives. The methodology combines: (Phase 1) systematic solution generation through extended goal programming and structured preference exploration; (Phase 2) robustness-aware enrichment and profiling under weight sensitivity analysis; (Phase 3) filtering and dominance-based reduction … Read more
This book chapter is a gentle introduction to the mathematics of congressional apportionment. It emphasizes the connections between mathematical optimization and the classical apportionment methods (e.g., Jefferson, Adams, Hamilton, Webster, Huntington-Hill, Dean). CitationPrepared for a forthcoming book edited by Bruce Golden and Doug ShierArticleDownload View PDF
Classical Benders decomposition fails when the subproblem is a mixed-integer program, due to the absence of strong duality. We propose a novel class of dual-free indicator cuts that are applicable to all Benders-decomposable problems with a pure-integer master problem and mixed-integer linear programming (MILP) subproblems. These cuts are derived from the monotonicity property of the … Read more
We consider a broad class of bilevel linear programs in which the follower’s decisions are all continuous, while the leader’s decisions may include integrality restrictions. Solving such problems to optimality is known to be NP-hard. A classical approach in bilevel optimization for constructing lower and upper bounds is based on a single-level relaxation, in which … Read more
We present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective … Read more
We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the Coordinate Optimality Reformulation (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable … Read more
We address the problem of optimal route planning for competitive orien- teering on real terrain. A Geographic Information System (GIS) pipeline transforms orienteering map data and digital terrain models into a fully asymmetric cost matrix that captures directional slope costs (via the Minetti metabolic model) and cumulative athlete fatigue. The resulting problem, the Asymmetric Orienteering … Read more