# Category: Mathematical optimization

Bilinear program
In mathematics, a bilinear program is a nonlinear optimization problem whose objective or constraint functions are bilinear. An example is the .
Dual norm
In functional analysis, the dual norm is a measure of size for a continuous linear function defined on a normed vector space.
Mean field annealing
Mean field annealing is a deterministic approximation to the simulated annealing technique of solving optimization problems. This method uses mean field theory and is based on Peierls' inequality.
Optimal apportionment
Optimal apportionment is an approach to apportionment that is based on mathematical optimization. In a problem of apportionment, there is a resource to allocate, denoted by . For example, it can be an
The dead-end elimination algorithm (DEE) is a method for minimizing a function over a discrete set of independent variables. The basic idea is to identify "dead ends", i.e., combinations of variables
Local optimum
In applied mathematics and computer science, a local optimum of an optimization problem is a solution that is optimal (either maximal or minimal) within a neighboring set of candidate solutions. This
Low-rank approximation
In mathematics, low-rank approximation is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable),
Maximum theorem
The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers with respect to its parameters. The statement was first proven by Claude Berge in 1959
Relaxation (approximation)
In mathematical optimization and related fields, relaxation is a modeling strategy. A relaxation is an approximation of a difficult problem by a nearby problem that is easier to solve. A solution of t
Open energy system models
Open energy system models are energy system models that are open source. However, some of them may use third party proprietary software as part of their workflows to input, process, or output data. Pr
Central composite design
In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to
Utility maximization problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should
Stress majorization
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of -dimensional data items, a configuration of points in -dimensional space is sought that minim
Vector optimization
Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to
Mathematical programming with equilibrium constraints
Mathematical programming with equilibrium constraints (MPEC) is the study of constrained optimization problems where the constraints include variational inequalities or complementarities. MPEC is rela
Pareto efficiency
Pareto efficiency or Pareto optimality is a situation where no individual or preference criterion can be made better off without making at least one individual or preference criterion worse off. The c
Response surface methodology
In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and
PDE-constrained optimization
PDE-constrained optimization is a subset of mathematical optimization where at least one of the constraints may be expressed as a partial differential equation. Typical domains where these problems ar
No free lunch in search and optimization
In computational complexity and optimization the no free lunch theorem is a result that states that for certain types of mathematical problems, the computational cost of finding a solution, averaged o
Smoothed analysis
In theoretical computer science, smoothed analysis is a way of measuring the complexity of an algorithm. Since its introduction in 2001, smoothed analysis has been used as a basis for considerable res
Robust optimization
Robust optimization is a field of mathematical optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented
Sion's minimax theorem
In mathematics, and in particular game theory, Sion's minimax theorem is a generalization of John von Neumann's minimax theorem, named after Maurice Sion. It states: Let be a compact convex subset of
Energy minimization
In the field of computational chemistry, energy minimization (also called energy optimization, geometry minimization, or geometry optimization) is the process of finding an arrangement in space of a c
Pseudo-Boolean function
In mathematics and optimization, a pseudo-Boolean function is a function of the form where B = {0, 1} is a Boolean domain and n is a nonnegative integer called the arity of the function. A Boolean fun
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 presen
Rosenbrock function
In mathematical optimization, the Rosenbrock function is a non-convex function, introduced by Howard H. Rosenbrock in 1960, which is used as a performance test problem for optimization algorithms. It
Bayesian efficiency
Bayesian efficiency is an analog of Pareto efficiency for situations in which there is incomplete information. Under Pareto efficiency, an allocation of a resource is Pareto efficient if there is no o
Linear search problem
In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman and independently considered by Anatole Beck.
Rastrigin function
In mathematical optimization, the Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms. It is a typical example of non-linear multimodal function.
Basis pursuit
Basis pursuit is the mathematical optimization problem of the form where x is a N-dimensional solution vector (signal), y is a M-dimensional vector of observations (measurements), A is a M × N transfo
Mixed complementarity problem
Mixed Complementarity Problem (MCP) is a problem formulation in mathematical programming. Many well-known problem types are special cases of, or may be reduced to MCP. It is a generalization of nonlin
Multiple-criteria decision analysis
Multiple-criteria decision-making (MCDM) or multiple-criteria decision analysis (MCDA) is a sub-discipline of operations research that explicitly evaluates multiple conflicting criteria in decision ma
S-procedure
The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality. The S-procedure was developed
Nearest neighbor search
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically
Applicable mathematics
No description available.
Variational Monte Carlo
In computational physics, variational Monte Carlo (VMC) is a quantum Monte Carlo method that applies the variational method to approximate the ground state of a quantum system. The basic building bloc
In mathematical optimization, a quadratically constrained quadratic program (QCQP) is an optimization problem in which both the objective function and the constraints are quadratic functions. It has t
Proportional-fair rule
In operations research and social choice, the proportional-fair (PF) rule is a rule saying that, among all possible alternatives, one should pick an alternative that cannot be improved, where "improve
Superiorization
Superiorization is an iterative method for constrained optimization. It is used for improving the efficacy of an iterative method whose convergence is resilient to certain kinds of perturbations. Such
Karush–Kuhn–Tucker conditions
In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first derivative tests (sometimes called first-order necessary conditions) for a so
VIKOR method
The VIKOR method is a multi-criteria decision making (MCDM) or multi-criteria decision analysis method. It was originally developed by Serafim Opricovic to solve decision problems with conflicting and
Utilitarian rule
In social choice and operations research, the utilitarian rule (also called the max-sum rule) is a rule saying that, among all possible alternatives, society should pick the alternative which maximize
TOPSIS
The Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is a multi-criteria decision analysis method, which was originally developed by Ching-Lai Hwang and Yoon in 1981 with fur
Hydrological optimization
Hydrological optimization applies mathematical optimization techniques (such as dynamic programming, linear programming, integer programming, or quadratic programming) to water-related problems. These
Linear complementarity problem
In mathematical optimization theory, the linear complementarity problem (LCP) arises frequently in computational mechanics and encompasses the well-known quadratic programming as a special case. It wa
Descent direction
In optimization, a descent direction is a vector that, in the sense below, moves us closer towards a local minimum of our objective function . Suppose we are computing by an iterative method, such as
Fritz John conditions
The Fritz John conditions (abbr. FJ conditions), in mathematics, are a necessary condition for a solution in nonlinear programming to be optimal. They are used as lemma in the proof of the Karush–Kuhn
Simulation-based optimization
Simulation-based optimization (also known as simply simulation optimization) integrates optimization techniques into simulation modeling and analysis. Because of the complexity of the simulation, the
Analysis of Boolean functions
In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on or (such functions are sometimes known as pseudo-Boolean functions) from a spect
Distributed constraint optimization
Distributed constraint optimization (DCOP or DisCOP) is the distributed analogue to constraint optimization. A DCOP is a problem in which a group of agents must distributedly choose values for a set o
Bauer maximum principle
Bauer's maximum principle is the following theorem in mathematical optimization: Any function that is convex and continuous, and defined on a set that is convex and compact, attains its maximum at som
Wolfe conditions
In the unconstrained minimization problem, the Wolfe conditions are a set of inequalities for performing inexact line search, especially in quasi-Newton methods, first published by Philip Wolfe in 196
Bilevel optimization
Bilevel optimization is a special kind of optimization where one problem is embedded (nested) within another. The outer optimization task is commonly referred to as the upper-level optimization task,
Convex optimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets
Highly optimized tolerance
In applied mathematics, highly optimized tolerance (HOT) is a method of generating power law behavior in systems by including a global optimization principle. It was developed by Jean M. Carlson and J
JuMP
JuMP is an algebraic modeling language and a collection of supporting packages for mathematical optimization embedded in the Julia programming language. JuMP is used by companies, government agencies,
Faustmann's formula
Faustmann's formula, or the Faustmann model, gives the present value of the income stream for forest rotation. It was derived by the German forester in 1849. The rotation problem, deciding when to cut
Optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has nume
Energy modeling
Energy modeling or energy system modeling is the process of building computer models of energy systems in order to analyze them. Such models often employ scenario analysis to investigate different ass
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take re
Multidisciplinary design optimization
Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplin
MCACEA
MCACEA (Multiple Coordinated Agents Coevolution Evolutionary Algorithm) is a general framework that uses a single evolutionary algorithm (EA) per agent sharing their optimal solutions to coordinate th
Elbow of a curve
No description available.
Wing-shape optimization
Wing-shape optimization is a software implementation of shape optimization primarily used for aircraft design. This allows for engineers to produce more efficient and cheaper aircraft designs.
P versus NP problem
The P versus NP problem is a major unsolved problem in theoretical computer science. In informal terms, it asks whether every problem whose solution can be quickly verified can also be quickly solved.
Similarity-based-TOPSIS
Similarity based TOPSIS is a multi-criteria decision-making method. The name TOPSIS is shortening from the Technique for Order Performance by Similarity to Ideal Solution. It is based on the idea of f
Jeep problem
The jeep problem, desert crossing problem or exploration problem is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The jeep
Ordinal priority approach
Ordinal priority approach (OPA) is a multiple-criteria decision analysis method that aids in solving the group decision-making problems based on preference relations.
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 cri
Optimal design
In the design of experiments, optimal designs (or optimum designs) are a class of experimental designs that are optimal with respect to some statistical criterion. The creation of this field of statis
Shape optimization
Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given const
Clarke's generalized Jacobian
In mathematics, Clarke's generalized Jacobian is a generalization of the Jacobian matrix of a smooth function to non-smooth functions. It was introduced by Clarke.
Lazy caterer's sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that
Complementarity theory
A complementarity problem is a type of mathematical optimization problem. It is the problem of optimizing (minimizing or maximizing) a function of two vector variables subject to certain requirements
Chebyshev center
In geometry, the Chebyshev center of a bounded set having non-empty interior is the center of the minimal-radius ball enclosing the entire set , or alternatively (and non-equivalently) the center of l
Multicriteria classification
In multiple criteria decision aiding (MCDA), multicriteria classification (or sorting) involves problems where a finite set of alternative actions should be assigned into a predefined set of preferent
Measuring attractiveness by a categorical-based evaluation technique (MACBETH)
Measuring attractiveness through a categorical-based evaluation technique (MACBETH) is a multiple-criteria decision analysis (MCDA) method that evaluates options against multiple criteria. MACBETH was
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition tha
Topological derivative
The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in highe
Sum-of-squares optimization
A sum-of-squares optimization program is an optimization problem with a linear cost function and a particular type of constraint on the decision variables. These constraints are of the form that when
Continuous optimization
Continuous optimization is a branch of optimization in applied mathematics. As opposed to discrete optimization, the variables used in the objective function are required to be continuous variables—th
Ackley function
In mathematical optimization, the Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation.
Steiner's calculus problem
Steiner's problem, asked and answered by , is the problem of finding the maximum of the function It is named after Jakob Steiner. The maximum is at , where e denotes the base of the natural logarithm.
Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos
Basis pursuit denoising
In applied mathematics and statistics, basis pursuit denoising (BPDN) refers to a mathematical optimization problem of the form where is a parameter that controls the trade-off between sparsity and re
Goal programming
Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA). It can be thought of as an extension or generalisation of linear prog
Online optimization
Online optimization is a field of optimization theory, more popular in computer science and operations research, that deals with optimization problems having no or incomplete knowledge of the future (
Himmelblau's function
In mathematical optimization, Himmelblau's function is a multi-modal function, used to test the performance of optimization algorithms. The function is defined by: It has one local maximum at and wher
Cake number
In mathematics, the cake number, denoted by Cn, is the maximum number of regions into which a 3-dimensional cube can be partitioned by exactly n planes. The cake number is so-called because one may im
Mathematical Optimization Society
The Mathematical Optimization Society (MOS), known as the Mathematical Programming Society until 2010, is an international association of researchers active in optimization. The MOS encourages the res
Knee of a curve
In mathematics, a knee of a curve (or elbow of a curve) is a point where the curve visibly bends, specifically from high slope to low slope (flat or close to flat), or in the other direction. This is
Binary constraint
A binary constraint, in mathematical optimization, is a constraint that involves exactly two variables. For example, consider the n-queens problem, where the goal is to place n chess queens on an n-by
Deterministic global optimization
Deterministic global optimization is a branch of numerical optimization which focuses on finding the global solutions of an optimization problem whilst providing theoretical guarantees that the report
Multi-attribute global inference of quality
Multi-attribute global inference of quality (MAGIQ) is a multi-criteria decision analysis technique. MAGIQ is based on a hierarchical decomposition of comparison attributes and rating assignment using
Oracle complexity (optimization)
In mathematical optimization, oracle complexity is a standard theoretical framework to study the computational requirements for solving classes of optimization problems. It is suitable for analyzing i
NP-completeness
In computational complexity theory, a problem is NP-complete when: 1. * it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-for
Geometric median
In geometry, the geometric median of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the
Superiority and inferiority ranking method
The superiority and inferiority ranking method (or SIR method) is a multi-criteria decision making model (MCDA) which can handle real data and provides six different preference structures for the syst
Paper bag problem
In geometry, the paper bag problem or teabag problem is to calculate the maximum possible inflated volume of an initially flat sealed rectangular bag which has the same shape as a cushion or pillow, m
ÉLECTRE
ÉLECTRE is a family of multi-criteria decision analysis (MCDA) methods that originated in Europe in the mid-1960s. The acronym ÉLECTRE stands for: ÉLimination Et Choix Traduisant la REalité ("Eliminat
Mixed linear complementarity problem
In mathematical optimization theory, the mixed linear complementarity problem, often abbreviated as MLCP or LMCP, is a generalization of the linear complementarity problem to include free variables.
Mathematical optimization
Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. I
Minimax theorem
In the mathematical area of game theory, a minimax theorem is a theorem providing conditions that guarantee that the max–min inequality is also an equality. The first theorem in this sense is von Neum
Shekel function
The Shekel function is a multidimensional, multimodal, continuous, deterministic function commonly used as a test function for testing optimization techniques. The mathematical form of a function in d
Map segmentation
In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into smaller sub-regions in order to achieve a certai
Proximal operator
In mathematical optimization, the proximal operator is an operator associated with a proper, lower semi-continuous convex function from a Hilbert space to , and is defined by: For any function in this
Trajectory optimization
Trajectory optimization is the process of designing a trajectory that minimizes (or maximizes) some measure of performance while satisfying a set of constraints. Generally speaking, trajectory optimiz
Compressed sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solu
Dispersive flies optimisation
Dispersive flies optimisation (DFO) is a bare-bones swarm intelligence algorithm which is inspired by the swarming behaviour of flies hovering over food sources. DFO is a simple optimiser which works
Mirror descent
In mathematics, mirror descent is an iterative optimization algorithm for finding a local minimum of a differentiable function. It generalizes algorithms such as gradient descent and multiplicative we
Corner solution
A corner solution is a special solution to an agent's maximization problem in which the quantity of one of the arguments in the maximized function is zero. In non-technical terms, a corner solution is
Signomial
A signomial is an algebraic function of one or more independent variables. It is perhaps most easily thought of as an algebraic extension of multivariable polynomials—an extension that permits exponen
Separation oracle
A separation oracle (also called a cutting-plane oracle) is a concept in the mathematical theory of convex optimization. It is a method to describe a convex set that is given as an input to an optimiz
Feasible region
In mathematical optimization, a feasible region, feasible set, search space, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem tha
Guess value
In mathematical modeling, a guess value is more commonly called a starting value or initial value. These are necessary for most optimization problems which use search algorithms, because those algorit
Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest valu
Topology optimization
Topology optimization (TO) is a mathematical method that optimizes material layout within a given design space, for a given set of loads, boundary conditions and constraints with the goal of maximizin
Hyperparameter optimization
In machine learning, hyperparameter optimization or tuning is the problem of choosing a set of optimal hyperparameters for a learning algorithm. A hyperparameter is a parameter whose value is used to
Backtracking line search
In (unconstrained) mathematical optimization, a backtracking line search is a line search method to determine the amount to move along a given search direction. Its use requires that the objective fun
Stochastic multicriteria acceptability analysis
Stochastic multicriteria acceptability analysis (SMAA) is a multiple-criteria decision analysis method for problems with missing or incomplete information.
Discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.
International Society for Structural and Multidisciplinary Optimization
The International Society for Structural and Multidisciplinary Optimization is a learned society in the field of multidisciplinary design optimization that was founded in October 1991. It has more tha
Keynes–Ramsey rule
In macroeconomics, the Keynes–Ramsey rule is a necessary condition for the optimality of intertemporal consumption choice. Usually it is express as a differential equation relating the rate of change
Walrasian auction
A Walrasian auction, introduced by Léon Walras, is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The pri