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Cross-entropy Method
The cross-entropy (CE) method is a Monte Carlo method for importance sampling and optimization. It is applicable to both combinatorial and continuous problems, with either a static or noisy objective. The method approximates the optimal importance sampling estimator by repeating two phases:Rubinstein, R.Y. and Kroese, D.P. (2004), The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning, Springer-Verlag, New York . #Draw a sample from a probability distribution. #Minimize the ''cross-entropy'' between this distribution and a target distribution to produce a better sample in the next iteration. Reuven Rubinstein developed the method in the context of ''rare-event simulation'', where tiny probabilities must be estimated, for example in network reliability analysis, queueing models, or performance analysis of telecommunication systems. The method has also been applied to the traveling salesman, quadratic assignment, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monte Carlo Method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution. In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model, interacting particle systems, McKean–Vlasov processes, kinetic models of gases). Other examples include modeling phenomena with significant uncertainty in inputs such as the calculation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its compleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heuristics
A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, short-term goal or approximation. Where finding an optimal solution is impossible or impractical, heuristic methods can be used to speed up the process of finding a satisfactory solution. Heuristics can be mental shortcuts that ease the cognitive load of making a decision. Examples that employ heuristics include using trial and error, a rule of thumb or an educated guess. Heuristics are the strategies derived from previous experiences with similar problems. These strategies depend on using readily accessible, though loosely applicable, information to control problem solving in human beings, machines and abstract issues. When an individual applies a heuristic in practice, it generally performs as expected. However it can alternatively cre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Importance Sampling
Importance sampling is a Monte Carlo method for evaluating properties of a particular distribution, while only having samples generated from a different distribution than the distribution of interest. Its introduction in statistics is generally attributed to a paper by Teun Kloek and Herman K. van Dijk in 1978, but its precursors can be found in statistical physics as early as 1949. Importance sampling is also related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both. Basic theory Let X\colon \Omega\to \mathbb be a random variable in some probability space (\Omega,\mathcal,P). We wish to estimate the expected value of ''X'' under ''P'', denoted E 'X;P'' If we have statistically independent random samples x_1, \ldots, x_n, generated according to ''P'', then an empirical estimate of E 'X;P''is : \widehat_ ;P= \frac \sum_^n x_i \quad \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Randomized Algorithm
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. One has to distinguish between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite ( Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result ( Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross Entropy
In information theory, the cross-entropy between two probability distributions p and q over the same underlying set of events measures the average number of bits needed to identify an event drawn from the set if a coding scheme used for the set is optimized for an estimated probability distribution q, rather than the true distribution p. Definition The cross-entropy of the distribution q relative to a distribution p over a given set is defined as follows: :H(p, q) = -\operatorname_plog q/math>, where E_p cdot/math> is the expected value operator with respect to the distribution p. The definition may be formulated using the Kullback–Leibler divergence D_(p \parallel q), divergence of p from q (also known as the ''relative entropy'' of p with respect to q). :H(p, q) = H(p) + D_(p \parallel q), where H(p) is the entropy of p. For discrete probability distributions p and q with the same support \mathcal this means The situation for continuous distributions is analogous. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Evolution Strategy
Natural evolution strategies (NES) are a family of numerical optimization algorithms for black box problems. Similar in spirit to evolution strategies, they iteratively update the (continuous) parameters of a ''search distribution'' by following the natural gradient towards higher expected fitness. Method The general procedure is as follows: the ''parameterized'' search distribution is used to produce a batch of search points, and the fitness function is evaluated at each such point. The distribution’s parameters (which include ''strategy parameters'') allow the algorithm to adaptively capture the (local) structure of the fitness function. For example, in the case of a Gaussian distribution, this comprises the mean and the covariance matrix. From the samples, NES estimates a search gradient on the parameters towards higher expected fitness. NES then performs a gradient ascent step along the natural gradient, a second order method which, unlike the plain gradient, renormalizes t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tabu Search
Tabu search is a metaheuristic search method employing local search methods used for mathematical optimization. It was created by Fred W. Glover in 1986 and formalized in 1989. Local (neighborhood) searches take a potential solution to a problem and check its immediate neighbors (that is, solutions that are similar except for very few minor details) in the hope of finding an improved solution. Local search methods have a tendency to become stuck in suboptimal regions or on plateaus where many solutions are equally fit. Tabu search enhances the performance of local search by relaxing its basic rule. First, at each step ''worsening'' moves can be accepted if no improving move is available (like when the search is stuck at a strict local minimum). In addition, ''prohibitions'' (henceforth the term ''tabu'') are introduced to discourage the search from coming back to previously-visited solutions. The implementation of tabu search uses memory structures that describe the visited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Of Distribution Algorithm
''Estimation of distribution algorithms'' (EDAs), sometimes called ''probabilistic model-building genetic algorithms'' (PMBGAs), are stochastic optimization methods that guide the search for the optimum by building and sampling explicit probabilistic models of promising candidate solutions. Optimization is viewed as a series of incremental updates of a probabilistic model, starting with the model encoding an uninformative prior over admissible solutions and ending with the model that generates only the global optima. EDAs belong to the class of evolutionary algorithms. The main difference between EDAs and most conventional evolutionary algorithms is that evolutionary algorithms generate new candidate solutions using an ''implicit'' distribution defined by one or more variation operators, whereas EDAs use an ''explicit'' probability distribution encoded by a Bayesian network, a multivariate normal distribution, or another model class. Similarly as other evolutionary algorithms, EDAs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmony Search
This is a chronologically ordered list of metaphor-based metaheuristics and swarm intelligence algorithms, sorted by decade of proposal. Algorithms 1980s-1990s Simulated annealing (Kirkpatrick et al., 1983) Simulated annealing is a probabilistic algorithm inspired by annealing, a heat treatment method in metallurgy. It is often used when the search space is discrete (e.g., all tours that visit a given set of cities). For problems where finding the precise global optimum is less important than finding an acceptable local optimum in a fixed amount of time, simulated annealing may be preferable to alternatives such as gradient descent. The analogue of the slow cooling of annealing is a slow decrease in the probability of simulated annealing accepting worse solutions as it explores the solution space. Accepting worse solutions is a fundamental property of metaheuristics because it allows for a more extensive search for the optimal solution. Ant colony optimization (ACO) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genetic Algorithms
In computer science and operations research, a genetic algorithm (GA) is a metaheuristic inspired by the process of natural selection that belongs to the larger class of evolutionary algorithms (EA). Genetic algorithms are commonly used to generate high-quality solutions to optimization and search problems by relying on biologically inspired operators such as mutation, crossover and selection. Some examples of GA applications include optimizing decision trees for better performance, solving sudoku puzzles, hyperparameter optimization, etc. Methodology Optimization problems In a genetic algorithm, a population of candidate solutions (called individuals, creatures, organisms, or phenotypes) to an optimization problem is evolved toward better solutions. Each candidate solution has a set of properties (its chromosomes or genotype) which can be mutated and altered; traditionally, solutions are represented in binary as strings of 0s and 1s, but other encodings are also possibl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simulated Annealing
Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic to approximate global optimization in a large search space for an optimization problem. It is often used when the search space is discrete (for example the traveling salesman problem, the boolean satisfiability problem, protein structure prediction, and job-shop scheduling). For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be preferable to exact algorithms such as gradient descent or branch and bound. The name of the algorithm comes from annealing in metallurgy, a technique involving heating and controlled cooling of a material to alter its physical properties. Both are attributes of the material that depend on their thermodynamic free energy. Heating and cooling the material affects both the temperature and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
