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Configuration Linear Program
The configuration linear program (configuration-LP) is a particular linear programming used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock problem.Gilmore P. C., R. E. Gomory (1961). A linear programming approach to the cutting-stock problem'. Operations Research 9: 849-859 Later, it has been applied to bin packing and job scheduling. In the configuration-LP, there is a variable for each possible ''configuration'' - each possible multiset of items that can fit in a single bin (these configurations are also known as ''patterns'') . Usually, the number of configurations is exponential in the problem size, but in some cases it is possible to attain approximate solutions using only a polynomial number of configurations. In bin packing The integral LP In the bin packing problem, there are ''n'' items with different sizes. The goal is to pack the items into a minimum number of bins, where each bin can contain at most ''B''. A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knapsack Problem
The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible. It derives its name from the problem faced by someone who is constrained by a fixed-size knapsack and must fill it with the most valuable items. The problem often arises in resource allocation where the decision-makers have to choose from a set of non-divisible projects or tasks under a fixed budget or time constraint, respectively. The knapsack problem has been studied for more than a century, with early works dating as far back as 1897. The name "knapsack problem" dates back to the early works of the mathematician Tobias Dantzig (1884–1956), and refers to the commonplace problem of packing the most valuable or useful items without overloading the luggage. Applications Knapsack problems ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Job Scheduling
A job scheduler is a computer application for controlling unattended background program execution of jobs. This is commonly called batch scheduling, as execution of non-interactive jobs is often called batch processing, though traditional ''job'' and ''batch'' are distinguished and contrasted; see that page for details. Other synonyms include batch system, distributed resource management system (DRMS), distributed resource manager (DRM), and, commonly today, workload automation (WLA). The data structure of jobs to run is known as the job queue. Modern job schedulers typically provide a graphical user interface and a single point of control for definition and monitoring of background executions in a distributed network of computers. Increasingly, job schedulers are required to orchestrate the integration of real-time business activities with traditional background IT processing across different operating system platforms and business application environments. ''Job scheduling'' s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Partitioning
In computer science, multiway number partitioning is the problem of partitioning a multiset of numbers into a fixed number of subsets, such that the sums of the subsets are as similar as possible. It was first presented by Ronald Graham in 1969 in the context of the Identical-machines scheduling problem. The problem is parametrized by a positive integer ''k'', and called ''k''-way number partitioning. The input to the problem is a multiset ''S'' of numbers (usually integers), whose sum is ''k*T''. The associated decision problem is to decide whether ''S'' can be partitioned into ''k'' subsets such that the sum of each subset is exactly ''T''. There is also an optimization problem: find a partition of ''S'' into ''k'' subsets, such that the ''k'' sums are "as near as possible". The exact optimization objective can be defined in several ways: * Minimize the difference between the largest sum and the smallest sum. This objective is common in papers about multiway number partitioning, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bin Packing
The bin packing problem is an optimization problem, in which items of different sizes must be packed into a finite number of bins or containers, each of a fixed given capacity, in a way that minimizes the number of bins used. The problem has many applications, such as filling up containers, loading trucks with weight capacity constraints, creating file backups in media and technology mapping in Field-programmable gate array, FPGA semiconductor chip design. Computationally, the problem is NP-hard, and the corresponding decision problem - deciding if items can fit into a specified number of bins - is NP-complete. Despite its worst-case hardness, optimal solutions to very large instances of the problem can be produced with sophisticated algorithms. In addition, many approximation algorithms exist. For example, the First-fit bin packing, first fit algorithm provides a fast but often non-optimal solution, involving placing each item into the first bin in which it will fit. It require ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
High-multiplicity Bin Packing
High-multiplicity bin packing is a special case of the bin packing problem, in which the number of different item-sizes is small, while the number of items with each size is large. While the general bin-packing problem is NP-hard, the high-multiplicity setting can be solved in polynomial time, assuming that the number of different sizes is a fixed constant. Problem definition The inputs to the problem are positive integers: * ''d'' - the number of different sizes (also called the ''dimension'' of the problem); *''B'' - the bin capacity. * ''s''1, ..., s''d'' - the sizes. The vector of sizes is denoted by s. * ''n''1, ..., n''d'' - the multiplicities; n''i'' is the number of items with size ''si''. The vector of multiplicities is denoted by n. **''n'' denotes the total number of items, that is, ''n = n''1+...+''nd''. **''V'' denotes the largest integer appearing in the description of the problem, that is, ''V'' = max(''s''1, ..., s''d'', ''n''1, ..., ''nd'', B) The output is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unrelated-machines Scheduling
Unrelated-machines scheduling is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. We need to schedule ''n'' jobs ''J''1, ''J''2, ..., ''Jn'' on ''m'' different machines, such that a certain objective function is optimized (usually, the makespan should be minimized). The time that machine ''i'' needs in order to process job j is denoted by ''pi,j''. The term ''unrelated'' emphasizes that there is no relation between values of ''pi,j'' for different ''i'' and ''j''. This is in contrast to two special cases of this problem: uniform-machines scheduling - in which ''pi,j'' = ''pi'' / ''sj'' (where ''sj'' is the speed of machine ''j''), and identical-machines scheduling - in which ''pi,j'' = ''pi'' (the same run-time on all machines). In the standard three-field notation for optimal job scheduling problems, the unrelated-machines variant is denoted by R in the first field. For example, the problem denoted by " R, , C_\max" i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrality Gap
In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form :x_i\in\. The relaxation of the original integer program instead uses a collection of linear constraints :0 \le x_i \le 1. The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program. Example Consider the set cover problem, the linear programming relaxation of which was first considered by . In this problem, one is given as input a family of sets ''F'' = ; the task is to find a subfamily, with as few sets as possible, having the same union as ''F''. To formula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separation Oracle
A separation oracle (also called a cutting-plane oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimization algorithm. Separation oracles are used as input to ellipsoid methods. Definition Let ''K'' be a convex and compact set in R''n''. A strong separation oracle for ''K'' is an oracle (black box) that, given a vector ''y'' in R''n'', returns one of the following: M. Grötschel, L. Lovász, A. Schrijver: ''Geometric Algorithms and Combinatorial Optimization'', Springer, 1988. *Assert that ''y'' is in ''K''. * Find a hyperplane that separates ''y'' from ''K'': a vector a in R''n'', such that a\cdot y > a\cdot x for all ''x'' in ''K''. A strong separation oracle is completely accurate, and thus may be hard to construct. For practical reasons, a weaker version is considered, which allows for small errors in the boundary of ''K'' and the inequalities. Given a small error toleran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorial Optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead. Combinatorial optimization is related to operations research, algorithm theory, and computational complexity theory. It has important applications in several fields, including artificial intelligence, machine learning, auction theory, software engineering, VLSI, applied mathematics and theoretical computer science. Some research literature considers discrete o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ellipsoid Method
In mathematical optimization, the ellipsoid method is an iterative method for convex optimization, minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds an optimal solution in a number of steps that is polynomial in the input size. The ellipsoid method generates a sequence of ellipsoids whose volume uniformly decreases at every step, thus enclosing a minimizer of a convex function. History The ellipsoid method has a long history. As an iterative method, a preliminary version was introduced by Naum Z. Shor. In 1972, an approximation algorithm for real convex optimization, convex minimization was studied by Arkadi Nemirovski and David B. Yudin (Judin). As an algorithm for solving linear programming problems with rational data, the ellipsoid algorithm was studied by Leonid Khachiyan; Khachiyan's achievement was to prove the Polynomial time, polynomial-time solvability of li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
