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# Mathematical optimization

In mathematics, computer science and operations research, mathematical optimization or mathematical programming, alternatively spelled optimisation, is the selection of a best element (with regard to some criterion) from some set of available alternatives. [1]

## Active filter

An active filter is a type of analog circuit implementing an electronic filter using active components, typically an amplifier.

## Aerospace engineering

Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft.

## Agent (economics)

In economics, an agent is an actor and more specifically a decision maker in a model of some aspect of the economy.

## Albert W. Tucker

Albert William Tucker (28 November 1905 – 25 January 1995) was a Canadian mathematician who made important contributions in topology, game theory, and non-linear programming.

## Alexander Schrijver

Alexander (Lex) Schrijver (born 4 May 1948 in Amsterdam) is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in Amsterdam.

## Algorithm

In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems.

## Andrzej Piotr Ruszczyński

Andrzej Piotr Ruszczyński (born July 29, 1951) is a Polish-American applied mathematician, noted for his contributions to mathematical optimization, in particular, stochastic programming and risk-averse optimization.

## Applied mathematics

Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.

## Argument of a function

In mathematics, an argument of a function is a specific input in the function, also known as an independent variable.

Arkadi Nemirovski (born March 14, 1947) is a professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology.

## Artificial bee colony algorithm

In computer science and operations research, the artificial bee colony algorithm (ABC) is an optimization algorithm based on the intelligent foraging behaviour of honey bee swarm, proposed by Karaboga in 2005.

## Artificial intelligence

Artificial intelligence (AI, also machine intelligence, MI) is intelligence demonstrated by machines, in contrast to the natural intelligence (NI) displayed by humans and other animals.

## Asset pricing

In financial economics, asset pricing refers to a formal treatment and development of two main pricing principles, outlined below.

## Automated reasoning

Automated reasoning is an area of computer science and mathematical logic dedicated to understanding different aspects of reasoning.

## Bellman equation

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming.

## Bernard Koopman

Bernard Osgood Koopman (1900 – August 18, 1981) was a French-born American mathematician, known for his work in ergodic theory, the foundations of probability, statistical theory and operations research.

## Bounded set

In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size.

## Brachistochrone curve

In mathematics and physics, a brachistochrone curve, or curve of fastest descent, is the one lying on plane between a point A and a lower point B, where B is not directly below A, on which a bead slides frictionlessly under the influence of a uniform gravitational field to a given end point in the shortest time.

## Broyden–Fletcher–Goldfarb–Shanno algorithm

In numerical optimization, the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is an iterative method for solving unconstrained nonlinear optimization problems.

Business is the activity of making one's living or making money by producing or buying and selling products (goods and services).

## Calculus of variations

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.

## Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

## Claude Berge

Claude Jacques Berge (5 June 1926 – 30 June 2002) was a French mathematician, recognized as one of the modern founders of combinatorics and graph theory.

## Claude Lemaréchal

Claude Lemaréchal is a French applied mathematician, and former senior researcher (directeur de recherche) at INRIA near Grenoble, France.

## Combinatorial optimization

In applied mathematics and theoretical computer science, combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.

## Comparative statics

In economics, comparative statics is the comparison of two different economic outcomes, before and after a change in some underlying exogenous parameter.

## Complementarity theory

A complementarity problem is a type of mathematical optimization problem.

## Computational complexity theory

Computational complexity theory is a branch of the theory of computation in theoretical computer science that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other.

## Computer programming

Computer programming is the process of building and designing an executable computer program for accomplishing a specific computing task.

## Computer science

Computer science deals with the theoretical foundations of information and computation, together with practical techniques for the implementation and application of these foundations.

## Computer vision

Computer vision is a field that deals with how computers can be made for gaining high-level understanding from digital images or videos.

## Concave function

In mathematics, a concave function is the negative of a convex function.

## Conformational isomerism

In chemistry, conformational isomerism is a form of stereoisomerism in which the isomers can be interconverted just by rotations about formally single bonds (refer to figure on single bond rotation).

## Conic optimization

Conic optimization is a subfield of convex optimization that studies problems consisting of minimizing a convex function over the intersection of an affine subspace and a convex cone.

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite.

## Constraint (mathematics)

In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy.

## Constraint programming

In computer science, constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints.

## Constraint satisfaction

In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution to a set of constraints that impose conditions that the variables must satisfy.

## Consumer

A consumer is a person or organization that use economic services or commodities.

## Control theory

Control theory in control systems engineering deals with the control of continuously operating dynamical systems in engineered processes and machines.

## Convex function

In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.

## Convex optimization

Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.

## Convex set

In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.

## Coordinate descent

Coordinate descent is an optimization algorithm that successively minimizes along coordinate directions to find the minimum of a function.

## Critical point (mathematics)

In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.

## Cuckoo search

In operations research, cuckoo search is an optimization algorithm developed by Xin-she Yang and Suash Deb in 2009.

## Curve fitting

Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints.

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

## Derivative test

In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point.

## 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 reported solution is indeed the global one, within some predefined tolerance.

## Differential evolution

In evolutionary computation, differential evolution (DE) is a method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality.

## Dimension

In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.

## Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

## Domain of a function

In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.

## Donald Goldfarb

Donald Goldfarb (born August 14, 1941 in New York City) is an American mathematician who specializes in mathematical optimization and numerical analysis.

## Duality (optimization)

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.

## Dynamic programming

Dynamic programming is both a mathematical optimization method and a computer programming method.

## Dynamic relaxation

Dynamic relaxation is a numerical method, which, among other things, can be used do "form-finding" for cable and fabric structures.

## Dynamic stochastic general equilibrium

Dynamic stochastic general equilibrium modeling (abbreviated as DSGE, or DGE, or sometimes SDGE) is a method in macroeconomics that attempts to explain economic phenomena, such as economic growth and business cycles, and the effects of economic policy, through econometric models based on applied general equilibrium theory and microeconomic principles.

## Economic equilibrium

In economics, economic equilibrium is a state where economic forces such as supply and demand are balanced and in the absence of external influences the (equilibrium) values of economic variables will not change.

## Economics

Economics is the social science that studies the production, distribution, and consumption of goods and services.

## Electrical engineering

Electrical engineering is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism.

## Ellipsoid method

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions.

## Engineering optimization

Engineering optimization is the subject which uses optimization techniques to achieve design goals in engineering.

## Envelope theorem

The envelope theorem is a result about the differentiability properties of the objective function of a parameterized optimization problem.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Evolutionary algorithm

In artificial intelligence, an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization algorithm.

## Expenditure minimization problem

In microeconomics, the expenditure minimization problem is the dual of the utility maximization problem: "how much money do I need to reach a certain level of happiness?".

## Extreme value theorem

In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval, then f must attain a maximum and a minimum, each at least once.

## 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 that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints.

## Fermat's theorem (stationary points)

In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of the function is a stationary point (the function derivative is zero at that point).

## Finite difference

A finite difference is a mathematical expression of the form.

## Flow network

In graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow.

## Fractional programming

In mathematical optimization, fractional programming is a generalization of linear-fractional programming.

## Frank–Wolfe algorithm

The Frank–Wolfe algorithm is an iterative first-order optimization algorithm for constrained convex optimization.

## Fritz John

Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems.

## Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

## Function of a real variable

In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers, or a subset of that contains an interval of positive length.

## Functional (mathematics)

In mathematics, the term functional (as a noun) has at least two meanings.

## Game theory

Game theory is "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers".

## Genetic algorithm

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).

## Geometric programming

A geometric program (GP) is an optimization problem of the form In the context of geometric programming (unlike all other disciplines), a monomial is a function h:\mathbb_^n \to \mathbb defined as where c > 0 \ and a_i \in \mathbb.

## Geophysics

Geophysics is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis.

## George Dantzig

George Bernard Dantzig (November 8, 1914 – May 13, 2005) was an American mathematical scientist who made important contributions to operations research, computer science, economics, and statistics.

## Global optimization

Global optimization is a branch of applied mathematics and numerical analysis that deals with the global optimization of a function or a set of functions according to some criteria.

## Goal programming

Goal programming is a branch of multiobjective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA).

In mathematics, the gradient is a multi-variable generalization of the derivative.

Gradient descent is a first-order iterative optimization algorithm for finding the minimum of a function.

## Harold W. Kuhn

Harold William Kuhn (July 29, 1925 – July 2, 2014) was an American mathematician who studied game theory.

## Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

## Heuristic (computer science)

In computer science, artificial intelligence, and mathematical optimization, a heuristic (from Greek εὑρίσκω "I find, discover") is a technique designed for solving a problem more quickly when classic methods are too slow, or for finding an approximate solution when classic methods fail to find any exact solution.

## Hill climbing

In numerical analysis, hill climbing is a mathematical optimization technique which belongs to the family of local search.

## Infinite-dimensional optimization

In certain optimization problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function or the shape of a body.

## Infinity

Infinity (symbol) is a concept describing something without any bound or larger than any natural number.

## Integer

An integer (from the Latin ''integer'' meaning "whole")Integer&#x2009;'s first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

## Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.

## Interior-point method

Interior-point methods (also referred to as barrier methods) are a certain class of algorithms that solve linear and nonlinear convex optimization problems.

International trade theory is a sub-field of economics which analyzes the patterns of international trade, its origins, and its welfare implications.

## Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

## Interval (mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

## Isaac Newton

Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, astronomer, theologian, author and physicist (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution.

## Iterative method

In computational mathematics, an iterative method is a mathematical procedure that uses an initial guess to generate a sequence of improving approximate solutions for a class of problems, in which the n-th approximation is derived from the previous ones.

## James B. Orlin

James Berger Orlin (born April 19, 1953), accessed 2011-03-05.

## JEL classification codes

Articles in economics journals are usually classified according to the JEL classification codes, a system originated by the Journal of Economic Literature.

## John Geanakoplos

John Geanakoplos (born March 18, 1955) is an American economist, and the current James Tobin Professor of Economics at Yale University.

## John von Neumann

John von Neumann (Neumann János Lajos,; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, and polymath.

## Jon Lee (mathematician)

Jon Lee (born 1960) is an American mathematician and operations researcher, the G. Lawton and Louise G. Johnson Professor of Engineering at the University of Michigan.

## Joseph-Louis Lagrange

Joseph-Louis Lagrange (or;; born Giuseppe Lodovico Lagrangia, Encyclopædia Britannica or Giuseppe Ludovico De la Grange Tournier, Turin, 25 January 1736 – Paris, 10 April 1813; also reported as Giuseppe Luigi Lagrange or Lagrangia) was an Italian Enlightenment Era mathematician and astronomer.

## Journal of Economic Literature

The Journal of Economic Literature is a peer-reviewed academic journal, published by the American Economic Association, that surveys the academic literature in economics.

## Karl Weierstrass

Karl Theodor Wilhelm Weierstrass (Weierstraß; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis".

## Karush–Kuhn–Tucker conditions

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are First-order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied.

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## Labour economics

Labour economics seeks to understand the functioning and dynamics of the markets for wage labour.

## Lagrange multiplier

In mathematical optimization, the method of Lagrange multipliers (named after Joseph-Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variables).

## Lagrangian relaxation

In the field of mathematical optimization, Lagrangian relaxation is a relaxation method which approximates a difficult problem of constrained optimization by a simpler problem.

## Lawrence E. Blume

Lawrence E. Blume is a Goldwin Smith Professor of Economics and Professor of Information Science at Cornell University, USA.

## László Lovász

László Lovász (born March 9, 1948) is a Hungarian mathematician, best known for his work in combinatorics, for which he was awarded the Wolf Prize and the Knuth Prize in 1999, and the Kyoto Prize in 2010.

## Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns.

## Leonid Kantorovich

Leonid Vitaliyevich Kantorovich (a) (19 January 19127 April 1986) was a Soviet mathematician and economist, known for his theory and development of techniques for the optimal allocation of resources.

## Leonid Khachiyan

Leonid Genrikhovich Khachiyan (Լեոնիդ Գենրիխովիչ Խաչիյան; Леонид Генрихович Хачиян; May 3, 1952 – April 29, 2005) was a Soviet mathematician of Armenian descent who taught Computer Science at Rutgers University.

## Lev Pontryagin

Lev Semyonovich Pontryagin (Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician.

## Line search

In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum \mathbf^* of an objective function f:\mathbb R^n\to\mathbb R. The other approach is trust region.

## 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.

## Linear programming

Linear programming (LP, also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.

## Linear-fractional programming

In mathematical optimization, linear-fractional programming (LFP) is a generalization of linear programming (LP).

## Lionel Robbins

Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist, and prominent member of the economics department at the London School of Economics.

## LIONsolver

LIONsolver is an integrated software for data mining, business intelligence, analytics, and modeling Learning and Intelligent OptimizatioN and reactive business intelligence approach.

## Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

## Logistics

Logistics is generally the detailed organization and implementation of a complex operation.

## Loss function

In mathematical optimization, statistics, econometrics, decision theory, machine learning and computational neuroscience, a loss function or cost 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.

## Macroeconomics

Macroeconomics (from the Greek prefix makro- meaning "large" and economics) is a branch of economics dealing with the performance, structure, behavior, and decision-making of an economy as a whole.

## Margaret H. Wright

Margaret H. Wright (born February 18, 1944) is an American computer scientist and mathematician.

## Mathematical model

A mathematical model is a description of a system using mathematical concepts and language.

## 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.

## 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.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

## 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 value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

## Maximum theorem

The maximum theorem provides conditions for the continuity of an optimized function and the set of its maximizers as a parameter changes.

## Memetic algorithm

Memetic algorithms (MAs) represent one of the recent growing areas of research in evolutionary computation.

## Metaheuristic

In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity.

## Microwave

Microwaves are a form of electromagnetic radiation with wavelengths ranging from one meter to one millimeter; with frequencies between and.

## Mineral physics

Mineral physics is the science of materials that compose the interior of planets, particularly the Earth.

## Model predictive control

Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints.

## Monomial

In mathematics, a monomial is, roughly speaking, a polynomial which has only one term.

## 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.

## Narendra Karmarkar

Narendra Krishna Karmarkar (born 1957) is an Indian mathematician, who developed Karmarkar's algorithm.

## Naum Z. Shor

Naum Zuselevich Shor (Наум Зуселевич Шор) (1 January 1937 – 26 February 2006) was a Soviet and Ukrainian Jewish mathematician specializing in optimization.

The Nelder–Mead method or downhill simplex method or amoeba method is a commonly applied numerical method used to find the minimum or maximum of an objective function in a multidimensional space.

## Newton's method in optimization

In calculus, Newton's method is an iterative method for finding the roots of a differentiable function (i.e. solutions to the equation). In optimization, Newton's method is applied to the derivative of a twice-differentiable function to find the roots of the derivative (solutions to), also known as the stationary points of.

## Nonlinear programming

In mathematics, nonlinear programming is the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.

## Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

## Operations research

Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions.

## Optimal control

Optimal control theory deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved.

## Optimization problem

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions.

## Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

## Parameter

A parameter (from the Ancient Greek παρά, para: "beside", "subsidiary"; and μέτρον, metron: "measure"), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

## Pareto efficiency

Pareto efficiency or Pareto optimality is a state of allocation of resources from which it is impossible to reallocate so as to make any one individual or preference criterion better off without making at least one individual or preference criterion worse off.

## Particle swarm optimization

In computer science, particle swarm optimization (PSO) is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality.

## Pattern search (optimization)

Pattern search (also known as direct search, derivative-free search, or black-box search) is a family of numerical optimization methods that does not require a gradient.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Pi

The number is a mathematical constant.

## Pierre de Fermat

Pierre de Fermat (Between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.

## Polyhedron

In geometry, a polyhedron (plural polyhedra or polyhedrons) is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices.

## Polytope

In elementary geometry, a polytope is a geometric object with "flat" sides.

## Portfolio (finance)

In finance, a portfolio is a collection of investments held by an investment company, hedge fund, financial institution or individual.

## Positive-definite matrix

In linear algebra, a symmetric real matrix M is said to be positive definite if the scalar z^Mz is strictly positive for every non-zero column vector z of n real numbers.

## Posynomial

A posynomial, also known as a posinomial in some literature, is a function of the form where all the coordinates x_i and coefficients c_k are positive real numbers, and the exponents a_ are real numbers.

## Princeton University Press

Princeton University Press is an independent publisher with close connections to Princeton University.

## Process optimization

Process optimization is the discipline of adjusting a process so as to optimize some specified set of parameters without violating some constraint.

## Profit (economics)

In economics, profit in the accounting sense of the excess of revenue over cost is the sum of two components: normal profit and economic profit.

Quadratic programming (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables.

## Quantum optimization algorithms

Mathematical optimization deals with finding the best solution to a problem (according to some criteria) from a set of possible solutions.

## Quasi-Newton method

Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method.

## Quasiconvex function

In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set.

## R. Tyrrell Rockafellar

Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics.

In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If is an open subset of '''R'''''n'' and &thinsp;&thinsp; is Lipschitz continuous, then &thinsp; is differentiable almost everywhere in; that is, the points in at which &thinsp; is not differentiable form a set of Lebesgue measure zero.

## Random variable

In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable whose possible values are outcomes of a random phenomenon.

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## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Relaxation (approximation)

In mathematical optimization and related fields, relaxation is a modeling strategy.

## Resource leveling

In project management, resource levelling is defined by A Guide to the Project Management Body of Knowledge (PMBOK Guide) as "A technique in which start and finish dates are adjusted based on resource constraints with the goal of balancing demand for resources with the available supply." When performing project planning activities, the manager will attempt to schedule certain tasks simultaneously.

## Richard E. Bellman

Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and important contributions in other fields of mathematics.

## Rigid body dynamics

Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.

## Risk aversion

In economics and finance, risk aversion is the behavior of humans (especially consumers and investors), when exposed to uncertainty, in attempting to lower that uncertainty.

## Robert B. Schnabel

Robert (“Bobby”) Schnabel (born December 18, 1950) is an American Computer Scientist, and is executive director and CEO of the Association for Computing Machinery (ACM), a position he has held since November 1, 2015.

## Robust optimization

Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution.

## Roger Fletcher (mathematician)

Roger Fletcher FRS FRSE (29 January 1939 – 15 July 2016) was a British mathematician and professor at University of Dundee.

## Roger J-B Wets

Roger Jean-Baptiste Robert Wets (born February 1937) is a "pioneer" in stochastic programming and a leader in variational analysis who publishes as Roger J-B Wets.

## Ronald A. Howard

Ronald Arthur Howard (born August 27, 1934) is a professor in the Department of Engineering-Economic Systems (now the Department of Management Science and Engineering) in the School of Engineering at Stanford University.

In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.

## Satisfiability

In mathematical logic, satisfiability and validity are elementary concepts of semantics.

## Scarcity

Scarcity refers to the limited availability of a commodity, which may be in demand in the market.

## Search theory

In microeconomics, search theory studies buyers or sellers who cannot instantly find a trading partner, and must therefore search for a partner prior to transacting.

## Second-order cone programming

A second-order cone program (SOCP) is a convex optimization problem of the form where the problem parameters are f \in \mathbb^n, \ A_i \in \mathbb^, \ b_i \in \mathbb^, \ c_i \in \mathbb^n, \ d_i \in \mathbb, \ F \in \mathbb^, and g \in \mathbb^p.

## Seismology

Seismology (from Ancient Greek σεισμός (seismós) meaning "earthquake" and -λογία (-logía) meaning "study of") is the scientific study of earthquakes and the propagation of elastic waves through the Earth or through other planet-like bodies.

## Semidefinite programming

Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.

Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization.

## Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

## Simplex algorithm

In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.

## Simulated annealing

Simulated annealing (SA) is a probabilistic technique for approximating the global optimum of a given function.

## Simulation-based optimization

Simulation-based optimization integrates optimization techniques into simulation analysis.

## Simultaneous perturbation stochastic approximation

Simultaneous perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters.

## Slack variable

In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality.

## Space mapping

The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993.

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

## Stationary point

In mathematics, particularly in calculus, a stationary point or critical point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero.

## Stochastic optimization

Stochastic optimization (SO) methods are optimization methods that generate and use random variables.

## Stochastic process

--> In probability theory and related fields, a stochastic or random process is a mathematical object usually defined as a collection of random variables.

## Stochastic programming

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty.

## Stochastic tunneling

In numerical analysis, stochastic tunneling (STUN) is an approach to global optimization based on the Monte Carlo method-sampling of the function to be objective minimized in which the function is nonlinearly transformed to allow for easier tunneling among regions containing function minima.

## Structure of the Earth

The interior structure of the Earth is layered in spherical shells: an outer silicate solid crust, a highly viscous asthenosphere and mantle, a liquid outer core that is much less viscous than the mantle, and a solid inner core.

## Subderivative

In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to functions which are not differentiable.

Subgradient methods are iterative methods for solving convex minimization problems.

## Subroutine

In computer programming, a subroutine is a sequence of program instructions that performs a specific task, packaged as a unit.

## Subset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.

## Surrogate model

A surrogate model is an engineering method used when an outcome of interest cannot be easily directly measured, so a model of the outcome is used instead.

## System

A system is a regularly interacting or interdependent group of items forming an integrated whole.

## Tabu search

Tabu search, created by Fred W. Glover in 1986 and formalized in 1989, is a metaheuristic search method employing local search methods used for mathematical optimization.

## Test functions for optimization

In applied mathematics, test functions, known as artificial landscapes, are useful to evaluate characteristics of optimization algorithms, such as.

## The American Economic Review

The American Economic Review is a peer-reviewed academic journal of economics.

## The New Palgrave Dictionary of Economics

The New Palgrave Dictionary of Economics (2008), 2nd ed., is an eight-volume reference work on economics, edited by Steven N. Durlauf and Lawrence E. Blume and published by Palgrave Macmillan.

## Thomas L. Magnanti

Thomas L. Magnanti (born 1945) is an American engineer and Institute Professor and former Dean of the School of Engineering at the Massachusetts Institute of Technology.

## Trust region

Trust region is a term used in mathematical optimization to denote the subset of the region of the objective function that is approximated using a model function (often a quadratic).

## Utility

Within economics the concept of utility is used to model worth or value, but its usage has evolved significantly over time.

## Utility maximization problem

In microeconomics, the utility maximization problem is the problem consumers face: "how should I spend my money in order to maximize my utility?" It is a type of optimal decision problem.

## Value (mathematics)

In mathematics, value may refer to several, strongly related notions.

## Variational inequality

In mathematics, a variational inequality is an inequality involving a functional, which has to be solved for all possible values of a given variable, belonging usually to a convex set.

## 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 certain constraints.

## Vehicle routing problem

The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver to a given set of customers?".

## William J. Cook

William John Cook (born October 18, 1957 in New Jersey) is an American operations researcher and mathematician, professor in Combinatorics and Optimization at the University of Waterloo since June 2013, after being for 10 years the Chandler Family Chair Professor of Industrial and Systems Engineering and Adjunct Professor of Mathematics at the Georgia Institute of Technology.

## William Karush

William Karush (1 March 1917 – 22 February 1997) was a professor of California State University at Northridge and was a mathematician best known for his contribution to Karush–Kuhn–Tucker conditions.

## William R. Pulleyblank

William R. Pulleyblank is a Canadian and American operations researcher.

## Yurii Nesterov

Yurii Nesterov is a Russian mathematician, an internationally recognized expert in convex optimization, especially in the development of efficient algorithms and numerical optimization analysis.

## References

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