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Topic: Simplex method

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Simplex algorithm

Linear programming

Numerical analysis

Lagrange multipliers

	PlanetMath: simplex algorithm
		The simplex algorithm is used as part of the simplex method (due to George B. Dantzig) to solve linear programming problems.
		The simplex algorithm is used as one phase of the simplex method.
		This is version 19 of simplex algorithm, born on 2003-04-24, modified 2006-10-30.
	planetmath.org /encyclopedia/SimplexMethod.html (320 words)

	Simplex algorithm - Wikipedia, the free encyclopedia
		In both cases, the method uses the concept of a simplex, which is a polytope of N + 1 vertices in N dimensions: a line segment on a line, a triangle on a plane, a tetrahedron in three-dimensional space and so forth.
		The simplex algorithm begins at a starting vertex and moves along the edges of the polytope until it reaches the vertex of the optimum solution.
		Since the simplex algorithm is concerned only with finding a single optimal point (even if other equally-optimal points exist), it is possible to look solely at moves skirting the edge of a simplex, ignoring the interior.
	en.wikipedia.org /wiki/Simplex_algorithm (1184 words)

	Simplex Method -- from Wolfram MathWorld
		A different type of methods for linear programming problems are interior point methods, whose complexity is polynomial for both average and worst case.
		These methods construct a sequence of strictly feasible points (i.e., lying in the interior of the polytope but never on its boundary) that converges to the solution.
		In practice, one of the best interior-point methods is the predictor-corrector method of Mehrotra (1992), which is competitive with the simplex method, particularly for large-scale problems.
	mathworld.wolfram.com /SimplexMethod.html (450 words)

	The Simplex Method
		The term 'simplex' arises because the feasible solutions for the parameters may be represented by a polytope figure called a "simplex." The simplex for the case of a function of N rates is stored as an (N+1)xN rectangular array.
		The Simplex method differs from the well-known and widely used Levenberg-Marquardt and Gauss-Newton methods in that it does not use derivatives, which confers safer convergence properties to the Simplex method since it is much less prone to finding false minima.
		The simplex method was compared to Levenberg-Marquardt or Gauss-Newton methods using derivatives of the function with respect to the parameters.
	olisweb.com /software/simplex.php (391 words)

	Optimization mehtods
		The simplex method has been improved by active workers in the field, to what is called the modified simplex method (see e.g.
		A method to form a joint response measure, from the individual response variables, is therefore needed.
		The sequential simplex methods used in the MultiSimplex software should not be confused with the simplex method for linear programming (a method to solve a linear program by progressing from one extreme point of the feasible polyhedron to an adjacent one).
	www.grabitech.com /methods.htm (599 words)

	Dr. Dobb's \| Simplex Optimization \| October 1, 2003 (Site not responding. Last check: 2007-10-17)
		Often overlooked, simplex optimization is a reasonable and quick-to-implement solution, useful at least for the initial stages of rapid development and for testing methodologies.
		Simplex just walks around in a stylized way, and you are guaranteed that you will at least have the best answer you have ever stumbled upon.
		The simplex optimization method begins with an initial (often guessed) point (that is, a set of parameter values) and constructs a simplex near that guessed point.
	www.ddj.com /dept/cpp/184401716?pgno=6 (2099 words)

	Simplex Method (Site not responding. Last check: 2007-10-17)
		The simplex method is an efficient iterative algorithm to solve unconstrained minimization problems numerically for several but not too many variables.
		The operations of changing the simplex optimally with respect to the minimal/maximal function values found at the corners of the simplex are contraction, expansion and reflection, each determining new simplex corner points by linear combinations of selected existing corner points.
		A mathematical discussion of the downhill simplex method and of a (quite different) simplex method used in linear programming problems, is found in [Press95].
	br.endernet.org /~akrowne/handbook/AN16pp/node262.html (232 words)

	Downhill Simplex Method for Many (~20) Dimensions
		A simplex is the geometrical figure consisting, in N dimensions, of N+1 points (or vertices) and all their interconnecting line segments, polygonal faces, etc. In two dimensions, a simplex is a triangle.
		The downhill simplex method takes a series of steps, most steps just moving the highest point (the point of simplex where the function is largest) through the opposite face of the simplex to a supposed lower point (Pr).
		The procedure writen in C language bellow is an implementation of the improved simplex method.
	paula.univ.gda.pl /~dokgrk/simplex.html (1086 words)

	The Simplex Method
		The Simplex method is a search procedure that sifts through the set of basic feasible solutions, one at a time, until the optimal basic feasible solution (whenever it exists) is identified.
		Although the method does not require any explicit reliance on the graph, we will first explain the basic idea behind the Simplex method graphically; and we will do this using the example from the last section.
		In each iteration of the Simplex method, the primary algebraic task is to transform, using Gaussian elimination, the constraint equations from a given configuration to a new configuration that corresponds to the next basic feasible solution.
	www.utdallas.edu /~scniu/OPRE-6201/documents/LP4-Simplex.html (1050 words)

	Refined Simplex Method for Data Fitting
		The refined method is applied to Zernike polynomials in Cartesian coordinates, which describe an optical surface or wavefront in terms of aberrations.
		When the algorithm of the simplex method was re-examined, a redundant mechanism in the process was found.
		The simplex method is used for curve fitting to Zernike polynomials in Cartesian coordinates.
	www.cv.nrao.edu /adass/adassVI/kimys.html (1092 words)

	Chapter 2: The Simplex Method
		That variable replaces one of its compatriots that is most severely restricting it, thus moving the Simplex Method to a different corner of the solution set and closer to the final solution.
		Where the Revised Simplex Method called for the inverse of the basis to be multiplied, the Bartels-Golub substitutes two backsolves involving the upper and lower triangular basis factors.
		Reid's Method significantly reduces the growth of the number of eta matrices in the Sparse Bartels-Golub Method, and the basis should not need to be decomposed nearly as often.
	www.cise.ufl.edu /~davis/Morgan/chapter2.htm (2800 words)

	Towards the Simplex Method
		However, each tableau in the simplex method corresponds to a movement from one basic variable set BVS (extreme or corner point) to another, making sure that the objective function improves at each iteration until the optimal solution is reached.
		The simplex method always starts at the origin (which is a corner point) and then jumps from a corner point to the neighboring corner point until it reaches the optimal corner point (if bounded).
		Numerical Recipes states that the Simplex algorithm is 'almost always' O(max(N,M)), which means that the number of iteration is a factor of number of variables or number of constraints, whichever is larger.
	home.ubalt.edu /ntsbarsh/opre640A/partIV.htm (3027 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-17)
		The simplex method is a way of moving from one vertex of the solid to another, to another, always increasing the value of F. When this can't be done any more, the solution has been found.
		Dantzig, who's still a rather old professor in the Dept. of Operations Research at Stanford.) The simplex method relies on noticing that the maximum of the objective function must occur on a corner of the space bounded by the constraints (the "feasible region").
		It then starts at a corner of the feasible region and keeps trying to increase the objective function by finding an edge to move in which the function will increase, and does this until there is no direction to go, at which point it's at the optimum.
	mathforum.org /library/drmath/view/51466.html (610 words)

	The modified simplex method
		The modified simplex method has much in common with the basic method, but can adjust its shape and size depending of the response in each step.
		The procedures for expansion and contraction enable the modified simplex both to accelerate along a successful track of improvement and to home in on the optimum conditions.
		The degree of contraction depends on how unfavorable the new response is. The next figure illustrates the different moves with the modified simplex method.
	www.multisimplex.com /simplex_m.htm (371 words)

	More Simplex Method
		In the previous lecture we described a method to solve a linear programming problem.
		The process of the simplex method in this form works exactly like it did before.
		To move to another point in the simplex method, we must identify entering and leaving variables.
	www.math.kent.edu /~tsutton/Simplex2.htm (721 words)

	The basic simplex method
		The shapes of the simplex in a one, a two and a three variable search space, are a line, a triangle or a tetrahedron.
		The reevaluation rule avoids the simplex to be stuck around a false favorable response.
		Instead a very unfavorable response is applied, forcing the simplex to move away from the boundary.
	www.multisimplex.com /simplex_b.htm (419 words)

	Simplex Method for Standard Problems
		The IDENTITY SUB-MATRIX (ISM) is an identity matrix located in the slack variable columns of the starting tableau, but moving to other columns during simplex method.
		Simplex method will move the ISM, one column at a time; after each such move, we arrive at (or "hop" to) a new corner point (basic solution) with bigger objective value.
		After a finite number of such repetitions (usually 2-3), simplex method must terminate at step 8.
	math.uww.edu /faculty/mcfarlat/simplex1.htm (561 words)

	Chapter 11.6 - Nelder-Mead (Simplex) Method (Site not responding. Last check: 2007-10-17)
		A totally different method that is quite commonly used in nonlinear regression programs is the Nelder-Mead or Simplex method.
		The simplex method is relatively robust and numerically less complicated but it can be inefficient (slow) for simple problems.
		Actually the method I use most of the time involves starting with the simplex method and (automatically) continuing with the Damping-Gauss-Newton method.
	www.boomer.org /c/p3/c11/c1106.html (247 words)

	Primal-Dual Interior-Point Methods
		A thorough description of the theoretical properties of these methods is given, as are a discussion of practical and computational aspects and a summary of current software.
		The book is useful for graduate students and researchers in the sciences and engineering who are interested in using large-scale optimization techniques in their research, including those interested in original research in interior-point methods.
		Researchers and students in the field of interior-point methods will find the book invaluable as a reference to the key results, the basic analysis in the area, and the current state of the art.
	www.siam.org /books/swright (928 words)

	Linear Programming - Simplex Method
		The simplex method starts at the origin and follows a path along the edges of the polytope to the vertex where the maximum occurs.
		The goal of the simplex method is to exchange some of the columns of 1's and 0's of the slack variables into columns of 1's and 0's of the decision variables.
		The above subroutines are for pedagogical purposes to illustrate the simplex method.
	math.fullerton.edu /mathews/n2003/LinearProgrammingMod.html (466 words)

	Simplex algorithm method - Wikipedia, the free encyclopedia
		This article deals with one of many methods for solving a linear programming problem using the simplex algorithm.
		No information will be given on why this method works, for this refer to the simplex algorithm page.
		This application of the simplex algorithm uses tables, called tableaux, to represent calculations and intermediate steps to completing a problem.
	en.wikipedia.org /wiki/Simplex_algorithm_method (638 words)

	DPLP, AN "EASY" DISTRIBUTED SIMPLEX METHOD (Site not responding. Last check: 2007-10-17)
		The simplex algorithm of linear programming is one of the most important algorithms in applications, but distributed implementations have been few and difficult.
		By changing from today's popular form of the simplex method, the "revised" form, to the earlier "standard" form we have been able to implement an effective coarse grained distributed algorithm whi ch is a simple extension of the standard form of the simplex method.
		We will briefly review linear programming problems and the simplex method for solving them, mainly emphasizing the relation between the original version of the algorithm, the standard method, and the approach more common today, the revised simplex method.
	cis.poly.edu /seminars/fall00/abstract1fall00.htm (225 words)

	Simplex -- from Wolfram MathWorld
		A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to
		The simplex is so-named because it represents the simplest possible polytope in any given space.
		The content (i.e., hypervolume) of a simplex can be computed using the Cayley-Menger determinant.
	mathworld.wolfram.com /Simplex.html (200 words)

	A Hybrid Approach to Modeling Metabolic Systems Using Genetic Algorithm and Simplex Method - Yen, Liao, Randolph, Lee ... (Site not responding. Last check: 2007-10-17)
		To alleviate this difficulty, we developed a hybrid approach that combines GA with a stochastic variant of the simplex method in function optimization.
		used a hybrid GA simplex method to solve the parameters in a dynamic metabolic model.
		Yen, J. Liao, D. Randolph, and B. Lee, "A hybrid approach to modeling metabolic systems using genetic algorithm and simplex method," In Proceedings of the 11th IEEE Conference on Artificial Intelligence for Applications (CAIA95), pp.
	citeseer.ist.psu.edu /yen95hybrid.html (937 words)

	Linear Programming FAQ
		These methods derive from techniques for nonlinear programming that were developed and popularized in the 1960s by Fiacco and McCormick, but their application to linear programming dates back only to Karmarkar's innovative analysis in 1984.
		Among the SLATEC library routines is a Fortran sparse implementation of the simplex method, SPLP.
		Simplex Tool that demonstrates the workings of the simplex method on small user-entered problems.
	www-unix.mcs.anl.gov /otc/Guide/faq/linear-programming-faq.html (13600 words)

	The Simplex Java Applet
		There are a series of windows that take you through the simplex method.
		This button is disabled if you are in the middle of the process of stepping through the simplex method.
		This problem requires both phases of the simplex method (the feasibility phase and the optimality phase).
	www-fp.mcs.anl.gov /otc/Guide/CaseStudies/simplex/applet/SimplexTool.html (502 words)

	The Simplex Method
		The simplex method is a method of solving linear programming problems by moving from corner point to corner point of the feasible region in the direction of greatest profit.
		The simplex method works by switching basic and non-basic variables to step from corner point to corner point to achieve an optimal solution.
		The simplex method is an algorithm, or set of steps, to solve linear programming problems that have been put into augmented form.
	www.math.kent.edu /~tsutton/Simplex1.htm (1342 words)

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