Simplex algorithm
[3][4][5][6] The simplicial cones in question are the corners (i.e., the neighborhoods of the vertices) of a geometric object called a polytope.George Dantzig worked on planning methods for the US Army Air Force during World War II using a desk calculator.Dantzig's core insight was to realize that most such ground rules can be translated into a linear objective function that needs to be maximized.[8] After Dantzig included an objective function as part of his formulation during mid-1947, the problem was mathematically more tractable.Dantzig realized that one of the unsolved problems that he had mistaken as homework in his professor Jerzy Neyman's class (and actually later solved), was applicable to finding an algorithm for linear programs.The column geometry used in this thesis gave Dantzig insight that made him believe that the Simplex method would be very efficient.This continues until the maximum value is reached, or an unbounded edge is visited (concluding that the problem has no solution).The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded above.Conversely, given a basic feasible solution, the columns corresponding to the nonzero variables can be expanded to a nonsingular matrix.Additional row-addition transformations can be applied to remove the coefficients cTB from the objective function.This process is called pricing out and results in a canonical tableau where zB is the value of the objective function at the corresponding basic feasible solution.If all the entries in the objective row are less than or equal to 0 then no choice of entering variable can be made and the solution is in fact optimal.First, only positive entries in the pivot column are considered since this guarantees that the value of the entering variable will be nonnegative.If there are no positive entries in the pivot column then the entering variable can take any non-negative value with the solution remaining feasible.[20] If there is more than one row for which the minimum is achieved then a dropping variable choice rule[22] can be used to make the determination.So a new objective function, equal to the sum of the artificial variables, is introduced and the simplex algorithm is applied to find the minimum; the modified linear program is called the Phase I problem.The simplex algorithm can then be applied to find the solution; this step is called Phase II.[13][14][24] Consider the linear program This is represented by the (non-canonical) tableau Introduce artificial variables u and v and objective function W = u + v, giving a new tableau The equation defining the original objective function is retained in anticipation of Phase II.The storage and computation overhead is such that the standard simplex method is a prohibitively expensive approach to solving large linear programming problems.These observations motivate the "revised simplex algorithm", for which implementations are distinguished by their invertible representation of B.When this is always the case no set of basic variables occurs twice and the simplex algorithm must terminate after a finite number of steps.When several such pivots occur in succession, there is no improvement; in large industrial applications, degeneracy is common and such "stalling" is notable.Worse than stalling is the possibility the same set of basic variables occurs twice, in which case, the deterministic pivoting rules of the simplex algorithm will produce an infinite loop, or "cycle".However, in 1972, Klee and Minty[32] gave an example, the Klee–Minty cube, showing that the worst-case complexity of simplex method as formulated by Dantzig is exponential time.[33] In 2014, it was proved[citation needed] that a particular variant of the simplex method is NP-mighty, i.e., it can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm's execution.[34] At about the same time it was shown that there exists an artificial pivot rule for which computing its output is PSPACE-complete.[citation needed] Another method to analyze the performance of the simplex algorithm studies the behavior of worst-case scenarios under small perturbation – are worst-case scenarios stable under a small change (in the sense of structural stability), or do they become tractable?This area of research, called smoothed analysis, was introduced specifically to study the simplex method.Indeed, the running time of the simplex method on input with noise is polynomial in the number of variables and the magnitude of the perturbations.[15] The Big-M method is an alternative strategy for solving a linear program, using a single-phase simplex.
Nelder–Mead methodmathematical optimizationDantzigalgorithmlinear programmingsimplexT. S. MotzkinpolytopeconstraintsGeorge DantzigWassily LeontiefJerzy NeymanLagrange multipliersLebesgue integralssystem of linear inequalitiesvertexcanonical formmatrix transposefeasible regionconvex polytopebasic feasible solutionslack variableidentity matrixDevex algorithmRevised simplex algorithmsparse matrixPadbergBland's rulecriss-cross algorithmZadeh's ruleCunningham's ruleFourier–Motzkin eliminationKlee–Minty cubeexponential timepolynomial timeNP-hardPSPACE-completeaverage-case complexityprobability distributionsrandom matricestypical phenomenaBaire category theorygeneral topologystructural stabilitysmoothed analysislinear-programmingKhachiyanellipsoidal algorithmKarmarkarprojective algorithmpath-following algorithmsBig-M methodLinear-fractional programmingLinear–fractional programminglinear functionPivoting rule of BlandCutting-plane methodGradient descentKarmarkar's algorithmNelder–Mead simplicial heuristicLoss FunctionsStrang, GilbertThe Mathematical IntelligencerGeorge B. DantzigAlbert W. TuckerDantzig, George B.Mitra, GautamBland, Robert G.Mathematics of Operations Researchoriented matroidKlee, VictorMinty, George J.Zwick, UriCiteSeerXChristos, PapadimitriouAlexander SchrijverTeng, Shang-HuaFukuda, KomeiTerlaky, TamásSIAM ReviewBibcodecomputer scienceoperations researchThomas H. CormenCharles E. LeisersonRonald L. RivestClifford SteinOptimizationAlgorithmsmethodsheuristicsUnconstrained nonlinearFunctionsGolden-section searchPowell's methodLine searchSuccessive parabolic interpolationGradientsConvergenceTrust regionWolfe conditionsQuasi–NewtonBerndt–Hall–Hall–HausmanBroyden–Fletcher–Goldfarb–ShannoL-BFGSDavidon–Fletcher–PowellSymmetric rank-one (SR1)Other methodsConjugate gradientGauss–NewtonGradientMirrorLevenberg–MarquardtPowell's dog leg methodTruncated NewtonHessiansNewton's methodConstrained nonlinearBarrier methodsPenalty methodsAugmented Lagrangian methodsSequential quadratic programmingSuccessive linear programmingConvex optimizationConvex minimizationReduced gradient (Frank–Wolfe)