This paper proposes a new arc-search interior-point algorithm for the nonlinear constrained optimization problem. The proposed algorithm uses the second-order derivatives to construct a search arc that approaches the optimizer. Because the arc stays in the interior set longer than any straight line, it is expected that the scheme will generate a better new iterate … Read more
Many retail distribution networks use a hierarchical structure in which regional distribution centers replenish local distribution centers that are closer to customers. In such networks, products are often shipped in large quantities, as case packs, cases or pallets, to the regional distribution center, whereas local distribution centers may require quantities at the individual-unit level. Case … Read more
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
In nonsmooth optimization, identifiable sets describe the local region eventually reached by sequences converging to a prescribed critical point. When such a set is a \(C^2\) manifold on which the objective restricts to a \(C^2\) function, it is called an identifiable manifold. Their appeal lies in what they enable: many first-order methods identify these manifolds … Read more
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
Wasserstein distributionally robust optimization (DRO) is a versatile and widely adopted framework for decision-making under uncertainty, yet its standard deterministic reformulations generally contain non-convex inner subproblems that are challenging to solve. To address this issue, we propose a shrinkage path heuristic that reduces the solution of a DRO problem to a one-dimensional search over the … Read more
We study clean energy transition planning for campus-scale integrated electricity-heat systems under both strategic level and operational level uncertainties. We formulate a multi-stage stochastic mixed-integer program that jointly optimizes investment and operational decisions for renewable generation, storage, and heat-transfer technologies whose costs and efficiencies evolve stochastically across stages. To account for short-term operational uncertainty, we … Read more