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Topic: Integer programming

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Integer computer science

positive integers

algebraic integers

Integer partition

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On Line Encyclopedia of Integer Sequences

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	Teach Integer Programming
		Integer programming uses a model similar to linear programming in that the objective function and constraint functions are linear.
		In integer programming, however, some or all the variables are required to be integer.
		This teaching add-in illustrates three primary methods of solving integer programming problems, the branch and bound method, the cutting plane method and Benders' algorithm.
	www.me.utexas.edu /~jensen/ORMM/methods/unit/integer/subunits/teach_int (57 words)

	Mathematical Programming Glossary - I (Site not responding. Last check: 2007-10-09)
		This solves a mathematical program by a sequence of approximations whose feasible regions are contained in the original feasible region.
		In linear programming, the dual of the dual is the primal.
		Its significance in mathematical programming is that it led to Lagrange's multiplier theorem.
	carbon.cudenver.edu /~hgreenbe/glossary/I.html (1523 words)

	Linear programming - Wikipedia, the free encyclopedia
		Linear programming problems must be converted into augmented form before being solved by the simplex algorithm.
		Every linear programming problem, referred to as a primal problem, can be converted into an equivalent dual problem.
		Alexander Schrijver: "Theory of Linear and Integer Programming".
	en.wikipedia.org /wiki/Linear_programming (1241 words)

	Combinatorial and Integer Optimization (Site not responding. Last check: 2007-10-09)
		The difficulty arises from the fact that unlike linear programming, for example, whose feasible region is a convex set, in combinatorial problems, one must search a lattice of feasible points or, in the mixed-integer case, a set of disjoint halflines or line segments to find an optimal solution.
		The underlying idea of polyhedral combinatorics is to replace the constraint set of an integer programming problem by an alternative convexification of the feasible points and extreme rays of the problem.
		The major components of this algorithm consist of automatic reformulation procedures, heuristics which provide "good" feasible integer solutions, and cutting plane procedures which tighten the linear programming relaxation to the combinatorial problem under consideration -- all of which is embedded into a tree-search framework as in the branch-and-bound approach to integer programming.
	iris.gmu.edu /~khoffman/papers/newcomb1.html (5076 words)

	Linear Programming FAQ
		It may not be obvious that integer programming is a very much harder problem than ordinary linear programming, but that is nonetheless the case, in both theory and practice.
		Integer models are known by a variety of names and abbreviations, according to the generality of the restrictions on their variables.
		The network linear programming problem is to minimize the (linear) total cost of flows along all arcs of a network, subject to conservation of flow at each node, and upper and/or lower bounds on the flow along each arc.
	www.faqs.org /faqs/linear-programming-faq (11971 words)

	Amazon.com: Integer Programming: Books (Site not responding. Last check: 2007-10-09)
		Integer Programming stands out from other textbooks by explaining in clear and simple terms how to construct custom-made algorithms or use existing commercial software to obtain optimal or near-optimal solutions for a variety of real-world problems, such as airline timetables, production line schedules, or electricity production on a regional or national scale.
		Integer Programming is an ideal text for courses in integer/mathematical programming-whether in operations research, mathematics, engineering, or computer science departments.
		Integer programming is one of the most interesting subjects and this book captures the beauty of it through the use of nice explanations and a neat and organised notation.
	www.amazon.com /exec/obidos/tg/detail/-/0471283665?v=glance (1009 words)

	Eudoxus Systems - Practical Integer Programming
		Integer Programming (IP) thus provides a powerful technique for tackling problems which have a mixture of continuous and discrete activities, including those involving scheduling.
		Next the building blocks of integer programming - integer variables, special ordered sets, etc - are introduced and demonstrated with examples.
		There follows a discussion of how integer programming codes work and tactics which can be used to assist in finding solutions.
	www.eudoxus.com /lect5a.html (480 words)

	CIM Thematic Term on Optimization
		The School is composed by five set of lectures, designed to introduce young researchers to the more recent advances on geometric and algebraic approaches for integer programming.
		Integer programming is a field of optimization with recognized scientific and economical relevance.
		The usual approach to solve integer programming problems is to use linear programming within a branch-and-bound or branch-and-cut framework, using whenever possible polyhedral results about the set of feasible solutions.
	www.mat.uc.pt /tt2005/ss (158 words)

	Parallel Integer Goal Programming (Site not responding. Last check: 2007-10-09)
		In this situation linear programming and round-off techniques fall far short because the scale of operations is seldom large enough to absorb the effects of the round-off gracefully.
		The coding method we employ to achieve the kind of discontinuous progress required by integer goal programming is to use the if/not combination which is similar to double negation but which also allows for alternative action if the problem is not feasible.
		Second, whenever the 2LP program reaches a point in the search tree where a branching decision must be made (called a choice point), it must call a routine so that the alternatives not taken at that point are made available to the global workorder stash as a possible new task.
	www.sci.brooklyn.cuny.edu /~arnow/RSCH/DP/pig.html (4154 words)

	Integer Programming (Site not responding. Last check: 2007-10-09)
		A reasonable idea is to make the integer variables correspond to Boolean variables and have constraints serve the same role as the clauses do in the original problem.
		Our translated integer programming problem will have twice as many variables as the SAT instance, one for each variable and one for its complement.
		Integer programming is hard because satisfying a set of constraints is hard.
	www.cs.toronto.edu /~yuana/AlgorithmManual/BOOK/BOOK3/NODE114.HTM (616 words)

	Stochastic Integer Programming Bibliography
		A finite branch and bound algorithm for two-stage stochastic integer programs.
		Solving stochastic programs with integer recourse by enumeration: a framework using Gröbner basis reductions.
		The integer programming background of a stochastic integer programming algorithm of Dentcheva-Prekopa-Ruszczynski.
	mally.eco.rug.nl /biblio/sip.html (3015 words)

	- SHOP.COM
		Integer programming is a subject with extensive applications and a large collection of solution methods and algorithms.
		Books on integer programming are often narrowly written for experts only, leaving a large interested audience without a text to follow.
		This book, written with the non-expert in mind, is a valuable collection of the latest developments in the area of integer programming, including theory, algorithms, and applications.
	www.shop.com /op/aprod-p23162653 (233 words)

	Amazon.com: Integer and Combinatorial Optimization (Wiley Interscience Series in Discrete Mathematics and ... (Site not responding. Last check: 2007-10-09)
		Integer and combinatorial optimization deals with problems of maximizing or minimizing a function of many variables subject to (a) inequality and equality constraints and (b) integrality restrictions on some or all of the variables.
		Though the title says "Integer and Combinatorial", the authors go a step ahead and present topics in advanced linear programming, computational complexity, polyhedral theory in a fashion appropriate to the learning of this subject.
		This book starts with the mathematical basics behind linear programming and develops on these introducing new techniques like Bender's decomposition, various cuts, etc. The way the mathematics is dealt is flawless but I thought the methods required more examples for better understanding.
	www.amazon.com /exec/obidos/tg/detail/-/047182819X?v=glance (1269 words)

	Centered Text by Integer Programming
		For this example, there are exact integer solutions to the equation ax+by=c, but these solutions all have one of x and y negative, which makes no sense for our application.
		If the variables are allowed to be real numbers, linear programs can be solved in polynomial time (in the number of variables and inequalities, and in the number of bits needed to store the various numbers defining the problem) using complicated methods such as Karmarkar's algorithm.
		The NC equivalence of planar integer linear programming and Euclidean GCD.
	www.ics.uci.edu /~eppstein/gene/algs/approximsum (1759 words)

	MIPLIB 3.0 (Site not responding. Last check: 2007-10-09)
		In response to the needs of researchers for access to real-world mixed integer programs a group of researchers at Rice University created MIPLIB, an electronically available library of both pure and mixed integer programs.
		Since mixed integer programming is such an active research area, and the performance of optimizers keeps improving, we anticipate that this update will not be the last.
		We encourage both researchers and practitioners in integer programming to submit real-world instances for consideration and possible inclusion in MIPLIB.
	www.caam.rice.edu /~bixby/miplib/miplib.html (233 words)

	Mixed Integer Programming
		A mixed-integer program is the minimization or maximization of a linear function subject to linear constraints.
		However when some or all of the variables must be integer, corresponding to pure integer and mixed integer programming respectively, the problem becomes NP-complete (formally intractible).
		The optimal solution to this linear program is at a corner of the polytope which represents the feasible region (the set of all variable settings which satisfy the constraints).
	www.cs.sandia.gov /opt/survey/mip.html (690 words)

	Integer programming formulation examples
		For the integer programming problem given before related to capital budgeting suppose now that we have the additional condition that either project 1 or project 2 must be chosen (i.e.
		The model given above is a very small zero-one integer programming problem with just 10 variables and 7 constraints and should be very easy to solve.
		Formulate the problem of how many units (if any) of each product to produce next week as an integer program in which all the constraints are linear.
	people.brunel.ac.uk /~mastjjb/jeb/or/moreip.html (4441 words)

	Linear Programming FAQ
		If you just want to solve a linear or integer program, however, then you should give some thought in advance to the work that will be involved in compiling a solver library and writing a program to generate your optimization problem, call the solver routines, and process the results.
		It is a hard (NP-complete) problem just like integer programming, but the obvious integer programming formulations of it are not especially useful in getting good solutions within a reasonable amount of time.
		Schrijver, A., Theory of Linear and Integer Programming, Wiley, 1986.
	www-unix.mcs.anl.gov /otc/Guide/faq/linear-programming-faq.html (13600 words)

	Stochastic Integer Programming
		We consider two-stage stochastic mixed-integer programs with nonnegative first-stage objective functions and expected recourse functions and give a factor 1/2 approximation algorithm.
		SPAR: Stochastic Programming with Adversarial Recourse by Matthew D. Bailey, Steven M. Shechter and Andrew J. Schaefer (submitted for publication).
		Two-Stage Integer Programs with Stochastic Right-Hand Sides: A Superadditive Dual Approach by Nan Kong, Andrew J. Schaefer and Brady Hunsaker (to appear in Mathematical Programming Series B).
	www.ie.pitt.edu /~schaefer/SIP.htm (691 words)

	Solver Technology - Mixed-Integer and Constraint Programming
		Strong Branching is a method used to estimate the impact of branching on each integer variable on the objective function (its pseudocost), by performing a few iterations of the Dual Simplex method.
		Such pseudocosts are used to guide the choice of the next subproblem to explore, and the next integer variable to branch upon, throughout the Branch and Bound process.
		In both of these solvers, integer variables and permutations are represented directly, and candidate solutions are generated that always satisfy integer and alldifferent constraints.
	www.solver.com /technology3.htm (924 words)

	INTEGER PROGRAMMING (Site not responding. Last check: 2007-10-09)
		An integer programming problem in which all variables are required to take only integer values is called a Pure Integer Programming (PIP) problem.
		The case where, in formulating IP's, we limitaed the integer variables to take the integer values 0 or 1 is called Pure (mixed) 0-1 Programming or Pure (mixed) Binary integer programming.Typically it it concern the "decisional situations", that is, for instance, do a project or not do a project (i.e.
		The coefficients of a linear problem have to be integer, then we have to multiply the first 4 constraints by 100.
	www.som.umd.umich.edu /romagnoli/int_prog.html (1798 words)

	Integer Programming (Site not responding. Last check: 2007-10-09)
		In many applications, the solution of an optimization problem makes sense only if certain of the unknowns are integers.
		We restrict ourselves to the pure integer case, bearing in mind that the software can also handle mixed problems with little additional complication of the underlying algorithm.
		Integer programming problems, such as the fixed-charge network flow problem and the famous traveling salesman problem, are often expressed in terms of binary variables.
	www-fp.mcs.anl.gov /otc/Guide/OptWeb/discrete/integerprog (157 words)

	Mixed integer programming - Wikipedia, the free encyclopedia
		It has been suggested that this article or section be merged with integer programming.
		In optimization theory, mixed integer linear programming (MILP or MIP) problems are linear programming problems in which some variables are additionally required to be integers.
		This page was last modified 00:44, 16 April 2005.
	en.wikipedia.org /wiki/Mixed_integer_programming (73 words)

	Integer Programming
		MOMIP (Modular Optimizer for Mixed Integer Programming) is a branch-and-bound solver for middle size mixed integer linear programming problems.
		Discrete variable types recognized are: zero/one (a switch), bivalent (zero or something else), general integer (pick an integer from a range of values), semicontinuous (must be zero or greater than a lower bound), SOS1 (pick an activity from a set of alternatives), and SOS2 (piecewise linearization of a curve).
		Descriptions of the branch and bound algorithm incorporated into this program is included in the opening sections of the library User's Guide.
	www.ici.ro /camo/hint.htm (610 words)

	Integer Linear Programming Model
		A mathematical programming problem is one in which there is a particular function to be maximized or minimized subject to several constraints.
		Here are two typical real-world applications that can be modeled as a pure integer programming problem.
		Now, the number of hearings of court cases is generally integer valued.
	cda.mrs.umn.edu /~pehng/Model/modelp1/node1.html (689 words)

	Dynamic Ideas - Optimization Over Integers
		The purpose of this book is to provide a unified, insightful, and modern treatment of the theory of integer optimization with an eye towards the future.
		We depart from earlier treatments of integer optimization by placing significant emphasis on strong formulations, duality, algebra and most importantly geometry.
		Both areas are practically significant as real world problems have very often both continuous and discrete variables and have elements of uncertainty that need to be addressed in a tractable manner.
	www.dynamic-ideas.com /Books/097591462/optoverint.html (440 words)

	Non-Standard Approaches to Integer Programming - Aardal, Weismantel, Wolsey (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		is to use linear programming within a branch-and-bound or branch-and-cut framework, using whenever possible polyhedral results about the structure of conv(X) or approximations to conv(X).
		43 A geometric Buchberger algorithm for integer programming - Thomas - 1995 ACM
		21 integer programming: optimization and augmentation are equiv..
	citeseer.ist.psu.edu /283638.html (1501 words)

	[No title]
		This mini-workshop is concerned with the use of stochastic models to study network interdiction problems and the study of solution methods for stochastic integer programs.
		Multiple variants of our basic stochastic integer program are developed to handle smugglers with different levels of information.
		Abstract: The traditional mean-value-based linear stochastic program with recourse can be extended towards risk aversion by adding suitable dispersion terms to the objective.
	www.math.ucdavis.edu /~deloera/stochasticip.html (718 words)

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