George Bernard Dantzig was an American mathematical scientist who made contributions to industrial engineering, operations research, computer science, economics, and statistics.
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming.
In mathematical optimization, Dantzig's simplex algorithm is a popular algorithm for linear programming.
AMPL is an algebraic modeling language to describe and solve high-complexity problems for large-scale mathematical computing . It was developed by Robert Fourer, David Gay, and Brian Kernighan at Bell Laboratories. AMPL supports dozens of solvers, both open source and commercial software, including CBC, CPLEX, FortMP, MINOS, IPOPT, SNOPT, KNITRO, and LGO. Problems are passed to solvers as nl files. AMPL is used by more than 100 corporate clients, and by government agencies and academic institutions.
Karmarkar's algorithm is an algorithm introduced by Narendra Karmarkar in 1984 for solving linear programming problems. It was the first reasonably efficient algorithm that solves these problems in polynomial time. The ellipsoid method is also polynomial time but proved to be inefficient in practice.
Computational Infrastructure for Operations Research (COIN-OR), is a project that aims to "create for mathematical software what the open literature is for mathematical theory." The open literature provides the operations research (OR) community with a peer-review process and an archive. Papers in operations research journals on mathematical theory often contain supporting numerical results from computational studies. The software implementations, models, and data used to produce the numerical results are typically not published. The status quo impeded researchers needing to reproduce computational results, make fair comparisons, and extend the state of the art.
Algebraic modeling languages (AML) are high-level computer programming languages for describing and solving high complexity problems for large scale mathematical computation. One particular advantage of some algebraic modeling languages like AIMMS, AMPL, GAMS, Gekko, MathProg, Mosel, and OPL is the similarity of their syntax to the mathematical notation of optimization problems. This allows for a very concise and readable definition of problems in the domain of optimization, which is supported by certain language elements like sets, indices, algebraic expressions, powerful sparse index and data handling variables, constraints with arbitrary names. The algebraic formulation of a model does not contain any hints how to process it.
In mathematical optimization, Bland's rule is an algorithmic refinement of the simplex method for linear optimization.
Margaret H. Wright is an American computer scientist and mathematician. She is a Silver Professor of Computer Science and former Chair of the Computer Science department at Courant Institute of Mathematical Sciences, New York University, with research interests in optimization, linear algebra, and scientific computing. She was elected to the National Academy of Engineering in 1997 for development of numerical optimization algorithms and for leadership in the applied mathematics community. She was elected to the National Academy of Sciences in 2005. She was the first woman to serve as President of the Society for Industrial and Applied Mathematics.
Dantzig–Wolfe decomposition is an algorithm for solving linear programming problems with special structure. It was originally developed by George Dantzig and Philip Wolfe and initially published in 1960. Many texts on linear programming have sections dedicated to discussing this decomposition algorithm.
FortMP is a software package for solving large-scale optimization problems. It solves linear programming problems, quadratic programming problems and mixed integer programming problems. Its robustness has been explored and published in the Mathematical Programming journal. FortMP is available as a standalone executable that accepts input in MPS format and as a library with interfaces in C and Fortran. It is also supported in the AMPL modeling system.
In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming problems, quadratic-programming problems, and linear complementarity problems.
The Klee–Minty cube or Klee–Minty polytope is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". On the three-dimensional version, the simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case.
The FICO Xpress optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed integer counterparts. Xpress includes a general purpose non-linear solver, Xpress NonLinear, including a successive linear programming algorithm, and Artelys Knitro.
Philip Starr "Phil" Wolfe was an American mathematician and one of the founders of convex optimization theory and mathematical programming.
MINOS is a Fortran software package for solving linear and nonlinear mathematical optimization problems. MINOS may be used for linear programming, quadratic programming, and more general objective functions and constraints, and for finding a feasible point for a set of linear or nonlinear equalities and inequalities.
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied to general linear program of same dimensions.
Donald Goldfarb is an American mathematician, best known for his works in mathematical optimization and numerical analysis.
Tamás Terlaky is a Hungarian-Canadian-American professor of Industrial and Systems Engineering at Lehigh University. He is especially well known for his work on criss-cross algorithms, interior-point methods, Klee-Minty examples for path following algorithms, and optimization.
Werner Römisch is a German mathematician, professor emeritus at the Humboldt University of Berlin, most known for his pioneer work in the field of stochastic programming.
