apply column generation

In column generation, pricing subproblems heavily rely on application-specific structural properties, resulting in low reusability of customized solvers. To address this, we propose a domain-agnostic, general-purpose pricing framework that— for the first time—integrates dynamic programming into both the column generation and branch-and-price processes, serving as a transferable pricing solver independent of problem-specific algorithms. Our method uniformly handles diverse combinatorial structures without requiring redesign of pricing logic for each problem class. Evaluated on seven canonical integer programming problems, it consistently outperforms state-of-the-art commercial and open-source solvers in solution quality and runtime, achieving superior scalability and robustness. The framework significantly enhances the generality, reliability, and computational efficiency of large-scale exact optimization.

This paper addresses the challenge of simultaneously achieving scalability, local interpretability, and multi-attribute, multi-class fairness in rule-based classification models. To this end, we propose the first column generation–based rule learning framework. Methodologically, we introduce column generation—previously unexplored in rule learning—integrating a linear programming master problem, a decision-tree–inspired column generation heuristic, a surrogate pricing subproblem solver, and weighted rule optimization; we further formulate generalized fairness constraints supporting multiple sensitive attributes and multi-class outcomes. Our key contributions are: (1) enabling local interpretability via rule weights, and (2) unifying support for complex fairness constraints and scalable search over large rule spaces. Extensive experiments on benchmark datasets demonstrate that our approach achieves significant trade-off improvements among accuracy, interpretability, and fairness, substantially enhancing the practicality of rule models in real-world, large-scale applications.

This paper addresses the Combinatorial Pricing Problem (CPP), a canonical combinatorial bilevel programming problem where a leader sets item tolls to maximize revenue, while a follower solves a combinatorial optimization subproblem subject to the induced cost constraints. To overcome the scalability limitations of conventional value-function-based approaches, we propose the first single-level dual formulation that embeds CPP into a dynamic programming framework. Specifically, we reformulate the follower’s problem as a longest-path problem on a directed acyclic graph and introduce a novel “Selection Diagram” structure—a compact decision diagram encoding feasible follower choices. We further pioneer the integration of decision diagrams with cutting-plane methods for efficient solution. Our approach significantly outperforms state-of-the-art algorithms on three CPP variants and the knapsack interdiction problem, substantially expanding the scale of combinatorial bilevel programs solvable to provable optimality.

This work addresses the limitations of existing large language model (LLM)-driven heuristic design, which often relies solely on end-point evaluation while neglecting solution process efficiency and incurs high re-adaptation costs under distribution shifts. To overcome these issues, the authors propose DASH, a novel framework that introduces a convergence-dynamics-aware evaluation mechanism to jointly optimize search strategies and runtime scheduling. DASH further incorporates a profiled library retrieval module that enables efficient reuse across heterogeneous problem instances through solver profiling. Experimental results demonstrate that DASH achieves over a fourfold improvement in runtime efficiency across four combinatorial optimization problems and outperforms state-of-the-art methods in overall performance. Moreover, under distribution shift scenarios, DASH significantly reduces solution quality degradation and lowers LLM re-adaptation costs by approximately 90%.

Many optimization problems in manufacturing, logistics, and healthcare remain reliant on manual heuristics due to the high modeling barrier for Mixed-Integer Linear Programming (MILP). Method: This paper proposes the first end-to-end MILP automation framework driven by natural language descriptions. It introduces a modular large language model (LLM) architecture integrating natural language understanding, program synthesis, code debugging, solution quality verification, and feedback-driven iterative refinement. Additionally, it establishes NLP4LP—the first long-horizon, complex LP benchmark dataset derived from natural language problem specifications. Contribution/Results: Experiments demonstrate that our framework achieves an accuracy gain of +12.3% over state-of-the-art methods on easy instances and +8.7% on hard instances—including those in NLP4LP—significantly advancing automated modeling and efficient solving of large-scale real-world optimization problems.

This study addresses the technician dispatching problem faced by electric utilities, which involves scheduling a large number of intervention tasks within a limited time horizon, with the primary objective of maximizing the total duration of completed tasks and the secondary objective of minimizing operational costs. The work introduces lexicographic multi-objective optimization to this domain for the first time and proposes a sequential column generation algorithm that transforms the multi-objective problem into a sequence of single-objective problems via weighted summation and hierarchical optimization. The approach integrates an extended set-covering formulation, mixed-integer linear programming, and a dynamic programming–based labeling algorithm to solve the pricing subproblem. Experiments on real-world data from Électricité de France demonstrate that the method yields superior solutions on small instances and produces high-quality schedules within five minutes for large-scale instances, achieving lower average optimality gaps and improving upon several best-known solutions.

This work addresses the computational intractability of large-scale combinatorial optimization problems arising from their exponentially sized search spaces by proposing a structure-aware parallel decomposition framework. The approach constructs a constrained maximum-cut model based on variable interaction structures, reformulates it as a QUBO problem, and leverages an Ising machine to efficiently cluster variables for automatic problem decomposition. The resulting subproblems are then solved in parallel using mathematical optimization solvers. This method uniquely integrates structure-aware clustering with Ising-based computation, substantially reducing the effective problem size. Experimental results on the capacitated vehicle routing problem demonstrate up to a 95.32% reduction in variable count, achieving within one minute the solution quality that conventional methods require thirty minutes to attain, while significantly improving the rate of feasible solutions.

This work addresses the high computational cost of repeatedly solving large-scale mixed-integer programming (MIP) problems that share structural similarity but differ in parameters. The authors propose the BIPC framework, which pioneers the integration of backdoor variable identification with machine learning: a supervised learning model predicts values for a critical subset of variables, enabling the construction and solution of a reduced MIP model, followed by a feasibility correction step to recover a high-quality feasible solution. Situated within the “Learning to Optimize” paradigm, this approach significantly reduces solving time in applications such as power systems and transportation—domains requiring rapid responses to parameter perturbations—while incurring only minor losses in solution quality, thereby offering an efficient and practical method for parametric MIP solving.

This study addresses the Length-Constrained Cycle Partition problem (LCCP), which seeks to partition the nodes of a graph into the minimum number of cycles such that the length of each cycle does not exceed the critical time associated with its nodes. Building upon a set-partitioning formulation, the authors propose a numerically stable branch-price-and-cut framework that integrates a dynamic programming-based pricing algorithm, bidirectional search, and symmetry-breaking strategies to efficiently generate improving cycles and accelerate convergence. The approach yields exceptionally strong linear programming dual bounds, substantially outperforming existing methods. It successfully closes 14 large-scale benchmark instances that had remained open for years and extends the solvable problem size from 52 to 76 nodes.

To address the low computational efficiency of solving mixed-integer bilevel linear programs (MIBLPs), this paper proposes a novel unified modeling approach based on *improving directions*: a single subproblem simultaneously verifies bilevel feasibility and generates strong valid inequalities. Theoretically, we characterize the role of improving directions in encoding the follower’s optimality conditions, establish an optimality-based relaxation hierarchy, and extend the theory of continuous cutting-plane closures to the mixed-integer bilevel setting. Algorithmically, we integrate improving-direction analysis into a branch-and-cut framework, implementing it atop the open-source solver MibS. Computational experiments demonstrate that our method substantially enhances inequality strength and overall solution performance across standard benchmark instances.