integer programming

integer programming
Name	Integer programming
Type	Discrete optimization
Related	Linear programming, Combinatorial optimization, Mixed-integer programming
First proposed	1950s

integer programming

Integer programming is a class of mathematical optimization in which some or all decision variables are constrained to take integer values. Originating in mid-20th century operations research, it connects to linear programming, combinatorial optimization, and computational complexity and is central to practice in logistics, scheduling, and network design. Prominent developments were influenced by researchers and institutions associated with John von Neumann, George Dantzig, Harold Kuhn, Albert Tucker, H. W. Kuhn, and research centers such as Bell Labs, RAND Corporation, IBM Research, and Courant Institute.

Introduction

Integer programming models encode discrete choices using binary, bounded, or general integer variables and often extend linear programs studied by George Dantzig and John von Neumann. Important early results emerged from investigations at Princeton University and Stanford University and from algorithmic foundations laid by researchers affiliated with Massachusetts Institute of Technology and University of California, Berkeley. The field interacts with paradigm-defining topics studied at Institute for Advanced Study and industrial problems addressed by AT&T and General Electric.

Mathematical Formulation

A standard integer program imposes linear constraints and an objective function over integer vectors, generalizing linear programs used in models by George Dantzig and theoretical constructs analyzed at Bell Labs. Common formulations include pure integer programs, mixed-integer programs, and binary (0–1) programs; canonical examples arise in formulations of the Travelling Salesman Problem studied in combinatorial optimization literature and in network flow variants related to work at IBM Research. Polyhedral theory, advanced by researchers at Mathematical Sciences Research Institute and INRIA, characterizes feasible integer hulls via facets, cuts, and valid inequalities, tools developed alongside contributions from scholars linked to Cornell University and University of Waterloo.

Solution Methods

Exact solution methods combine branch-and-bound, branch-and-cut, and cutting-plane techniques; branch-and-bound was popularized in computational studies at RAND Corporation and Bell Labs, while cutting-plane methods trace to work inspired by researchers at Princeton University and Columbia University. Heuristics and metaheuristics—such as tabu search, simulated annealing, and genetic algorithms—were advanced by teams at Los Alamos National Laboratory and Sandia National Laboratories for large-scale combinatorial instances. Decomposition approaches like Benders decomposition and Dantzig–Wolfe decomposition derive from foundational work linked to George Dantzig and have been implemented in industrial settings at Shell and Siemens. Polyhedral cutting families—Gomory cuts, cover cuts, and clique cuts—originated from theoretical studies at IBM Research and are standard in modern solvers made by groups at Zuse Institute Berlin and ETH Zurich.

Complexity and Theory

Integer programming is NP-hard in general; theoretical boundaries were clarified through complexity theory advances at Princeton University and University of Chicago during the development of NP-completeness concepts by researchers influenced by Michael Garey and David S. Johnson. Parameterized complexity and fixed-parameter tractability results were investigated in academic centers such as Carnegie Mellon University and École Normale Supérieure, while approximation algorithms for special cases—set cover, knapsack, and scheduling—trace to seminal papers from Stanford University and Harvard University. Connections to polyhedral combinatorics, matroid theory, and integer lattices reflect collaborative research across University of Cambridge and University of Oxford.

Applications

Integer programming models are applied to routing and logistics problems like vehicle routing, facility location, and crew scheduling widely studied at FedEx and United Parcel Service, and to capital budgeting and portfolio selection problems in financial institutions such as Goldman Sachs and J.P. Morgan. Telecommunications capacity planning and network design applications appear in work by Cisco Systems and Nokia, while energy systems optimization and unit commitment models have been developed at General Electric and National Renewable Energy Laboratory. In manufacturing and supply chain contexts, integer programs underpin production planning and lot-sizing solutions employed by Toyota and Procter & Gamble. Integer formulations are also central to timetabling and rostering used by Amtrak and various academic institutions.

Software and Implementations

Commercial and open-source solvers implement advanced integer programming algorithms; prominent commercial packages include engines developed by IBM (CPLEX), FICO (Xpress), and Gurobi Optimization, while open-source projects such as COIN-OR and solvers emerging from Zuse Institute Berlin and ETH Zurich provide accessible implementations. Modeling environments and algebraic modeling languages—AMPL, GAMS, and Pyomo—originated in collaborations involving Cornell University and Sandia National Laboratories and interface with solver backends. Benchmarking and computational studies are regularly conducted at venues like DIMACS and INFORMS conferences, with large instance libraries maintained by research groups at NEOS Server and university centers.

Category:Mathematical optimization