Two‐stage stochastic minimum s − t cut problems: Formulations, complexity and decomposition algorithms

We introduce the two‐stage stochastic minimum s − t cut problem. Based on a classical linear 0‐1 programming model for the deterministic minimum s − t cut problem, we provide a mathematical programming formulation for the proposed stochastic extension. We show that its constraint matrix loses the total unimodularity property, however, preserves it if the considered graph is a tree. This fact turns out to be not surprising as we prove that the considered problem is NP‐hard in general, but admits a linear time solution algorithm when the graph is a tree. We exploit the special structure of the problem and propose a tailored Benders decomposition algorithm. We evaluate the computational efficiency of this algorithm by solving the Benders dual subproblems as max‐flow problems. For many tested instances, we outperform a standard Benders decomposition by two orders of magnitude with the Benders decomposition exploiting the max‐flow structure of the subproblems

On a class of data-driven combinatorial optimization problems under uncertainty: a distributionally robust approach

In this study we analyze linear combinatorial optimization problems where the cost vector is not known a priori, but is only observable through a finite data set. In contrast to the related studies, we presume that the number of observations with respect to particular components of the cost vector may vary. The goal is to find a procedure that transforms the data set into an estimate of the expected value of the objective function (which is referred to as a prediction rule) and a procedure that retrieves a candidate decision (which is referred to as a prescription rule). We aim at finding the least conservative prediction and prescription rules, which satisfy some specified asymptotic guarantees. We demonstrate that the resulting vector optimization problems admit a weakly optimal solution, which can be obtained by solving a particular distributionally robust optimization problem. Specifically, the decision-maker may optimize the worst-case expected loss across all probability distributions with given component-wise relative entropy distances from the empirical marginal distributions. Finally, we perform numerical experiments to analyze the out-of-sample performance of the proposed solution approach

On Bilevel Optimization with Inexact Follower

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader's actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower's use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems

On Maximum Degree-Based Γ-Quasi-Clique Problem: Complexity And Exact Approaches

We consider the problem of finding a degree-based γ-quasi-clique of maximum cardinality in a given graph for some fixed γ ∈ (0,1]. A degree-based γ-quasi-clique (often referred to as simply a quasi-clique) is a subgraph, where the degree of each vertex is at least γ times the maximum possible degree of a vertex in the subgraph. A degree-based γ-quasi-clique is a relative clique relaxation model, where the case of γ=1 corresponds to the well-known concept of a clique. In this article, we first prove that the problem is NP-hard for any fixed γ ∈ (0,1], which addresses one of the open questions in the literature. More importantly, we also develop new exact solution methods for solving the problem and demonstrate their advantages and limitations in extensive computational experiments with both random and real-world networks. Finally, we outline promising directions of future research including possible functional generalizations of the considered clique relaxation model. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(2), 136–152 2018

A Simple Greedy Heuristic For Linear Assignment Interdiction

We consider a bilevel extension of the classical linear assignment problem motivated by network interdiction applications. Specifically, given a bipartite graph with two different (namely, the leader’s and the follower’s) edge costs, the follower solves a linear assignment problem maximizing his/her own profit, whereas the leader is allowed to affect the follower’s decisions by eliminating some of the vertices from the graph. The leader’s objective is to minimize the total cost given by the cost of the interdiction actions plus the cost of the assignments made by the follower. The considered problem is strongly NP-hard. First, we formulate this problem as a linear mixed integer program (MIP), which can be solved by commercial MIP solvers. More importantly, we also describe a greedy-based construction heuristic, which provides (under some mild conditions) an optimal solution for the case, where the leader’s and the follower’s edge costs are equal to one. Finally, we present the results of our computational experiments comparing the proposed heuristic against an MIP solver

The equitable dispersion problem

Most optimization problems focus on efficiency-based objectives. Given the increasing awareness of system inequity resulting from solely pursuing efficiency, we conceptualize a number of new element-based equity-oriented measures in the dispersion context. We propose the equitable dispersion problem that maximizes the equity among elements based on the introduced measures in a system defined by inter-element distances. Given the proposed optimization framework, we develop corresponding mathematical programming formulations as well as their mixed-integer linear reformulations. We also discuss computational complexity issues, related graph-theoretic interpretations and provide some preliminary computational results.Equitable dispersion Maximum dispersion Equity Binary nonlinear programming

Optimal Condition-Based Maintenance via a Mobile Maintenance Resource

We consider the problem of performing condition-based maintenance on a set of geographically distributed assets via a single maintenance resource that travels between the assets’ locations. That is, we dynamically determine the optimal positioning of the maintenance resource and the optimal timing of condition-based maintenance interventions that the maintenance resource performs. These decisions are made as a function of the conditions of the assets and the current location of the maintenance resource to minimize total expected costs, which include downtime, travel, and maintenance expenses. This holistic approach enables us to study unique trade-offs, namely, maintaining an asset early if the maintenance resource is currently close by, or alternatively, optimally repositioning the maintenance resource or having it idle in key locations in anticipation of asset deterioration. We model the location of the maintenance resource and assets using a graph representation and the assets’ deterioration process as a discrete-time Markov chain. We formulate a Markov decision process to obtain the optimal policy for the maintenance resource (i.e., where to travel, idle, or repair). We explore the properties of the optimal policies (analytically and numerically) and how they are affected by the graph structure. Finally, we develop and analyze some implementation-friendly heuristic policies