In this study, we focus on finding exact binary optimal classification trees (OCTs) via mixed-integer (linear) programs (MIP). To alleviate the scalability issues of existing MIP-based OCT formulations, we propose a novel decomposable model that treats each leaf of the classification tree as a decision rule. The model exploits the order-independent nature of decision rules to reduce the problem size and the solution space, and lends itself to a Dantzig–Wolfe (extended) reformulation. We then implement a tailored branch-and-price solution with pricing subproblems designated for each leaf solved in parallel. We identify the conditions under which the extended formulation performs better in order to form a hybrid algorithm, OrderIndependentOCT, that decides when to invoke branch-and-price. On 18 benchmark datasets, OrderIndependentOCT proves optimality on 71.1% of instances within a one-hour time limit, compared to 60.0% for the well-known Benders decomposition approach (BendersOCT). In cases where neither method attains optimality, OrderIndependentOCT yields higher (or the same) training accuracy than BendersOCT in 50.68% (resp. 71.23% ) of the instances. Similarly, in 86.3% of all instances, OrderIndependentOCT matches or exceeds BendersOCT’s out-of-sample accuracy. These findings illustrate the efficiency of the proposed decomposition approach for the optimal classification tree training problem compared to the state-of-the-art.
We consider a fundamental question in political districting: How many counties can be kept whole (i.e., not split across multiple districts), while satisfying basic criteria like contiguity and population balance? To answer this question, we propose integer programming techniques based on combinatorial Benders decomposition. The main problem decides which counties to keep whole, and the subproblem coarsens the selected counties and then seeks a feasible plan. We apply the approach to all congressional and legislative instances across the USA, generating plans that are provably optimal. Finally, we conduct a case study for Tennessee’s state house districts and find that the plaintiffs’ case in Wygant v. Lee could have been stronger if our optimization methods had assisted in the drawing of demonstration plans. Our code and results are available on GitHub.
The transportation problem has recently gained renewed interest due to its applications in machine learning and artificial intelligence, particularly in measuring distances between images. A novel exact algorithm for the transportation problem, called Iterated Inside Out, was recently proposed showing superior performances with respect to the relevant literature. In this paper, we present an improved version of Iterated Inside Out tailored for image processing instances. The algorithm relies on an improved search of negative reduced cost variables embedded into a Matryoshka approach for computing heuristic high-quality initial solutions. We show the superiority of our algorithm against the algorithms in the literature on the DOTmark dataset, a benchmark for image processing and large-scale transportation problems.
The most cost-effective and energy-efficient way to handle many computationally expensive tasks in the modern computing environment is to offload those tasks to graphics processing units (GPUs). However, for the most expensive branch of mathematical optimization—deterministic global optimization—no existing mature solvers, to the best of our knowledge, use GPUs in any capacity. In this paper, we demonstrate that a core component of deterministic global optimization techniques—the generation of relaxations and relaxation subgradients—can be effectively performed on GPU hardware for arbitrarily complicated factorable expressions. This is accomplished using a source code generation approach for McCormick-based relaxation rules made available in the open-source software package SourceCodeMcCormick.jl. Across a benchmark test set of problems from MINLPLib, we achieve speedups in the calculation of relaxations and their subgradients in the range of one-to-four orders-of-magnitude over a typical CPU-based implementation of these rules at a level of parallelization that would be useful for a GPU-based deterministic global optimizer.
Feasibility Pump (FP) is a well-studied mixed integer linear programming primal heuristic that sequentially guides fractional solutions into integer feasibility by minimizing their distance to given target integer points, while preserving the linear constraints of the original problem. Due to its effectiveness and simplicity, FP has been embedded in most commercial solvers, and several variants of the basic method have been proposed, aiming at improving its convergence and the quality of the solutions obtained. This work introduces Kernel Pump (KP), a novel FP speed-up technique that relies on the intuition that not all binary variables are active (i.e., set to one) in a feasible solution, and can, thus, be sub-divided in a kernel search fashion. Precisely, KP identifies a subset of promising binary variables (the kernel) and distributes the rest into buckets ranked by their likelihood of composing a feasible/optimal solution. FP sub-problems are then solved iteratively, focusing on the current kernel and progressively adding variables from the buckets whenever necessary. Extensive computational experiments on benchmark instances from MIPLIB 2017, as well as two additional problems from the literature of vehicle routing and scheduling, reveal a considerable boost in success rate, also decreasing running times for the majority of instances tested. Notably, aside from outperforming FP, the new method was able to find feasible solutions in cases where CPLEX failed within an hour of search. To our knowledge, this is also the first study to introduce a decomposition approach within the FP framework.
