Dynamic programming is both a mathematical optimization method and an algorithmic paradigm. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
The calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
In statistics and control theory, Kalman filtering is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán.
Optimal control theory is a branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the Moon with minimum fuel expenditure. Or the dynamical system could be a nation's economy, with the objective to minimize unemployment; the controls in this case could be fiscal and monetary policy. A dynamical system may also be introduced to embed operations research problems within the framework of optimal control theory.
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions. This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from finance to transportation to energy optimization.
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines.
Decision tree learning is a supervised learning approach used in statistics, data mining and machine learning. In this formalism, a classification or regression decision tree is used as a predictive model to draw conclusions about a set of observations.
Mechanism design, sometimes called implementation theory or institutiondesign, is a branch of economics, social choice, and game theory that deals with designing game forms to implement a given social choice function. Because it starts with the end of the game and then works backwards to find a game that implements it, it is sometimes described as reverse game theory.
Markov decision process (MDP), also called a stochastic dynamic program or stochastic control problem, is a model for sequential decision making when outcomes are uncertain.
In mathematics, engineering, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions.
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem. Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem. Therefore, the solution to the primal is an upper bound to the solution of the dual, and the solution of the dual is a lower bound to the solution of the primal. This fact is called weak duality.
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.
GPOPS-II is a general-purpose MATLAB software for solving continuous optimal control problems using hp-adaptive Gaussian quadrature collocation and sparse nonlinear programming. The acronym GPOPS stands for "General Purpose OPtimal Control Software", and the Roman numeral "II" refers to the fact that GPOPS-II is the second software of its type.
Quantum optimization algorithms are quantum algorithms that are used to solve optimization problems. Mathematical optimization deals with finding the best solution to a problem from a set of possible solutions. Mostly, the optimization problem is formulated as a minimization problem, where one tries to minimize an error which depends on the solution: the optimal solution has the minimal error. Different optimization techniques are applied in various fields such as mechanics, economics and engineering, and as the complexity and amount of data involved rise, more efficient ways of solving optimization problems are needed. Quantum computing may allow problems which are not practically feasible on classical computers to be solved, or suggest a considerable speed up with respect to the best known classical algorithm.
Brachybacterium phenoliresistens is a species of Gram positive, facultatively anaerobic, yellow-pigmented bacterium. The cells are coccoid during the stationary phase, and irregular rods during the exponential phase. It was first isolated from oil-contaminated sand in Pingtung County, Taiwan. The species was first described in 2007, and its name refers to the species' ability to resist phenol. It is most closely related to B. nesterenkovii.
Janibacter limosus is a species of Gram positive, strictly aerobic, bacterium. The species was initially isolated from sludge from a wastewater treatment plant in Jena, Germany. The species was first described in 1997, and the species name is derived from Latin limosus (muddy). J. limosus was the first species assigned to Janibacter, and is the type species for the genus.
Terrabacter aeriphilus is a species of Gram-positive, nonmotile, non-endospore-forming bacteria. Cells are either rods or coccoid. It was initially isolated from an air sample in Taean County, South Korea. The species was first described in 2010, and its name is derived from Latin aer (air), and Greek philos (loving).
Terrabacter ginsenosidimutans is a species of Gram-positive, nonmotile, non-endospore-forming bacteria. Cells are short rods. It was initially isolated from ginseng soil from a farm near Pocheon, South Korea. The species was first described in 2010, and its name refers the bacteria's ability to transform ginsenosides into rare gypenosides.
Terrabacter koreensis is a species of Gram-positive, nonmotile, non-endospore-forming bacteria. Cells are rod-shaped. It was initially isolated from soil from a flowerbed in Bucheon, South Korea. The species was first described in 2014, and its name refers to its South Korean isolation location.
Terrabacter terrigena is a species of Gram-positive, nonmotile, non-endospore-forming bacteria. Cells are rod-shaped. It was initially isolated from soil from around a wastewater treatment plant in South Korea. The species was first described in 2009, and its name is derived from Latin terrigena referring to the isolation of the type strain from soil.
