Linear programming

Linear programming
Ylloh · CC0 · source
Name	Linear programming
Caption	Feasible region and optimal vertex
Field	Operations research; Mathematics
Inventor	George Dantzig
Year	1947
Related	Simplex algorithm; Interior-point methods; Linear algebra; Convex optimization

Linear programming Linear programming is a mathematical optimization technique for maximizing or minimizing a linear objective function subject to linear equality and inequality constraints. Developed in the mid‑20th century, it underpins modern decision science and has influenced institutions such as RAND Corporation, Bell Laboratories, and Brookings Institution. Key contributors and practitioners include George Dantzig, John von Neumann, Harold Kuhn, Albert Tucker, and Leonid Kantorovich.

Introduction

Linear programming models represent decision variables, linear objective functions, and linear constraints that form a convex polyhedron known as the feasible region. Foundational episodes in its development include wartime allocation problems addressed by United States Department of Defense planners and postwar industrial planning at General Electric. Influential publications appeared in venues associated with Proceedings of the National Academy of Sciences and journals edited by members of Institute for Operations Research and the Management Sciences and International Federation of Operational Research Societies. Practical adoption spread through corporations such as United Airlines, Procter & Gamble, and Ford Motor Company.

Mathematical Formulation

A canonical form expresses an objective c^T x to be maximized (or minimized) subject to A x ≤ b and x ≥ 0, where A is a matrix, b and c are vectors, and x represents variables. Matrix theory developed in contexts like Cambridge University and Moscow State University informs analysis of rank, degeneracy, and basis selection. Existence and uniqueness of optimal solutions rely on results associated with Helly's theorem and concepts formalized by researchers at Princeton University and Harvard University. Special cases include transportation problems studied by practitioners at United States Postal Service and assignment problems explored in collaborations involving Bell Laboratories.

Solution Methods

The simplex algorithm, introduced by George Dantzig, traverses vertices of the feasible polyhedron and saw early computational work on machines such as the ENIAC and systems at IBM. Alternative pivot rules and revisions were developed by teams including researchers from Cornell University and Stanford University. Interior-point methods, advanced through work by Narendra Karmarkar and followed by implementations influenced by groups at AT&T Bell Labs and Microsoft Research, traverse the interior of the feasible region using barrier functions. Other approaches include cutting-plane methods linked to studies by R. E. Gomory, branch-and-bound frameworks used by industrial planning groups like Siemens, and polynomial-time algorithms rooted in theoretical work by scholars affiliated with Princeton University and Massachusetts Institute of Technology.

Duality and Sensitivity Analysis

Every linear program has an associated dual problem; strong duality holds under conditions formalized in results attributed to researchers at University of Chicago and University of California, Berkeley. The dual provides economic interpretations used in shadow‑price analysis by consultants from McKinsey & Company and analysts at World Bank projects. Sensitivity and post‑optimality analysis examine how perturbations to coefficients affect optimal bases, leveraging matrix perturbation theory developed in departments at ETH Zurich and University of Cambridge. Complementary slackness conditions are taught in curricula at London School of Economics and applied in policy modeling by agencies such as United Nations.

Applications

Linear programming has extensive applications in resource allocation problems faced by corporations like Amazon (company), Walmart, and Toyota Motor Corporation; in energy sector scheduling managed by Federal Energy Regulatory Commission and National Grid (Great Britain); in telecommunications capacity planning undertaken by Verizon Communications and AT&T; in transportation and routing problems studied by Federal Aviation Administration and municipal transit authorities; and in finance for portfolio optimization implemented by firms such as Goldman Sachs and J.P. Morgan Chase. Other domains include cutting-stock problems addressed by manufacturers like ArcelorMittal, production planning at 3M, and workforce scheduling in hospitals accredited by Joint Commission reviewers.

Computational Complexity and Software

Theoretical complexity results link linear programming to classes studied in computational complexity research at Institute for Advanced Study and Carnegie Mellon University; the existence of polynomial-time interior-point algorithms resolved questions posed in conferences like International Congress of Mathematicians. Widely used software includes commercial solvers developed by companies such as IBM (CPLEX), Gurobi and FICO (Xpress), open-source projects from academic groups at University of Waikato and Université de Montréal (COIN-OR), and modeling languages produced by teams at MIT and Impetus. Benchmarking and industrial testbeds have been organized through collaborations involving National Institute of Standards and Technology and consortia with European Commission programs.

Category:Optimization