arXiv subjects

Christoph Buchheim

Publications and source records attributed to Christoph Buchheim.

At least 19 recordsLinked to original sources

arXiv2026

Complexity of partitioned-items response problems: matchings and perfect matchings

We consider bilevel optimization problems in which leader and follower jointly construct a feasible solution for an underlying combinatorial optimization problem. Response problems ask whether the leader can encourage -- or, in the pessimistic setting, enforce -- a reaction of the follower that includes a set of mandatory items while excluding a set of forbidden items. Our investigation focuses on tractability results for various cases which emerge from different combinations of the total number of mandatory, forbidden, and neutral items. After providing some results for response problems that hold for any underlying combinatorial optimization problem, we examine response problems over the maximum-weight matching problem and the minimum-weight perfect matching problem as illustrative and surprisingly varied examples. Among other results, we show that the response problem is hard for even a single given mandatory or forbidden edge. On the other hand, it is fixed-parameter tractable with respect to the total number of non-mandatory edges. If, however, each follower's edge is either mandatory or forbidden, the response problem for the perfect matching problem is solvable in polynomial time while it remains NP-hard for the maximum-weight matching problem.

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arXiv2026

A note on the complexity of two-stage stochastic linear optimization with small second stage

Two-stage stochastic linear optimization is known to be #P-hard when all involved random variables are independently and uniformly distributed over intervals, even with fixed recourse. We show that this problem is actually #P-hard in the strong sense. More surprisingly, this hardness persists when the random vector is one-dimensional, i.e., uniformly distributed over a single interval. To obtain this result, we show that computing the area of a two-dimensional polytope given by a compact extended formulation is strongly #P-hard. Furthermore, we obtain the same complexity result in case the number of second-stage constraints is fixed (for a problem in standard form), while fixing the number of second-stage variables leads to a weakly #P-hard problem. Finally, if both the dimension of the random vector and the number of second-stage constraints are fixed, the problem turns out to be tractable.

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arXiv2026

Switching Point Optimization for Abstract Parabolic Equations

This work is concerned with a switching point optimization problem governed by a semilinear parabolic equation in abstract function spaces. It is shown that the switching-point-to-control mapping is continuously Fr\'echet-differentiable when considered with values in the dual of H\"older continuous functions in time. By treating the state equation in weak form based on the concept of maximal parabolic regularity, one can then show that the reduced objective is continuously differentiable w.r.t. the switching points, which allows to use gradient-based methods like the proximal gradient method for its minimization. Numerical experiments confirm our theoretical findings, but also illustrate that such a method will in general not be able to solve the problem to global optimality due to the non-convex nature of the switching-point-to-control map. We therefore give a precise characterization of the convex hull of set of feasible switching functions in terms of an extended formulation. The latter might be useful for a branch-and-bound approach for the computation of global minimizers, but this is subject to future research.

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arXiv2025

On the feasibility of generalized inverse linear programs

We investigate the feasibility problem for generalized inverse linear programs. Given an LP with affinely parametrized objective function and right-hand side as well as a target set Y, the goal is to decide whether the parameters can be chosen such that there exists an optimal solution that belongs to Y (optimistic scenario) or such that all optimal solutions belong to Y (pessimistic scenario). We study the complexity of this decision problem and show how it depends on the structure of the set Y, the form of the LP, the adjustable parameters, and the underlying scenario. For a target singleton Y={y}, we show that the problem is tractable if the given LP is in standard form, but NP-hard if the LP is given in natural form. If instead we are given a target basis B, the problem in standard form becomes NP-complete in the optimistic case, while remaining tractable in the pessimistic case. For partially fixed target solutions, the problem gets almost immediately NP-hard, but for particular cases we prove tractability if the number of free target variables is fixed. Moreover, we give a rigorous proof of membership in NP for any polyhedral target set, and discuss how this property can be extended to more general target sets using an oracle-based approach.

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arXiv2024

Extended Formulations for Binary Optimal Control Problems

Extended formulations are an important tool in polyhedral combinatorics. Many combinatorial optimization problems require an exponential number of inequalities when modeled as a linear program in the natural space of variables. However, by adding artificial variables, one can often find a small linear formulation, i.e., one containing a polynomial number of variables and constraints, such that the projection to the original space of variables yields a perfect linear formulation. Motivated by binary optimal control problems with switching constraints, we show that a similar approach can be useful also for optimization problems in function space, in order to model the closed convex hull of feasible controls in a compact way. More specifically, we present small extended formulations for switches with bounded variation and for dwell-time constraints. For general linear switching point constraints, we devise an extended model linearizing the problem, but show that a small extended formulation that is compatible with discretization cannot exist unless P=NP.

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arXiv2024

Parabolic optimal control problems with combinatorial switching constraints -- Part III: Branch-and-bound algorithm

We present a branch-and-bound algorithm for globally solving parabolic optimal control problems with binary switches that have bounded variation and possibly need to satisfy further combinatorial constraints. More precisely, for a given tolerance $\varepsilon>0$, we show how to compute in finite time an $\varepsilon$-optimal solution in function space, independently of any prior discretization. The main ingredients in our approach are an appropriate branching strategy in infinite dimension, an a posteriori error estimation in order to obtain safe dual bounds, and an adaptive refinement strategy in order to allow arbitrary switching points in the limit. The performance of our approach is demonstrated by extensive experimental results.

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arXiv2023

Parabolic optimal control problems with combinatorial switching constraints -- Part I: Convex relaxations

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. While such combinatorial constraints are often seen as an additional complication that is treated in a heuristic postprocessing, the core of our approach is to investigate the convex hull of all feasible switching patterns in order to define a tight convex relaxation of the control problem. The convex relaxation is built by cutting planes derived from finite-dimensional projections, which can be studied by means of polyhedral combinatorics. A numerical example for the case of a bounded number of switchings shows that our approach can significantly improve the dual bounds given by the straightforward continuous relaxation, which is obtained by relaxing binarity constraints.

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arXiv2023

Parabolic optimal control problems with combinatorial switching constraints -- Part II: Outer approximation algorithm

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon, they can thus be seen as dynamic switches. The switching patterns may be subject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [arXiv:2203.07121], we describe the $L^p$-closure of the convex hull of feasible switching patterns as intersection of convex sets derived from finite-dimensional projections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in $L^2$ to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semi-smooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm.

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arXiv2023

Bilevel linear optimization belongs to NP and admits polynomial-size KKT-based reformulations

It is a well-known result that bilevel linear optimization is NP-hard. In many publications, reformulations as mixed-integer linear optimization problems are proposed, which suggests that the decision version of the problem belongs to NP. However, to the best of our knowledge, a rigorous proof of membership in NP has never been published, so we close this gap by reporting a simple but not entirely trivial proof. A related question is whether a large enough "big M" for the classical KKT-based reformulation can be computed efficiently, which we answer in the affirmative. In particular, our big M has polynomial encoding length in the original problem data.

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arXiv2022

Partitioned vs. Integrated Planning of Hinterland Networks for LCL Transportation

Utilizing existing transportation networks better and designing (parts of) networks involves routing decisions to minimize transportation costs and maximize consolidation effects. We study the concrete example of hinterland networks for the truck-transportation of less-than-container-load (LCL) ocean freight shipments: A set of LCL shipments is given. They have to be routed through the hinterland network to be transported to an origin port and finally to the destination port via ship. On their way, they can be consolidated in hubs to full-container-load (FCL) shipments. The overall transportation cost depends on the selection of the origin port and the routing and consolidation in the hinterland network. A problem of this type appears for the global logistics provider DB Schenker. We translate the business problem into a hub location problem, describe it mathematically, and discuss solution strategies. As a result, an integrated modeling approach has several advantages over solving a simplified version of the problem, although it requires more computational effort.

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arXiv2022

On the Complexity of the Bilevel Minimum Spanning Tree Problem

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an open question stated in by Shi et al. We prove that BMST remains hard even in the special case where the follower only controls a matching. Moreover, by a polynomial reduction from the vertex-disjoint Steiner trees problem, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a polynomial-time $(|V|-1)$-approximation algorithm for BMST, where $|V|$ is the number of vertices in the input graph. Moreover, considering the number of edges controlled by the follower as parameter, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.

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arXiv2022

The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower's Objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack, while the follower chooses a feasible packing maximizing his own profit. The leader's aim is to optimize a linear objective function in the capacity and in the follower's solution, but with respect to different item values. We address a stochastic version of this problem where the follower's profits are uncertain from the leader's perspective, and only a probability distribution is known. Assuming that the leader aims at optimizing the expected value of her objective function, we first observe that the stochastic problem is tractable as long as the possible scenarios are given explicitly as part of the input, which also allows to deal with general distributions using a sample average approximation. For the case of independently and uniformly distributed item values, we show that the problem is #P-hard in general, and the same is true even for evaluating the leader's objective function. Nevertheless, we present pseudo-polynomial time algorithms for this case, running in time linear in the total size of the items. Based on this, we derive an additive approximation scheme for the general case of independently distributed item values, which runs in pseudo-polynomial time.

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arXiv2022

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower's objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower's problem. More precisely, adopting the robust optimization approach and assuming that the follower's profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower's reaction from the leader's perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader's objective function.

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arXiv2022

An oracle-based framework for robust combinatorial optimization

We propose a general solution approach for min-max-robust counterparts of combinatorial optimization problems with uncertain linear objectives. We focus on the discrete scenario case, but our approach can be extended to other types of uncertainty sets such as polytopes or ellipsoids. Concerning the underlying certain problem, the algorithm is entirely oracle-based, i.e., our approach only requires a (primal) algorithm for solving the certain problem. It is thus particularly useful in case the underlying problem is hard to solve, or only defined implicitly by a given software addressing the certain case. The idea of our algorithm is to solve the convex relaxation of the robust problem by a simplicial decomposition approach, the main challenge being the non-differentiability of the objective function in the case of discrete or polytopal uncertainty. The resulting dual bounds are then used within a tailored branch-and-bound framework for solving the robust problem to optimality. By a computational evaluation, we show that our method outperforms straightforward linearization approaches on the robust minimum spanning tree problem. Moreover, using the Concorde solver for the certain oracle, our approach computes much better dual bounds for the robust traveling salesman problem in the same amount of time.

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arXiv2021

On the Complexity of Robust Bilevel Optimization With Uncertain Follower's Objective

We investigate the complexity of bilevel combinatorial optimization with uncertainty in the follower's objective, in a robust optimization approach. We show that the robust counterpart of the bilevel problem under interval uncertainty can be $Σ^{\text P}_2$-hard, even when the certain bilevel problem is NP-equivalent and the follower's problem is tractable. On the contrary, in the discrete uncertainty case, the robust bilevel problem is at most one level harder than the follower's problem.

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arXiv2019

SDP-based branch-and-bound for non-convex quadratic integer optimization

Semidefinite programming (SDP) relaxations have been intensively used for solving discrete quadratic optimization problems, in particular in the binary case. For the general non-convex integer case with box constraints, the branch-and-bound algorithm Q-MIST has been proposed by Buchheim and Wiegele (Math Program 141(1--2):435--452, 2013), which is based on an extension of the well-known SDP-relaxation for max-cut. For solving the resulting SDPs, Q-MIST uses an off-the-shelf interior point algorithm. In this paper, we present a tailored coordinate ascent algorithm for solving the dual problems of these SDPs. Building on related ideas of Dong (SIAM J Optim 26(3):1962--1985, 2016), it exploits the particular structure of the SDPs, most importantly a small rank of the constraint matrices. The latter allows both an exact line search and a fast incremental update of the inverse matrices involved, so that the entire algorithm can be implemented to run in quadratic time per iteration. Moreover, we describe how to extend this approach to a certain two-dimensional coordinate update. Finally, we explain how to include arbitrary linear constraints into this framework, and evaluate our algorithm experimentally.

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arXiv2018

An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles

We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-oder cone program (SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurobi.

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arXiv2017

A Frank-Wolfe Based Branch-and-Bound Algorithm for Mean-Risk Optimization

We present an exact algorithm for mean-risk optimization subject to a budget constraint, where decision variables may be continuous or integer. The risk is measured by the covariance matrix and weighted by an arbitrary monotone function, which allows to model risk-aversion in a very individual way. We address this class of convex mixed-integer minimization problems by designing a branch-and-bound algorithm, where at each node, the continuous relaxation is solved by a non-monotone Frank-Wolfe type algorithm with away-steps. Experimental results on portfolio optimization problems show that our approach can outperform the MISOCP solver of CPLEX 12.6 for instances where a linear risk-weighting function is considered.

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