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Pareto Front
In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. The concept is widely used in engineering. It allows the designer to restrict attention to the set of efficient choices, and to make tradeoffs within this set, rather than considering the full range of every parameter. Definition The Pareto frontier, ''P''(''Y''), may be more formally described as follows. Consider a system with function f: X \rightarrow \mathbb^m, where ''X'' is a compact set of feasible decisions in the metric space \mathbb^n, and ''Y'' is the feasible set of criterion vectors in \mathbb^m, such that Y = \. We assume that the preferred directions of criteria values are known. A point y^ \in \mathbb^m is preferred to (strictly dominates) another point y^ \in \mathbb^m, written as y^ \succ y^. The Pareto frontier is thus written as: : P(Y) = \. Marginal rate of substitution A significant aspect of the Pareto frontie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-objective Optimization
Multi-objective optimization (also known as multi-objective programming, vector optimization, multicriteria optimization, multiattribute optimization or Pareto optimization) is an area of multiple criteria decision making that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives. For a nontrivial multi-objective optimization problem, no single solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book '' Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice Fréchet in 1906, in his study of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skyline Operator
The skyline operator is the subject of an optimization problem, used in a query to filter results from a database to keep only those objects that are not worse than any other. This operator is an extension to SQL proposed by Börzsönyi et al. A classic example of application of the skyline operator involves selecting a hotel for a holiday. The user wants the hotel to be both cheap and close to the beach. However, hotels that are close to the beach may also be expensive. In this case, the skyline operator would only present those hotels that are not worse than any other hotel in both price and distance to the beach. Proposed syntax To give an example in SQL: Börzsönyi et al. proposed the following syntax for the skyline operator: SELECT ... FROM ... WHERE ... GROUP BY ... HAVING ... SKYLINE OF ISTINCTd1 MAX , DIFF ..., dm MAX , DIFFORDER BY ... where d1, ... dm denote the dimensions of the skyline and MIN, MAX and DIFF specify whether the value in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (including the design and implementation of hardware and software). Computer science is generally considered an area of academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing security vulnerabilities. Computer graphics and computational geometry address the generation of images. Programming language theory considers different ways to describe computational processes, and database theory concerns the management of repositories o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can perform automated deductions (referred to as automated reasoning) and use mathematical and logical tests to divert the code execution through various routes (referred to as automated decision-making). Using human characteristics as descriptors of machines in metaphorical ways was already practiced by Alan Turing with terms such as "memory", "search" and "stimulus". In contrast, a heuristic is an approach to problem solving that may not be fully specified or may not guarantee correct or optimal results, especially in problem domains where there is no well-defined correct or optimal result. As an effective method, an algorithm can be expressed within a finite amount of space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian Mechanics
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his 1788 work, '' Mécanique analytique''. Lagrangian mechanics describes a mechanical system as a pair (M,L) consisting of a configuration space M and a smooth function L within that space called a ''Lagrangian''. By convention, L = T - V, where T and V are the kinetic and potential energy of the system, respectively. The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point (a maximum, minimum, or saddle) throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations. Introduction Suppose there exists a bead sliding around on a wire, or a swinging sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Rate Of Substitution
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical. The marginal rate of substitution is one of the three factors from marginal productivity, the others being marginal rates of transformation and marginal productivity of a factor. As the slope of indifference curve Under the standard assumption of neoclassical economics that goods and services are continuously divisible, the marginal rates of substitution will be the same regardless of the direction of exchange, and will correspond to the slope of an indifference curve (more precisely, to the slope multiplied by −1) passing through the consumption bundle in question, at that point: mathematically, it is the implicit derivative. MRS of X for Y is the amount of Y which a consumer ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Efficient Frontier 1024x1024
Pareto may refer to: People * Vilfredo Pareto (1848–1923), Italian economist, political scientist, and philosopher, works named for him include: ** Pareto analysis, a statistical analysis tool in problem solving **Pareto distribution, a power-law probability distribution **Pareto efficiency **Pareto front, the set of all Pareto efficient solutions **Pareto principle, or the 80-20 rule * Bartolomeo Pareto, medieval priest and cartographer from Genoa * Graziella Pareto (1889–1973), Catalan soprano * Lorenzo Pareto (1800–1865), Italian geologist and statesman * Paula Pareto (born 1986), Argentine judoka * Benedetto Pareto, builder of the Shrine of Nostra Signora della Guardia in Liguria, Italy Other uses * Pareto, Piedmont, a town in Italy * Pareto Group, a Norwegian finance company See also * Paret, a surname * Pereto, a town in Italy * Perito, a town in Italy {{disambiguation, geo, surname Italian-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pareto Efficient
Pareto efficiency or Pareto optimality is a situation where no action or allocation is available that makes one individual better off without making another worse off. The concept is named after Vilfredo Pareto (1848–1923), Italian civil engineer and economist, who used the concept in his studies of economic efficiency and income distribution. The following three concepts are closely related: * Given an initial situation, a Pareto improvement is a new situation where some agents will gain, and no agents will lose. * A situation is called Pareto-dominated if there exists a possible Pareto improvement. * A situation is called Pareto-optimal or Pareto-efficient if no change could lead to improved satisfaction for some agent without some other agent losing or, equivalently, if there is no scope for further Pareto improvement. The Pareto front (also called Pareto frontier or Pareto set) is the set of all Pareto-efficient situations. Pareto originally used the word "optimal" for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Front Pareto
Front may refer to: Arts, entertainment, and media Films * ''The Front'' (1943 film), a 1943 Soviet drama film * '' The Front'', 1976 film Music *The Front (band), an American rock band signed to Columbia Records and active in the 1980s and early 1990s * The Front (Canadian band), a Canadian studio band from the 1980s Periodicals * ''Front'' (magazine), a British men's magazine * ''Front Illustrated Paper'', a publication of the Yugoslav People's Army Television * Front TV, a Toronto broadcast design and branding firm * "The Front" (''The Blacklist''), a 2014 episode of the TV series ''The Blacklist'' * "The Front" (''The Simpsons''), a 1993 episode of the TV series ''The Simpsons'' Military * Front (military), a geographical area where armies are engaged in conflict * Front (military formation), roughly, an army group, especially in eastern Europe Places * Front, California, former name of Brown, California * Front, Piedmont, an Italian municipality * The Front, now pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
