Test Functions For Optimization
In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as: * Convergence rate. * Precision. * Robustness. * General performance. Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective Pareto fronts for multi-objective optimization problems (MOP) are given. The artificial landscapes presented herein for single-objective optimization problems are taken from Bäck, Haupt et al. and from Rody Oldenhuis software. Given the number of problems (55 in total), just a few are presented here. The test functions used to evaluate the algorithms for MOP were taken from Deb,Deb, Kalyanmoy (2002) Multiobjective optimization us ...
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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 ...
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McCormick Contour
McCormick may refer to: Business * McCormick & Company, an American food company specializing in spices and flavorings * McCormick & Schmick's, an American restaurant chain specializing in seafood * McCormick Harvesting Machine Company, a manufacturer of the first mechanical reaper * McCormick Distilling Company, an American distillery * McCormick Tractors, a manufacturer of tractors, headquartered in Fabbrico, Italy Education * McCormick Observatory, at the University of Virginia * McCormick Theological Seminary, a Presbyterian school of theology in Chicago, Illinois * Robert R. McCormick School of Engineering and Applied Science, at Northwestern University, Evanston, Illinois Places in the United States * McCormick, Illinois * McCormick, South Carolina * McCormick, Washington * McCormick Gap, a windgap in Virginia Sports * McCormick Place, an exposition complex in Chicago, Illinois * McCormick Field, a Minor League Baseball park in Asheville, North Carolina Other uses * McCorm ...
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McCormick Function
McCormick may refer to: Business * McCormick & Company, an American food company specializing in spices and flavorings * McCormick & Schmick's, an American restaurant chain specializing in seafood * McCormick Harvesting Machine Company, a manufacturer of the first mechanical reaper * McCormick Distilling Company, an American distillery * McCormick Tractors, a manufacturer of tractors, headquartered in Fabbrico, Italy Education * McCormick Observatory, at the University of Virginia * McCormick Theological Seminary, a Presbyterian school of theology in Chicago, Illinois * Robert R. McCormick School of Engineering and Applied Science, at Northwestern University, Evanston, Illinois Places in the United States * McCormick, Illinois * McCormick, South Carolina * McCormick, Washington * McCormick Gap, a windgap in Virginia Sports * McCormick Place, an exposition complex in Chicago, Illinois * McCormick Field, a Minor League Baseball park in Asheville, North Carolina Other uses * McCorm ...
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Hölder Table Function
Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder mean * Jordan–Hölder theorem In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...