# George L. NemhauserGeorgia Institute of Technology | GT · School of Industrial and Systems Engineering

George L. Nemhauser

PhD

## About

326

Publications

88,238

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33,760

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Citations since 2017

Introduction

**Skills and Expertise**

Additional affiliations

September 1985 - present

## Publications

Publications (326)

The service network design problem is commonly used to represent the tactical decisions encountered by a consolidation carrier operating a hub‐and‐spoke network: what transportation services to operate between hubs and how to route commodities from their origin to their destination through the network. In most settings, the capacity at hubs is not...

We consider integer programming (IP) problems consisting of (possibly a large number of) subsystems and a small number of coupling constraints that link variables from different subsystems. Such problems are called loosely coupled or nearly decomposable. In this paper, we exploit the idea of resource-directive decomposition to reformulate the probl...

Finding a shortest path in a network is a fundamental optimization problem. We focus on settings in which the travel time on an arc in the network depends on the time at which traversal of the arc begins. In such settings, reaching the destination as early as possible is not the only objective of interest. Minimizing the duration of the path, that...

This paper considers a class of two-stage stochastic mixed-integer optimization problems where, for a given first-stage solution, we can determine the optimal values of recourse variables sequentially. This class of problems arises in a wide variety of applications. In the case of multivariate discrete distributions for uncertain parameters, a stan...

We consider a problem involving a set of agents who need to coordinate their actions to optimize the sum of their objectives while satisfying a common resource constraint. The objective functions of the agents are unknown to them a priori and are revealed in an online manner. The resulting problem is an online optimization problem to optimally allo...

We consider a multi-player optimization where each player has her own optimization problem and the individual problems are connected by a cardinality constraint on their shared resources. We give distributed algorithms that allow each player to solve their own optimization problem and still achieve a global optimization solution for problems that p...

Waiting at the right location at the right time can be critically important in certain variants of time-dependent shortest path problems. We investigate the computational complexity of time-dependent shortest path problems in which there is either a penalty on waiting or a limit on the total time spent waiting at a given subset of the nodes. We sho...

In \cite{siebert2019linear} the authors present a set of integer programs (IPs) for the Steiner tree problem, which can be used for both, the directed and the undirected setting of the problem. Each IP finds an optimal Steiner tree with a specific structure. A solution with the lowest cost, corresponds to an optimal solution to the entire problem....

We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all IPs provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is integral so that it can be solved as...

We present a set of integer programs (IPs) for the Steiner tree problem with the property that the best solution obtained by solving all, provides an optimal Steiner tree. Each IP is polynomial in the size of the underlying graph and our main result is that the linear programming (LP) relaxation of each IP is integral so that it can be solved as a...

We present an exact algorithm for the Minimum Duration Time-Dependent Shortest Path Problem with piecewise linear arc travel time functions. The algorithm iteratively refines a time-expanded network model, which allows for the computation of a lower and an upper bound, until - in a finite number of iterations - an optimal solution is obtained.

We present a parallel large neighborhood search framework for finding high quality primal solutions for general mixed-integer programs (MIPs). The approach simultaneously solves a large number of sub-MIPs with the dual objective of reducing infeasibility and optimizing with respect to the original objective. Both goals are achieved by solving restr...

This chapter presents an extensive computational study of simple, but prominent matheuristics (i.e., heuristics that rely on mathematical programming models) to find high quality ship schedules and inventory policies for a class of maritime inventory routing problems. Our computational experiments are performed on a test bed of the publicly availab...

Because semiconductor manufacturing is a complex and dynamic process, production scheduling in this industry typically relies on simple decision policies that use local rather than global information. Such myopic policies may lead to increased congestion in the material handling system and negatively impact throughput. In this paper, we propose a f...

We show how recently-defined abstract models of the Branch-and-Bound algorithm can be used to obtain information on how the nodes are distributed in B&B search trees. This can be directly exploited in the form of probabilities in a sampling algorithm given by Knuth that estimates the size of a search tree. This method reduces the offline estimation...

Primal heuristics'' are a key contributor to the improved performance of exact branch-and-bound solvers for combinatorial optimization and integer programming. Perhaps the most crucial question concerning primal heuristics is that of at which nodes they should run, to which the typical answer is via hard-coded rules or fixed solver parameters tuned...

This paper studies a Maritime Inventory Routing Problem with Time Windows (MIRPTW) for deliveries with uncertain disruptions. We consider disruptions that increase travel times between ports and ultimately affect the deliveries in one or more time windows. The objective is to find flexible solutions that can accommodate unplanned disruptions. We pr...

We present a new framework for finding feasible solutions to mixed integer programs (MIP). We use the feasibility pump heuristic coupled to a biased random-key genetic algorithm (BRKGA). The feasibility pump heuristic attempts to find a feasible solution to a MIP by first rounding a solution to the linear programming (LP) relaxation to an integer (...

We present a parallel local search approach for obtaining high quality solutions to the Fixed Charge Multicommodity Network Flow problem (FCMNF). The approach proceeds by improving a given feasible solution by solving restricted instances of the problem where flows of certain commodities are fixed to those in the solution while the other commoditie...

We study the minimum-concave-cost flow problem on a two-dimensional grid. We characterize the computational complexity of this problem based on the number of rows and columns of the grid, the number of different capacities over all arcs, and the location of sources and sinks. The concave cost over each arc is assumed to be evaluated through an orac...

The design of strategies for branching in Mixed Integer Programming (MIP) is guided by cycles of parameter tuning and offline experimentation on an extremely heterogeneous testbed, using the average performance. Once devised, these strategies (and their parameter settings) are essentially input-agnostic. To address these issues, we propose a machin...

The selection of branching variables is a key component of branch-and-bound
algorithms for solving Mixed-Integer Programming (MIP) problems since the
quality of the selection procedure is likely to have a significant effect on
the size of the enumeration tree. State-of-the-art procedures base the
selection of variables on their "LP gains", which is...

In linear programming based branch-and-bound algorithms, many heuristics have been developed to improve primal solutions, but on the dual side we rely solely on cutting planes to improve dual bounds. We design a dual heuristic which incorporates relaxation algorithms within a branch-and-bound framework to improve dual bounds. We study the effect of...

The minimum concave cost network flow problem (MCCNFP) is NP-hard, but efficient polynomial-time algorithms exist for some special cases such as the uncapacitated lot-sizing problem and many of its variants. We study the MCCNFP over a grid network with a general nonnegative separable concave cost function. We show that this problem is polynomially...

We show the importance of selecting good branching variables by exhibiting a family of instances for which an optimal solution is both trivial to find and provably optimal by a fixed-size branch-and-bound tree, but for which state-of-the-art Mixed Integer Programming solvers need an increasing amount of resources. The instances encode the edge-colo...

We present two decomposition algorithms for single product deep-sea maritime inventory routing problems (MIRPs) that possess a core substructure common in many real-world applications. The problem involves routing vessels, each belonging to a particular vessel class, between loading and discharging ports, each belonging to a particular region. Our...

The theme of this chapter is to use structure to determine strong valid inequalities for the constraint sets of some N-P hard integer programming problems. The determination of families of strong valid inequalities is more of an art than a formal methodology. Thus the presentation will largely be a series of examples that convey the basic ideas. Th...

This chapter discusses approaches for finding an optimal, or e-approximate, solution of the linear integer programming problem. It gives a fractional cutting-plane algorithm (FCPA) for general integer programs that uses C-G inequalities. The chapter presents a fractional cutting-plane algorithm (FCPA) for pure-integer programs that uses Gomory cuts...

This chapter describes a class of matrices and their subsets, including node-arc incidence matrices of digraphs. It provides a recognition algorithm for these matrices and observes that the associated linear programming problem can be solved by network flow algorithms. The chapter continues to describe matrices, in terms of forbidden submatrices, s...

Matroids and submodular functions are the foundations for some combinatorial optimization problems that generalize both network flow problems and the spanning tree problem. Matroids can be viewed as prototypes of independence systems and 0-1 integer programs with “nice” properties that can be used to obtain efficient algorithms for the correspondin...

Matching problems involve choosing a subset of the edges subject to degree constraints on the nodes. This chapter highlights that the weighted perfect 1-matching problem is a meaningful generalization of the assignment problem. An application of weighted 1-matching is to the postman problem. Matching problems are celebrated in the history of combin...

The theme of this chapter is to develop a theory for determining zIP, or at least a good upper bound on zIP, without explicitly solving IP. This can be considered to be a theory of optimality, since a tight bound on zIP provides the fundamental way of proving optimality of a feasible solution to IP. The chapter considers relaxations of the maximum-...

The structure invoked in this chapter is that the problems have only one constraint other than bounds and integrality on the variables. It considers the integer knapsack problem, the group problem, and the 0-1 knapsack problem. The chapter introduces the uncapacitated lot-size problem (ULS) using the formulation, and then reformulates it as an unca...

This chapter describes a theory of computational complexity that yields insights into how difficult a problem may be to solve. For most integer programming problems, no such algorithm is known. The chapter shows that there are integer programming problems much more specific than the general pure-integer programming problem with the following proper...

The three major reasons why a problem class may not be solved satisfactorily by a general algorithm are: (1) size of the formulation; (2) weakness of the bounds; and (3) speed of the algorithm. This chapter shows how structure can be used either to devise special-purpose algorithms or to improve the performance of general algorithms for several cla...

This chapter discusses representation of an integer program by a linear program that has the same optimal solution. It shows the relationship between the valid inequalities used by Gomory and the rounding procedure. The chapter develops a procedure for generating valid inequalities for T. Note that the C-G procedure does not work when there are con...

This chapter presents some polynomial-time algorithms for linear programming and discusses their consequences in combinatorial optimization. It talks about the ellipsoid algorithm, which was acclaimed on the front pages of newspapers throughout the world when it appeared in 1979. Although the algorithm turned out to be computationally impractical,...

This chapter discusses the basic problem which is: The Linear Equation Integer Feasibility Problem. It describes the euclidean algorithm to find gcd(a1, a2). The chapter establishes the connection between the euclidean algorithm and the continued fraction expansion of a rational number a1/a2. It introduces some basic properties of the lattice L(A)...

This chapter gives the terminology and some elementary results of graph theory. It defines some classical optimization problems on graphs and presents algorithms to solve them. All of these problems are linear programming problems and, excluding the minimum-weight spanning tree problem, are in the class of linear programming problems known as netwo...

A good understanding of the theory and algorithms of linear programming is essential for understanding integer programming. Integer programming is a much harder problem than linear programming, and neither the theory nor the computational aspects of integer programming are as developed as they are for linear programming. So, first of all, the theor...

Integer programming is different. Typically, a set S ϵ Zn + of feasible points are described implicitly, for example, the set of integer solutions to a linear inequality system S = {x ϵ Zn+: Ax < b}, the set of binary vectors corresponding to tours in a graph, and so on. One of the objectives is to find a linear inequality description of the set. F...

We study a deterministic maritime inventory routing problem with a long planning horizon. For instances with many ports and many vessels, mixed-integer linear programming (MIP) solvers often require hours to produce good solutions even when the planning horizon is 90 or 120 periods. Building on the recent successes of approximate dynamic programmin...

This paper presents a detailed description of a particular class of deterministic single product Maritime Inventory Routing Problems (MIRPs), which we call deep-sea MIRPs with inventory tracking at every port. This class involves vessel travel times between ports that are significantly longer than the time spent in port and require inventory levels...

In automated material handling systems (AMHS), such as those used to transport wafers in semiconductor manufacturing facilities, vehicular congestion leads to transport delay and reduced production efficiency. Through the use of a high-fidelity simulation, we demonstrate a congestion-aware dynamic routing strategy that efficiently reroutes vehicles...

Performance variability of modern mixed-integer programming solvers and possible ways of exploiting this phenomenon present an interesting opportunity in the development of algorithms to solve mixed-integer linear programs (MILPs). We propose a framework using multiple branch-and-bound trees to solve MILPs while allowing them to share information i...

This paper considers a cutting and scheduling problem of minimizing scrap motivated by float glass manufacturing and introduces the float glass scheduling problem. We relate it to classical problems in the scheduling literature such as no-wait hybrid flow shops and cyclic scheduling. We show that the problem is NP-hard, and identify when each of th...

Statement of Scope and Purpose Flat glass is approximately a $20 billion/year industry worldwide, with almost all flat glass products being manufactured on float glass lines. New technologies are allowing float glass manufacturers to increase the level of automation in their plants, but the question of how to effectively use the automation has give...

In the United States Army, regional brigades schedule and attend recruiting events. A set of centrally managed physical assets such as large-scale displays and branded vehicles provide supplemental support. The integer programming model presented here assigns assets to maximize recruitment value for sets of up to 500 events given event values, loca...

We apply branch-and-price guided search to a real-world maritime inventory routing problem, in which the inventory of a single product, which is produced and consumed at multiple sites, and its transport, which is done with a heterogeneous fleet of vessels, is managed over a finite horizon. Computational experiments demonstrate that branch-and-pric...

A highly desirable characteristic of methods for solving 0-1 mixed-integer programs is that they should be capable of producing high-quality solutions quickly. We introduce restrict-and-relax search, a branch-and-bound algorithm that explores the solution space not only by fixing variables (restricting), but also by freeing, or unfixing, previously...

We develop an exact algorithm for integer programs that uses restrictions of the problem to produce high-quality solutions quickly. Column generation is used both for generating these problem restrictions and for producing bounds on the value of the optimal solution. The performance of the algorithm is greatly enhanced by using structure, such as a...

The scheduling and routing of small planes for fly-in safaris is a challenging planning problem. Given a fleet of planes and a set of flight requests with bounds on the earliest departure and latest arrival times, the planes must be scheduled and routed so that all demands are satisfied. Capacity restrictions on the load and fuel also must be satis...

Adjusting prices to influence demand so as to increase revenue has become common practice. We investigate adjusting prices to influence demand so as to reduce cost. More specifically, we consider offering price discounts in return for production and delivery flexibility. We do so in the context of the single-item, single-level uncapacitated lot-siz...

Consider a network N =(N, A) and associate with each arc e ∈ A a fixed cost ce for using arc e, an interval [le, ue] (le, ue ∈ Z) specifying the range of allowable resource consumption quantities along arc e, and a per-unit cost $\bar{c}_e$ for resource consumed along e. Furthermore, for each node n ∈ N, let Un ∈ Z be the maximum amount of resource...

We introduce two new formulations for probabilistic constraints based on extended disjunctive formulations. Their strength results from considering multiple rows of the probabilistic constraints together. The properties of the first can be used to construct hierarchies of relaxations for probabilistic constraints, while the second provides computat...

When solving large-scale integer programming (IP) models, there are the conflicting goals of solution quality and solution time. Solving realistic-size instances of many problems to optimality is still beyond the capability of state-of-the-art solvers. However, by reducing the size of the solution space, such as by fixing variables, good primal sol...

The sell or hold problem (SHP) is to sell kk out of nn indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. We show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number...

A branch-price-and-cut algorithm is developed for a complex maritime inventory-routing problem with varying storage capacities and production/consumption rates at facilities. The resulting mixed-integer pricing problem is solved exactly and efficiently using a dynamic program that exploits certain "extremal" characteristics of the pricing problem....

We characterize the value function of a discounted infinite-horizon version of the single-item lot-sizing problem. As corollaries, we show that this value function inherits several properties of finite, mixed-integer program value functions; namely, it is subadditive, lower semicontinuous, and piecewise linear.

We provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff. Under a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triang...

We describe families of inequalities for 0--1 mixed-integer programming problems that are obtained by lifting cover and packing inequalities. We show that these inequalities can be separated from single rows of the simplex tableaux of their linear programming relaxations. We present the results of a computational study comparing their performance w...

Numerous planning models within the chemical, petroleum, and process industries involve coordi-nating the movement of raw materials in a distribution network so that they can be blended into final products. The uncapacitated fixed-charge transportation problem with blending (FCTPwB) studied in this paper captures a core structure encountered in man...

We present a relaxation-based dynamic programming algorithm for solving resource-constrained shortest-path problems (RCSPPs) in the context of column generation for the dial-a-flight problem. The resulting network formulation and pricing problem require solving RCSPPs on extremely large time-expanded networks having a huge number of local resource...

We present a time decomposition for inventory routing problems. The methodology is based on valuing inventory with a concave piecewise linear function and then combining solutions to single-period subproblems using dynamic programming techniques. Computational experiments show that the resulting value function accurately captures the inventory's va...

Advances in aviation technology including the development of relatively cheap, very light jets and the possibility of free-flight have led to the realization of a per-seat, on-demand air transportation business that operates without a published flight schedule. One of the decision problems this business faces is planning the scheduled maintenance t...

This paper studies two mixed-integer linear programming (MILP) formulations for piecewise linear functions considered in Li et al. [Li, H.-L., H.-C. Lu, C.-H. Huang, N.-Z. Hu. 2009. A superior representation method for piecewise linear functions. INFORMS J. Comput. 21 (2) 314-321]. Although the ideas used to construct one of these formulations are...

The use of auction mechanisms like the GSP in online advertising can lead to loss of both efficiency and revenue when advertisers have rich preferences: even simple forms of expressiveness like budget constraints can lead to suboptimal outcomes. This has led to the recognition of the value of (sequential and/or stochastic) optimization in ad alloca...

We develop a solution approach for the fixed charge network flow problem (FCNF) that produces provably high-quality solutions quickly. The solution approach combines mathematical programming algorithms with heuristic search techniques. To obtain high-quality solutions it relies on neighborhood search with neighborhoods that involve solving carefull...

We study the modeling of nonconvex piecewise-linear functions as mixed-integer programming (MIP) problems. We review several new and existing MIP formulations for continuous piecewise-linear functions with special attention paid to multivariate nonseparable functions. We compare these formulations with respect to their theoretical properties and th...

Linear programs with joint probabilistic constraints (PCLP) are known to be highly in- tractable due to the non-convexity of the feasible region. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We present a mixed integer programming formulation and study the relaxation...

Branching variable selection can greatly affect the effectiveness and efficiency of a branch-and-bound algorithm. Traditional
approaches to branching variable selection rely on estimating the effect of the candidate variables on the objective function.
We propose an approach which is empowered by exploiting the information contained in a family of...

We present a strategic planning model in which the activities to be planned, such as produc- tion and distribution in a supply network, require technology to be installed before they can be performed. The technology improves over time, so that a decision-maker has incentive to delay starting an activity to take advantage of better technology and lo...

Two minimum cardinality set covering problems of similar structure are presented as difficult test problems for evaluating
the computational efficiency of integer programming and set covering algorithms. The smaller problem has 117 constraints and
27 variables, and the larger one, constructed by H.J. Ryser, has 330 constraints and 45 variables. The...

We consider optimization problems with some binary variables, where the objective function is linear in these variables. The stability region of a given solution of such a problem is the polyhedral set of objective coefficients for which the solution is optimal. A priori knowledge of this set provides valuable information for sensitivity analysis a...

This paper addresses the problem of finding cutting planes for multi-stage stochastic integer programs. We give a general method for generating cutting planes for multi-stage stochastic integer programs based on combining inequalities that are valid for the individual scenarios. We apply the method to generate cuts for a stochastic version of a dyn...

Let N be a finite set and
a nonempty collection of subsets of N which have the property that
and F
2⊂F
1 imply
. A real-valued function z defined on the subsets of N that satifies z(S)≤z(T) for all S⊂T⊃-N and z(S)+z(T)≥(S∪T)+z(S∩T) for all S,T⊂-N is called nondecreasing and submodular. We consider the problem
, z(S) submodular and nondecreasing...

We develop an exact solution approach for integer programs that produces high- quality solutions quickly by solving well-chosen restrictions of the problem. Column generation is used both for generating these problem restrictions and for producing bounds on the value of an optimal solution to the problem. Obtaining primal solutions by solving probl...

The availability of relatively cheap small jet aircrafts suggests a new air transporta- tion business: dial-a-igh t, an on-demand service in which travelers call a few days in advance to schedule transportation. A successful on-demand air transportation ser- vice requires an eectiv e scheduling system to construct minimum cost pilot and jet itinera...

This paper develops a linear programming based branch-and-bound algorithm for mixed in- teger conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is dierent from other linear programming based branch-and-bound algo...

The availability of relatively cheap small jet planes has led to the creation of on-demand air transportation services in which travelers call a few days in advance to schedule a flight. A successful on-demand air transportation service requires an effective scheduling system to construct minimum-cost pilot and jet itineraries for a set of accepted...

Many combinatorial constraints over continuous variables such as SOS1 and SOS2 constraints can be interpreted as disjunctive
constraints that restrict the variables to lie in the union of m specially structured polyhedra. Known mixed integer binary formulations for these constraints have a number of binary variables
and extra constraints that is li...

A branch-and-cut algorithm for solving linear problems with continuous separable piecewise linear cost functions was developed in 2005 by Keha et al. This algorithm is based on valid inequalities for an SOS2 based formulation of the problem. In this paper we study the extension of the algorithm to the case where the cost function is only lower semi...

This paper reviews George Dantzig’s contributions to integer programming, especially his semin