arXiv subjects

Sergey S. Ketkov

Publications and source records attributed to Sergey S. Ketkov.

10 recordsLinked to original sources

arXiv2026

On the Tightness of Standard Relaxations for Mixed-Integer Bilevel Linear Programs

Exact algorithms for solving mixed-integer bilevel linear programs (MIBLPs) typically rely on sequences of lower and upper bounds that converge to the optimal value. These procedures are commonly initialized using the single-level relaxation (SLR), obtained by omitting the follower's optimality condition and solving the resulting single-level optimization problem. In this paper, we investigate whether, for broad classes of MIBLPs, the resulting standard bounds admit uniform improvements that can be computed within the same computational complexity regime. For pure continuous bilevel linear programs, we show that, unless $P = NP$, neither the SLR-based lower bound nor its associated upper bound can be uniformly improved in polynomial time, even for the class of min-max problems. We then extend this analysis to the class of pure integer min-max bilevel linear programs under the assumption that the polynomial hierarchy does not collapse. First, we show that the continuous relaxation of the SLR admits no uniform polynomial-time computable improvement. We then prove that neither the SLR itself nor its associated upper bound admits a uniform improvement by a polynomial-time algorithm with access to a mixed-integer linear programming (MILP) oracle. Importantly, this rules out uniform improvements by iterative MILP-based approaches, including cutting-plane-based and decomposition algorithms. Overall, our results demonstrate that the SLR-based bounds are, in a complexity-theoretic sense, unimprovable systematically within their natural computational regimes.

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arXiv2026

On the Complexity of Bilevel Linear and Quadratic Programs in Fixed Dimensions

It is well-known that general bilevel linear programs (BLPs) are strongly $NP$-hard, even when the leader's and the follower's objective functions are exact opposites. However, the complexity classification of BLPs remains incomplete when one of the decision-makers has a fixed number of variables or constraints. In this paper, we close the remaining gap in this complexity landscape. Thus, while optimistic BLPs are known to be polynomially solvable when the number of follower variables is fixed, we prove that the corresponding pessimistic problem is strongly $NP$-hard. To the best of our knowledge, this is the first result demonstrating that, under comparable assumptions, the pessimistic formulation can be computationally harder than its optimistic counterpart. In addition, we prove that BLPs remain polynomially solvable in both the optimistic and the pessimistic settings when the number of follower constraints is fixed. We further investigate whether these polynomial-time solvability results persist for bilevel convex quadratic programs. While the optimistic formulation remains polynomially solvable when the number of follower variables is fixed, we prove that the pessimistic formulation with a fixed number of follower constraints becomes $NP$-hard. In other words, unless $P = NP$, there is a strict complexity gap between bilevel programs with linear and convex quadratic objective functions. Finally, we show that replacing a convex quadratic follower objective with a nonconvex quadratic one renders the optimistic problem $NP$-hard, even when both follower dimensions are fixed.

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arXiv2026

On Big-M Reformulations of Bilevel Linear Programs: Hardness of A Posteriori Verification

A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary binary variables. This yields a single-level mixed-integer linear programming (MILP) model involving big-$M$ parameters. While sufficiently large and bilevel-correct big-$M$s can be computed in polynomial time, verifying a priori that given big-$M$s do not cut off any feasible or optimal lower-level solutions is known to be computationally difficult. In this paper, we establish two complementary hardness results. First, we show that, even with a single potentially incorrect big-$M$ parameter, it is $coNP$-complete to verify a posteriori whether the optimal solution of the resulting MILP model is bilevel optimal. In particular, this negative result persists for min-max problems without coupling constraints and applies to strong-duality-based reformulations of mixed-integer BLPs. Second, we show that verifying global big-$M$ correctness remains computationally difficult a posteriori, even when an optimal solution of the MILP model is available.

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arXiv2026

Data-driven interdiction with asymmetric cost uncertainty: a distributionally robust optimization approach

We consider a class of stochastic interdiction games between an upper-level decision-maker (the leader) and a lower-level decision-maker (the follower), where uncertainty lies in the follower's objective function coefficients. Specifically, the follower's profits (or costs) in our model comprise a random vector, whose probability distribution is estimated independently by the leader and the follower, based on their own data. To address the distributional uncertainty, we formulate a distributionally robust interdiction (DRI) model, where both decision-makers solve conventional distributionally robust optimization problems based on the Wasserstein metric. For this model, we prove asymptotic consistency and derive a polynomial-size mixed-integer linear programming (MILP) reformulation. Furthermore, in our bilevel optimization context, the leader may face uncertainty due to its incomplete knowledge of the follower's data. In this regard, we propose two distinct approximations of the true DRI model, where the leader has incomplete or no information about the follower's data. The first approach employs a pessimistic approximation, which turns out to be computationally challenging and requires a specialized reformulation amenable to a Benders-type decomposition algorithm. The second approach leverages a robust optimization approach from the leader's perspective. To address the resulting problem, we propose a scenario-based approximation that admits a potentially large single-level MILP reformulation and satisfies asymptotic robustness guarantees. Finally, for a class of randomly generated instances of the packing interdiction problem, we evaluate numerically how the information asymmetry and the decision-makers' risk preferences affect the models' out-of-sample performance.

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arXiv2025

On a class of interdiction problems with partition matroids: complexity and polynomial-time algorithms

In this study, we consider a class of linear matroid interdiction problems, where the feasible sets for the upper-level decision-maker (referred to as a leader) and the lower-level decision-maker (referred to as a follower) are induced by two distinct partition matroids with a common weighted ground set. Unlike classical network interdiction models where the leader is subject to a single budget constraint, in our setting, both the leader and the follower are subject to several independent capacity constraints and engage in a zero-sum game. While the problem of finding a maximum weight independent set in a partition matroid is known to be polynomially solvable, we prove that the considered bilevel problem is $NP$-hard even when the weights of ground elements are all binary. On a positive note, it is revealed that, if the number of capacity constraints is fixed for either the leader or the follower, then the considered class of bilevel problems admits several polynomial-time solution schemes. Specifically, these schemes are based on a single-level dual reformulation, a dynamic programming-based approach, and a greedy algorithm for the leader.

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arXiv2024

A study of distributionally robust mixed-integer programming with Wasserstein metric: on the value of incomplete data

This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains.

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arXiv2022

On the multi-stage shortest path problem under distributional uncertainty

In this paper we consider an ambiguity-averse multi-stage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker's objective is to maximize the user's expected loss by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the related studies, both the user and the attacker can dynamically adjust their decisions at particular nodes of the user's path. By observing the user's decisions, the attacker may reveal some additional distributional information associated with the arcs emanated from the current user's position. It is shown that the resulting multi-stage distributionally robust shortest path problem (DRSPP) admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network~instances.

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arXiv2022

On a class of data-driven mixed-integer programming problems under uncertainty: a distributionally robust approach

In this study we analyze linear mixed-integer programming problems, in which the distribution of the cost vector is only observable through a finite training data set. In contrast to the related studies, we assume that the number of random observations for each component of the cost vector may vary. Then the goal is to find a prediction rule that converts the data set into an estimate of the expected value of the objective function and a prescription rule that provides an associated estimate of the optimal decision. We aim at finding the least conservative prediction and prescription rules, which satisfy some specified asymptotic guarantees as the sample size tends to infinity. We demonstrate that under some mild assumption the resulting vector optimization problems admit a Pareto optimal solution with some attractive theoretical properties. In particular, this solution can be obtained by solving a distributionally robust optimization (DRO) problem with respect to all probability distributions with given component-wise relative entropy distances from the empirical marginal distributions. It turns out that the outlined DRO problem can be solved rather effectively whenever there exists an effective algorithm for the respective deterministic problem. In addition, we perform numerical experiments where the out-of-sample performance of the proposed approach is analyzed.

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arXiv2021

An approach to the distributionally robust shortest path problem

In this study we consider the shortest path problem, where the arc costs are subject to distributional uncertainty. Basically, the decision-maker attempts to minimize her worst-case expected loss over an ambiguity set (or a family) of candidate distributions that are consistent with the decision-maker's initial information. The ambiguity set is formed by all distributions that satisfy prescribed linear first-order moment constraints with respect to subsets of arcs and individual probability constraints with respect to particular arcs. Under some additional assumptions the resulting distributionally robust shortest path problem (DRSPP) admits equivalent robust and mixed-integer programming (MIP) reformulations. The robust reformulation is shown to be $NP$-hard, whereas the problem without the first-order moment constraints is proved to be polynomially solvable. We perform numerical experiments to illustrate the advantages of the considered approach; we also demonstrate that the MIP reformulation of DRSPP can be solved effectively using off-the-shelf solvers.

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arXiv2019

On Greedy and Strategic Evaders in Sequential Interdiction Settings with Incomplete Information

We consider a class of sequential network interdiction problem settings where the interdictor has incomplete initial information about the network while the evader has complete knowledge of the network including its structure and arc costs. In each decision epoch, the interdictor can block (for the duration of the epoch) at most $k$ arcs known to him/her. By observing the evader's actions, the interdictor learns about the network structure and costs and thus, can adjust his/her actions in subsequent decision epochs. It is known from the literature that if the evader is greedy (i.e., the shortest available path is used in each decision epoch), then under some assumptions the greedy interdiction policies that block $k$-most vital arcs in each epoch are efficient and have a finite regret. In this paper, we consider the evader's perspective and explore deterministic "strategic" evasion policies under the assumption that the interdictor is greedy. We first study the theoretical computational complexity of the evader's problem. Then we derive basic constructive properties of optimal evasion policies for two decision epochs when the interdictor has no initial information about the network structure. These properties are then exploited for the design of a heuristic algorithm for a strategic evader in a general setting with an arbitrary time horizon and any initial information available to the interdictor. Our computational experiments demonstrate that the proposed heuristic outperforms the greedy evasion policy on several classes of synthetic network instances under either perfect or noisy information feedback. Finally, some interesting insights from our theoretical and computational results conclude the paper.

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