arXiv subjects

Nagisa Sugishita

Publications and source records attributed to Nagisa Sugishita.

10 recordsLinked to original sources

arXiv2026

Complexity of Bilevel Linear and Bilevel Pricing Problems with a Single Upper-Level Variable

Bilevel linear programming (LP) is one of the simplest classes of bilevel optimization problems, yet it is known to be NP-hard in general. Specifically, determining whether the optimal objective value of a bilevel LP is at least as good as a given threshold, a standard decision version of the problem, is NP-complete. However, this decision problem becomes tractable when either the number of lower-level variables or the number of lower-level constraints is fixed, which prompts the question: What if restrictions are placed on the upper-level problem? In this paper, we address this gap by showing that the decision version of bilevel LP remains NP-complete even when there is only a single upper-level variable, no upper-level constraints (apart from the constraint enforcing optimality of the lower-level decision) and all variables are bounded between 0 and 1. This result implies that fixing the number of variables or constraints in the upper-level problem alone does not lead to tractability in general. On the positive side, we show that there is a polynomial-time algorithm that finds a locally optimal solution of such a rational bilevel LP instance. We also discuss an extension of our result to a bilevel pricing problem. These hardness results rely on auxiliary results of broad interest. Notably, we show that a lexicographic LP can be transformed to a single-objective LP by aggregating objectives with weights of encoding size polynomial in the sizes of cost coefficients and constraint matrix. In particular, these weights are valid uniformly for all right-hand sides.

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arXiv2026

On the Computational Complexity of Bilevel Integer Linear Programming

We investigate the computational complexity of bilevel integer linear programming. While Jeroslow~(1985) established that the decision version of this problem is $Σ^p_2$-complete when restricted to binary variables, we prove that this $Σ^p_2$-completeness persists even for general integer variables, settling a question that remained open for over 40 years. Furthermore, we analyze the impact of various structural assumptions on computational complexity. Notably, we strengthen the result of Köppe et al.~(2010) by proving polynomial-time solvability whenever the total number of upper- and lower-level variables is fixed, without any additional assumptions.

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arXiv2026

Price of Coupling in Multilevel Linear Programming

Multilevel programming is the standard framework for modeling hierarchical decision-making. In this paper, we characterize the computational complexity of deciding the existence of feasible and optimal solutions, as well as computing the optimal objective value in multilevel linear programming (LP). Our analysis considers various combinations of modeling assumptions, including the presence or absence of linking (coupling) constraints and whether all variables are bounded. In particular, we show the feasibility problem of $k$-level LP is $Σ^{p}_{k-1}$-complete for $k \ge 2$. Without linking constraints and unbounded variables, it is polynomial-time solvable for $k \le 4$ but becomes $Σ^{p}_{k-1}$-complete for $k \ge 5$, indicating a sharp jump in computational complexity assuming the polynomial hierarchy does not collapse. Combined with other results, one major implication is that no polynomial-time Turing machine can transform a bilevel LP instance with linking constraints into one without linking constraints while preserving feasibility unless P $=$ NP. In contrast, such machines exist for all $k \ge 5$. We observe similar phenomena with the decision of the existence of an optimal solution. In the bilevel case, feasibility and boundedness fully characterize the existence of an optimal solution, implying that the problem is DP-complete. However, these conditions are insufficient for $k \ge 3$ and the problem for $k \ge 3$ is $Δ^{p}_k$-complete. Similar to the feasibility problem, the problem becomes polynomially solvable for $k=2,3$ without linking constraints and unbounded variables. However, the problem is $Δ^{p}_k$-complete for $k \ge 4$, even with these simplifying assumptions. The computation of the optimal objective value is F$Δ^{p}_k$-complete for any $k \ge 2$, even without linking constraints and unbounded variables.

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arXiv2026

Decision Problems in Multilevel Linear Programming

We study the computational complexity of decision problems in $k$-level linear programming (LP). Seminal work by Jeroslow establishes that determining whether the optimal objective value of a $k$-level LP is at least as good as a given threshold is $Σ^{\mathrm{p}}_{k-1}$-hard. In this paper, we demonstrate the matching upper bound and thereby prove that this problem is $Σ^{\mathrm{p}}_{k-1}$-complete. To this end, we show that the feasible region of a $k$-level LP can be expressed as a union of sets defined by weak and strict linear inequalities. Moreover, we show that the decision of the unboundedness is $Σ^{\mathrm{p}}_{k-1}$-complete. Finally, we discuss the extension of our results to the mixed-binary cases. In short, this work closes lasting open questions in multilevel programming.

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arXiv2025

Competitive EV charging station location with queues

Electric vehicle (EV) public charging infrastructure planning faces significant challenges in competitive markets, where multiple service providers affect congestion and user behavior. This work extends existing modeling frameworks by incorporating the presence of competitors' stations and more realistic queueing systems. First, we analyze three finite queueing systems, M/M/1/K, M/M/s/K, and M/Er/s/K, with varying numbers of servers (charging outlets) and service time distributions, deriving analytic expressions for user behavior metrics. Second, we embed the queueing-based user behavior model into a bilevel program, where the upper level locates new charging stations to maximize accessibility (throughput), and the lower level captures users' station choices via a user equilibrium. Third, we apply a reformulation from competitive congested user-choice facility location models to approximately solve the bilevel problem and introduce a surrogate-based heuristic to enhance scalability. Fourth, we showcase our methodology on a real-world case study of an urban area in Montreal (Canada), offering managerial insights into how user-choice behavior assumptions and competition affect throughput and location decisions. The results demonstrate that our model yields (re)location strategies that outperform the existing network. More broadly, this approach provides a tool for incorporating charging service quality-through queueing metrics-and existing competition into station planning.

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arXiv2025

The Branch-and-Bound Tree Closure

This paper investigates the a-posteriori analysis of Branch-and-Bound~(BB) trees to extract structural information about the feasible region of mixed-binary linear programs. We introduce three novel outer approximations of the feasible region, systematically constructed from a BB tree. These are: a tight formulation based on disjunctive programming, a branching-based formulation derived from the tree's branching logic, and a mixing-set formulation derived from the on-off properties inside the tree. We establish an inclusion hierarchy, which ranks the approximations by their theoretical strength \wrt to the original feasible region. The analysis is extended to the generation of valid inequalities, revealing a separation-time hierarchy that mirrors the inclusion hierarchy in reverse. This highlights a trade-off between the tightness of an approximation and the computational cost of generating cuts from it. Motivated by the computational expense of the stronger approximations, we introduce a new family of valid inequalities called star tree inequalities. Although their closure forms the weakest of the proposed approximations, their practical appeal lies in an efficient, polynomial-time combinatorial separation algorithm. A computational study on multi-dimensional knapsack and set-covering problems empirically validates the theoretical findings. Moreover, these experiments confirm that computationally useful valid inequalities can be generated from BB trees obtained by solving optimization problems considered in practice.

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arXiv2025

The Strength of Flow Refueling Location Problem Formulations and an Extension to Cyclic Routing

The Flow Refueling Location Problem (FRLP) is a stylized model for determining the optimal placement of refueling stations for vehicles with limited travel ranges, such as hydrogen fuel cell vehicles and electric vehicles. A notable extension, the deviation FRLP, accounts for the possibility that drivers may deviate from their preferred routes to refuel or recharge. While solution techniques based on various mathematical programming formulations have been thoroughly explored for this extension, there is a lack of theoretical insights into the relationships and strengths of these formulations. In this work, for the deviation extension, we study two prominent FRLP formulations from the literature and compare their strengths in terms of linear programming (LP) relaxations. We show that the LP relaxation of one formulation yields a bound that is at least as tight as that of the other, which may explain its observed superior performance. Building on these insights, we address a common modeling assumption in the FRLP that requires drivers to use the same paths for their outbound and inbound trips. Specifically, we relax this assumption and introduce the cyclic FRLP, where drivers may use different paths in each direction. We show how existing formulations can be naturally extended to accommodate this setting and describe a branch-and-cut algorithm to solve the problem. We provide numerical experiments highlighting the benefits of such asymmetric routing. For example, in an instance based on the Californian network, the cyclic FRLP serves all demands using 30% fewer facilities than the original FRLP.

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arXiv2023

Use of Machine Learning Models to Warmstart Column Generation for Unit Commitment

The unit commitment problem is an important optimization problem in the energy industry used to compute the most economical operating schedules of power plants. Typically, this problem has to be solved repeatedly with different data but with the same problem structure. Machine learning techniques have been applied in this context to find primal feasible solutions. On the other hand, Dantzig-Wolfe decomposition with a column generation procedure has been shown to be successful in solving the unit commitment problem to tight tolerance. We propose the use of machine learning models not to find primal feasible solutions directly but to generate initial dual values for the column generation procedure. Our numerical experiments compare machine learning based methods for warmstarting the column generation procedure with three baselines: column pre-population, the linear programming relaxation and coldstart. The experiments reveal that the machine learning approaches are able to find both tight lower bounds and accurate primal feasible solutions in a shorter time compared to the baselines. Furthermore, these approaches scale well to handle large instances.

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arXiv2022

Primal heuristics for Dantzig-Wolfe decomposition for unit commitment

The unit commitment problem is a short-term planning problem in the energy industry. Dantzig-Wolfe decomposition is a popular approach to solve the problem. This paper focuses on primal heuristics used with Dantzig-Wolfe decomposition. We propose two primal heuristics: one based on decomposition and one based on machine learning. The first one uses the fractional solution to the restricted master problem to fix a subset of the integer variables. In each iteration of the column generation procedure, the primal heuristic obtains the fractional solution, checks whether each binary variable satisfies the integrality constraint and fix those which do. The remaining variables are then optimised quickly by a solver to find a feasible, near-optimal solution to the original instance. The other primal heuristic based on machine learning is of interest when the problems are to be solved repeatedly with different demand data but with the same problem structure. The primal heuristic uses a pre-trained neural network to fix a subset of the integer variables. In the training phase, a neural network is trained to predict for any demand data and for each binary variable how likely it is that the variable takes each of two possible values. After the training, given an instance to be solved, the prediction of the model is used with a rounding threshold to fix some binary variables. Our numerical experiments compare our methods with solving the undecomposed problem and also with other primal heuristics from the literature. The experiments reveal that the primal heuristic based on machine learning is superior when the suboptimality tolerance is relatively large, such as 0.5% or 0.25%, while the decomposition is the best when the tolerance is small, for example 0.1%.

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arXiv2021

Incremental cutting-plane method and its application

We consider regularized cutting-plane methods to minimize a convex function that is the sum of a large number of component functions. One important example is the dual problem obtained from Lagrangian relaxation on a decomposable problem. In this paper, we focus on an incremental variant of the regularized cutting-plane methods, which only evaluates a subset of the component functions in each iteration. We first consider a limited-memory setup where the method deletes cuts after a finite number of iterations. The convergence properties of the limited-memory methods are studied under various conditions on regularization. We then provide numerical experiments where the incremental method is applied to the dual problems derived from large-scale unit commitment problems. In many settings, the incremental method is able to find a solution of high precision in a shorter time than the non-incremental method.

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