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Penalty Method
Penalty methods are a certain class of algorithms for solving constrained optimization problems. A penalty method replaces a constrained optimization problem by a series of unconstrained problems whose solutions ideally converge to the solution of the original constrained problem. The unconstrained problems are formed by adding a term, called a penalty function, to the objective function that consists of a ''penalty parameter'' multiplied by a measure of violation of the constraints. The measure of violation is nonzero when the constraints are violated and is zero in the region where constraints are not violated. Example Let us say we are solving the following constrained problem: : \min f(\mathbf x) subject to : c_i(\mathbf x) \le 0 ~\forall i \in I. This problem can be solved as a series of unconstrained minimization problems : \min \Phi_k (\mathbf x) = f (\mathbf x) + \sigma_k ~ \sum_ ~ g(c_i(\mathbf x)) where : g(c_i(\mathbf x))=\max(0,c_i(\mathbf x ))^2. In the a ... [...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 Computational problem, 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 (computer science), 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 ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constrained Optimization
In mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized. Constraints can be either hard constraints, which set conditions for the variables that are required to be satisfied, or soft constraints, which have some variable values that are penalized in the objective function if, and based on the extent that, the conditions on the variables are not satisfied. Relation to constraint-satisfaction problems The constrained-optimization problem (COP) is a significant generalization of the classic constraint-satisfaction problem (CSP) model. COP is a CSP that includes an ''objective function'' to be optimized. Many algorithms are used to handle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Objective Function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Compression
Image compression is a type of data compression applied to digital images, to reduce their cost for storage or transmission. Algorithms may take advantage of visual perception and the statistical properties of image data to provide superior results compared with generic data compression methods which are used for other digital data. Lossy and lossless image compression Image compression may be lossy or lossless. Lossless compression is preferred for archival purposes and often for medical imaging, technical drawings, clip art, or comics. Lossy compression methods, especially when used at low bit rates, introduce compression artifacts. Lossy methods are especially suitable for natural images such as photographs in applications where minor (sometimes imperceptible) loss of fidelity is acceptable to achieve a substantial reduction in bit rate. Lossy compression that produces negligible differences may be called visually lossless. Methods for lossy compression: * Transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrier Method (mathematics)
In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the objective function that is easier to handle. The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated by their connection with primal-dual interior point methods. Motivation Consider the following constrained optimization problem: :minimize :subject to where is some constant. If one wishes to remove the inequality constraint, the problem can be re-formulated as :minimize , :where if , and zero otherwise. This problem is equivalent to the first. It gets rid of the inequality, but introduces the issue that the penalty function , and therefo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequential Quadratic Programming
Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable. SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to Newton's method for finding a point where the gradient of the objective vanishes. If the problem has only equality constraints, then the method is equivalent to applying Newton's method to the first-order optimality conditions, or Karush–Kuhn–Tucker conditions, of the problem. Algorithm basics Consider a nonlinear programming problem of the form: :\begin \min\limits_ & f(x) \\ \mbox & b(x) \ge 0 \\ & c(x) = 0. \end The Lagrangian for this problem is :\mathcal(x,\lambda,\sigma) = f(x) - \lambda b(x) - \sigma c(x), wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Successive Linear Programming
Successive Linear Programming (SLP), also known as Sequential Linear Programming, is an optimization technique for approximately solving nonlinear optimization problems. Starting at some estimate of the optimal solution, the method is based on solving a sequence of first-order approximations (i.e. linearizations) of the model. The linearizations are linear programming problems, which can be solved efficiently. As the linearizations need not be bounded, trust regions or similar techniques are needed to ensure convergence in theory. SLP has been used widely in the petrochemical industry since the 1970s. See also * Sequential quadratic programming * Sequential linear-quadratic programming * Augmented Lagrangian method Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems a ... References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequential Linear-quadratic Programming
Sequential linear-quadratic programming (SLQP) is an iterative method for nonlinear optimization problems where objective function and constraints are twice continuously differentiable. Similarly to sequential quadratic programming (SQP), SLQP proceeds by solving a sequence of optimization subproblems. The difference between the two approaches is that: * in SQP, each subproblem is a quadratic program, with a quadratic model of the objective subject to a linearization of the constraints * in SLQP, two subproblems are solved at each step: a linear program (LP) used to determine an active set, followed by an equality-constrained quadratic program (EQP) used to compute the total step This decomposition makes SLQP suitable to large-scale optimization problems, for which efficient LP and EQP solvers are available, these problems being easier to scale than full-fledged quadratic programs. Algorithm basics Consider a nonlinear programming problem of the form: :\begin \min\limits_ & f(x ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Point Method
Interior-point methods (also referred to as barrier methods or IPMs) are a certain class of algorithms that solve linear and nonlinear convex optimization problems. An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967 and reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, which runs in provably polynomial time and is also very efficient in practice. It enabled solutions of linear programming problems that were beyond the capabilities of the simplex method. Contrary to the simplex method, it reaches a best solution by traversing the interior of the feasible region. The method can be generalized to convex programming based on a self-concordant barrier function used to encode the convex set. Any convex optimization problem can be transformed into minimizing (or maximizing) a linear function over a convex set by converting to the epigraph form. The idea of encodi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmented Lagrangian Method
Augmented Lagrangian methods are a certain class of algorithms for solving constrained optimization problems. They have similarities to penalty methods in that they replace a constrained optimization problem by a series of unconstrained problems and add a penalty term to the objective; the difference is that the augmented Lagrangian method adds yet another term, designed to mimic a Lagrange multiplier. The augmented Lagrangian is related to, but not identical with the method of Lagrange multipliers. Viewed differently, the unconstrained objective is the Lagrangian of the constrained problem, with an additional penalty term (the augmentation). The method was originally known as the method of multipliers, and was studied much in the 1970 and 1980s as a good alternative to penalty methods. It was first discussed by Magnus Hestenes, and by Michael Powell in 1969. The method was studied by R. Tyrrell Rockafellar in relation to Fenchel duality, particularly in relation to proximal-poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barrier Function
In constrained optimization, a field of mathematics, a barrier function is a continuous function whose value on a point increases to infinity as the point approaches the boundary of the feasible region of an optimization problem. Such functions are used to replace inequality constraints by a penalizing term in the objective function that is easier to handle. The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions. Resumption of interest in logarithmic barrier functions was motivated by their connection with primal-dual interior point methods. Motivation Consider the following constrained optimization problem: :minimize :subject to where is some constant. If one wishes to remove the inequality constraint, the problem can be re-formulated as :minimize , :where if , and zero otherwise. This problem is equivalent to the first. It gets rid of the inequality, but introduces the issue that the penalty function , and therefo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alice E
Alice may refer to: * Alice (name), most often a feminine given name, but also used as a surname Literature * Alice (''Alice's Adventures in Wonderland''), a character in books by Lewis Carroll * ''Alice'' series, children's and teen books by Phyllis Reynolds Naylor * ''Alice'' (Hermann book), a 2009 short story collection by Judith Hermann Computers * Alice (computer chip), a graphics engine chip in the Amiga computer in 1992 * Alice (programming language), a functional programming language designed by the Programming Systems Lab at Saarland University * Alice (software), an object-oriented programming language and IDE developed at Carnegie Mellon * Alice mobile robot * Artificial Linguistic Internet Computer Entity, an open-source chatterbot * Matra Alice, a home micro-computer marketed in France * Alice, a brand name used by Telecom Italia for internet and telephone services Video games * '' Alice: An Interactive Museum'', a 1991 adventure game * ''American McGee's Alic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
