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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. It is generally divided into two subfields:
discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.
Scope
As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete variab ...
and
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—that is, to be chosen from a set of rea ...
. Optimization problems of sorts arise in all quantitative disciplines from
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
to
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
and
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, and the development of solution methods has been of interest in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
for centuries.
In the more general approach, an
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
consists of
maximizing or minimizing a
real function
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 \mathbb, or a subset of \mathbb that contains an interv ...
by systematically choosing
input values from within an allowed set and computing the
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
. More generally, optimization includes finding "best available" values of some objective function given a defined
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
(or input), including a variety of different types of objective functions and different types of domains.
Optimization problems
Optimization problems can be divided into two categories, depending on whether the
variables are
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
or
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
:
* An optimization problem with discrete variables is known as a ''
discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.
Scope
As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete variab ...
'', in which an
object
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ...
such as an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
,
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
or
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
must be found from a
countable set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
.
* A problem with continuous variables is known as a ''
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—that is, to be chosen from a set of rea ...
'', in which an optimal value from a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
must be found. They can include
constrained problems and multimodal problems.
An optimization problem can be represented in the following way:
:''Given:'' a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
from some
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
to the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s
:''Sought:'' an element such that for all ("minimization") or such that for all ("maximization").
Such a formulation is called an
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
or a mathematical programming problem (a term not directly related to
computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, but still in use for example in
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 function#As a polynomial function, li ...
– see
History
History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the History of writing#Inventions of writing, invention of writing systems is considered prehistory. "History" is an umbr ...
below). Many real-world and theoretical problems may be modeled in this general framework.
Since the following is valid
:
it suffices to solve only minimization problems. However, the opposite perspective of considering only maximization problems would be valid, too.
Problems formulated using this technique in the fields of
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
may refer to the technique as ''
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
minimization'', speaking of the value of the function as representing the energy of the
system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment (systems), environment, is described by its boundaries, ...
being
modeled. In
machine learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence.
Machine ...
, it is always necessary to continuously evaluate the quality of a data model by using a
cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.
Typically, is some
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, often specified by a set of ''
constraints'', equalities or inequalities that the members of have to satisfy. The
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of is called the ''search space'' or the ''choice set'', while the elements of are called ''
candidate solution
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, potent ...
s'' or ''feasible solutions''.
The function is called, variously, an ''objective function'', a ''
loss 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 ...
'' or ''cost function'' (minimization), a ''utility function'' or ''fitness function'' (maximization), or, in certain fields, an ''energy function'' or ''energy
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
''. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an ''optimal solution''.
In mathematics, conventional optimization problems are usually stated in terms of minimization.
A ''local minimum'' is defined as an element for which there exists some such that
:
the expression holds;
that is to say, on some region around all of the function values are greater than or equal to the value at that element.
Local maxima are defined similarly.
While a local minimum is at least as good as any nearby elements, a
global minimum
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 ran ...
is at least as good as every feasible element.
Generally, unless the objective function is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
in a minimization problem, there may be several local minima.
In a
convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.
A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem.
Global optimization
Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the max ...
is the branch of
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical s ...
and
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.
Notation
Optimization problems are often expressed with special notation. Here are some examples:
Minimum and maximum value of a function
Consider the following notation:
:
This denotes the minimum
value
Value or values may refer to:
Ethics and social
* Value (ethics) wherein said concept may be construed as treating actions themselves as abstract objects, associating value to them
** Values (Western philosophy) expands the notion of value beyo ...
of the objective function , when choosing from the set of
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s . The minimum value in this case is 1, occurring at .
Similarly, the notation
:
asks for the maximum value of the objective function , where may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is "
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
" or "undefined".
Optimal input arguments
Consider the following notation:
:
or equivalently
:
This represents the value (or values) of the
Argument of a function, argument in the
interval that minimizes (or minimize) the objective function (the actual minimum value of that function is not what the problem asks for). In this case, the answer is , since is infeasible, that is, it does not belong to the
feasible set
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, potent ...
.
Similarly,
:
or equivalently
:
represents the pair (or pairs) that maximizes (or maximize) the value of the objective function , with the added constraint that lie in the interval (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form and , where ranges over all
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s.
Operators and are sometimes also written as and , and stand for ''argument of the minimum'' and ''argument of the maximum''.
History
Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he i ...
and
Lagrange found calculus-based formulae for identifying optima, while
Newton
Newton most commonly refers to:
* Isaac Newton (1642–1726/1727), English scientist
* Newton (unit), SI unit of force named after Isaac Newton
Newton may also refer to:
Arts and entertainment
* ''Newton'' (film), a 2017 Indian film
* Newton ( ...
and
Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
proposed iterative methods for moving towards an optimum.
The term "
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 function#As a polynomial function, li ...
" for certain optimization cases was due to
George B. Dantzig, although much of the theory had been introduced by
Leonid Kantorovich
Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet ...
in 1939. (''Programming'' in this context does not refer to
computer programming
Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, but comes from the use of ''program'' by the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
military to refer to proposed training and
logistics
Logistics is generally the detailed organization and implementation of a complex operation. In a general business sense, logistics manages the flow of goods between the point of origin and the point of consumption to meet the requirements of ...
schedules, which were the problems Dantzig studied at that time.) Dantzig published the
Simplex algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
in 1947, and
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
developed the theory of
duality in the same year.
Other notable researchers in mathematical optimization include the following:
*
Richard Bellman
Richard Ernest Bellman (August 26, 1920 – March 19, 1984) was an American applied mathematician, who introduced dynamic programming in 1953, and made important contributions in other fields of mathematics, such as biomathematics. He founde ...
*
Dimitri Bertsekas
Dimitri Panteli Bertsekas (born 1942, Athens, el, Δημήτρης Παντελής Μπερτσεκάς) is an applied mathematician, electrical engineer, and computer scientist, a McAfee Professor at the Department of Electrical Engineering ...
*
Michel Bierlaire
Michel Bierlaire (born 1967 in Namur, Belgium) is a Belgian- Swiss applied mathematician specialized in transportation modeling and optimization. He is a professor at EPFL (École Polytechnique Fédérale de Lausanne) and the head of the Transp ...
*
Roger Fletcher
*
Ronald A. Howard
*
Fritz John
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a 1 ...
*
Narendra Karmarkar
Narendra Krishna Karmarkar (born Circa 1956) is an Indian Mathematician. Karmarkar developed Karmarkar's algorithm. He is listed as an ISI highly cited researcher.
He invented one of the first provably polynomial time algorithms for linear prog ...
*
William Karush
William Karush (1 March 1917 – 22 February 1997) was an American professor of mathematics at California State University at Northridge and was a mathematician best known for his contribution to Karush–Kuhn–Tucker conditions. In his master's ...
*
Leonid Khachiyan
Leonid Genrikhovich Khachiyan (; russian: Леони́д Ге́нрихович Хачия́н; May 3, 1952April 29, 2005) was a Soviet and American mathematician and computer scientist.
He was most famous for his ellipsoid algorithm (1979) for ...
*
Bernard Koopman
Bernard Osgood Koopman (January 19, 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.
Education and work
Af ...
*
Harold Kuhn
*
László Lovász
László Lovász (; born March 9, 1948) is a Hungarian mathematician and professor emeritus at Eötvös Loránd University, best known for his work in combinatorics, for which he was awarded the 2021 Abel Prize jointly with Avi Wigderson. He ...
*
Arkadi Nemirovski
Arkadi Nemirovski (born March 14, 1947) is a professor at the H. Milton Stewart School of Industrial and Systems Engineering at the Georgia Institute of Technology. He has been a leader in continuous optimization and is best known for his work o ...
*
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. He is currently a professor at the University of Lou ...
*
Lev Pontryagin
Lev Semenovich Pontryagin (russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September 1908 – 3 May 1988) was a Soviet mathematician. He was born in Moscow and lost his eyesight completely due ...
*
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. He is the author of four major books including the landmark ...
*
Naum Z. Shor
*
Albert Tucker
Major subfields
*
Convex programming
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). Many classes of convex optimization pro ...
studies the case when the objective function is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
(minimization) or
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
* The concavity
In ca ...
(maximization) and the constraint set is
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytope ...
. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
**
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 function#As a polynomial function, li ...
(LP), a type of convex programming, studies the case in which the objective function ''f'' is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
or a
polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -d ...
if it is
bounded.
**
Second-order cone programming
A second-order cone program (SOCP) is a convex optimization problem of the form
:minimize \ f^T x \
:subject to
::\lVert A_i x + b_i \rVert_2 \leq c_i^T x + d_i,\quad i = 1,\dots,m
::Fx = g \
where the problem parameters are f \in \mathbb^n, ...
(SOCP) is a convex program, and includes certain types of quadratic programs.
**
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 ...
(SDP) is a subfield of convex optimization where the underlying variables are
semidefinite matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. It is a generalization of linear and convex quadratic programming.
**
Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
**
Geometric programming A geometric program (GP) is an optimization problem of the form
:
\begin
\mbox & f_0(x) \\
\mbox & f_i(x) \leq 1, \quad i=1, \ldots, m\\
& g_i(x) = 1, \quad i=1, \ldots, p,
\end
where f_0,\dots,f_m are posynomials and g_1,\dots,g_p are monomials. I ...
is a technique whereby objective and inequality constraints expressed as
posynomials A posynomial, also known as a posinomial in some literature, is a function of the form
: f(x_1, x_2, \dots, x_n) = \sum_^K c_k x_1^ \cdots x_n^
where all the coordinates x_i and coefficients c_k are positive real numbers, and the exponents a_ are ...
and equality constraints as
monomials
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
can be transformed into a convex program.
*
Integer programming
An integer programming problem is a mathematical optimization or Constraint satisfaction problem, feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer linear programmin ...
studies linear programs in which some or all variables are constrained to take on
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
values. This is not convex, and in general much more difficult than regular linear programming.
*
Quadratic programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constra ...
allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
*
Fractional programming In mathematical optimization, fractional programming is a generalization of linear-fractional programming. The objective function in a fractional program is a ratio of two functions that are in general nonlinear. The ratio to be optimized often desc ...
studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
*
Nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or sta ...
studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
*
Stochastic programming
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, ...
studies the case in which some of the constraints or parameters depend on
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s.
*
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 as deterministic variability in the value of the ...
is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties defined by an uncertainty set.
*
Combinatorial optimization
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combi ...
is concerned with problems where the set of feasible solutions is discrete or can be reduced to a
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
*Discrete group, a ...
one.
*
Stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective funct ...
is used with random (noisy) function measurements or random inputs in the search process.
*
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. Such a problem is an infinite-dimensional optimization problem, becaus ...
studies the case when the set of feasible solutions is a subset of an infinite-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al space, such as a space of functions.
*
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, ...
and
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 optimizati ...
s make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
*
Constraint satisfaction In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through
a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for th ...
studies the case in which the objective function ''f'' is constant (this is used in
artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
, particularly in
automated reasoning
In computer science, in particular in knowledge representation and reasoning and metalogic, the area of automated reasoning is dedicated to understanding different aspects of reasoning. The study of automated reasoning helps produce computer progra ...
).
**
Constraint programming
Constraint programming (CP) is a paradigm for solving combinatorial problems that draws on a wide range of techniques from artificial intelligence, computer science, and operations research. In constraint programming, users declaratively state th ...
is a programming paradigm wherein relations between variables are stated in the form of constraints.
* Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
*
Space mapping
The space mapping methodology for modeling and design optimization of engineering systems was first discovered by John Bandler in 1993. It uses relevant existing knowledge to speed up model generation and design optimization of a system. The know ...
is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or
surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
.
In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
*
Calculus of variations
The calculus of variations (or Variational Calculus) 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 t ...
Is concerned with finding the best way to achieve some goal, such as finding a surface whose boundary is a specific curve, but with the least possible area.
*
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 numerous applications in science, engineering and ...
theory is a generalization of the calculus of variations which introduces control policies.
*
Dynamic programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics.
I ...
is the approach to solve the
stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective funct ...
problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the
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. It writes the "value" of a decision problem at a certain point in time ...
.
*
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 related to the Stackelberg game.
MPEC is used ...
is where the constraints include
variational inequalities 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. The mathematical theory of variational inequalities was initial ...
or
complementarities.
Multi-objective optimization
Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that improve upon one criterion at the expense of another is known as the
Pareto set. The curve created plotting weight against stiffness of the best designs is known as the
Pareto frontier
In multi-objective optimization, the Pareto front (also called Pareto frontier or Pareto curve) is the set of all Pareto efficient solutions. The concept is widely used in engineering. It allows the designer to restrict attention to the set of effi ...
.
A design is judged to be "Pareto optimal" (equivalently, "Pareto efficient" or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.
The choice among "Pareto optimal" solutions to determine the "favorite solution" is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.
Multi-objective optimization problems have been generalized further into
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. A multi-objective optimiz ...
problems where the (partial) ordering is no longer given by the Pareto ordering.
Multi-modal or global optimization
Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.
Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm.
Common approaches to
global optimization
Global optimization is a branch of applied mathematics and numerical analysis that attempts to find the global minima or maxima of a function or a set of functions on a given set. It is usually described as a minimization problem because the max ...
problems, where multiple local extrema may be present include
evolutionary algorithm
In computational intelligence (CI), an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic population-based metaheuristic optimization algorithm. An EA uses mechanisms inspired by biological evolution, such as reproduc ...
s, Bayesian optimization and simulated annealing.
Classification of critical points and extrema
Feasibility problem
The ''satisfiability problem'', also called the ''feasibility problem'', is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.
Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to Relaxation (approximation), relax the feasibility conditions using a slack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until the slack is null or negative.
Existence
The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.
Necessary conditions for optimality
Fermat's theorem (stationary points), One of Fermat's theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero (see first derivative test). More generally, they may be found at Critical point (mathematics), critical points, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.
Optima of equality-constrained problems can be found by the Lagrange multiplier method. The optima of problems with equality and/or inequality constraints can be found using the 'Karush–Kuhn–Tucker conditions'.
Sufficient conditions for optimality
While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the Hessian matrix#Bordered Hessian, bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then the satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.
Sensitivity and continuity of optima
The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes. The process of computing this change is called comparative statics.
The maximum theorem of Claude Berge (1963) describes the continuity of an optimal solution as a function of underlying parameters.
Calculus of optimization
For unconstrained problems with twice-differentiable functions, some critical point (mathematics), critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points). More generally, a zero subgradient certifies that a local minimum has been found for convex optimization, minimization problems with convex convex function, functions and other Rademacher's theorem, locally Lipschitz functions.
Further, critical points can be classified using the positive definite matrix, definiteness of the Hessian matrix: If the Hessian is ''positive'' definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.
Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers. Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.
When the objective function is a convex function, then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.
Global convergence
More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of subgradient method, non-differentiable optimization. Usually, a global optimizer is much slower than advanced local optimizers (such as BFGS method, BFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.
Computational optimization techniques
To solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methods that converge to a solution (on some specified class of problems), or heuristic algorithm, heuristics that may provide approximate solutions to some problems (although their iterates need not converge).
Optimization algorithms
*
Simplex algorithm
In mathematical optimization, Dantzig's simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The name of the algorithm is derived from the concept of a simplex and was suggested by T. S. Motzkin. Simplices are n ...
of George Dantzig, designed for
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 function#As a polynomial function, li ...
* Extensions of the simplex algorithm, designed for quadratic programming and for linear-fractional programming
* Variants of the simplex algorithm that are especially suited for flow network, network optimization
* Combinatorial optimization, Combinatorial algorithms
* Quantum optimization algorithms
Iterative methods
The iterative methods used to solve problems of nonlinear programming differ according to whether they subroutine, evaluate Hessian matrix, Hessians, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the Computational complexity theory, computational complexity (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.
One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables. The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd-order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.
* Methods that evaluate Hessians (or approximate Hessians, using finite differences):
** Newton's method in optimization, Newton's method
** Sequential quadratic programming: A Newton-based method for small-medium scale ''constrained'' problems. Some versions can handle large-dimensional problems.
** Interior point methods: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians.
* Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
** Coordinate descent methods: Algorithms which update a single coordinate in each iteration
** Conjugate gradient methods: Iterative methods for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
** Gradient descent (alternatively, "steepest descent" or "steepest ascent"): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
** Subgradient methods: An iterative method for large Rademacher's theorem, locally Lipschitz continuity, Lipschitz functions using subgradient, generalized gradients. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
** Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for convex optimization, convex minimization problems (similar to conjugate gradient methods).
** Ellipsoid method: An iterative method for small problems with quasiconvex function, quasiconvex objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
** Frank–Wolfe algorithm, Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear constraints, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
** Quasi-Newton methods: Iterative methods for medium-large problems (e.g. N<1000).
** Simultaneous perturbation stochastic approximation (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
* Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
** Interpolation methods
** Pattern search (optimization), Pattern search methods, which have better convergence properties than the Nelder–Mead method, Nelder–Mead heuristic (with simplices), which is listed below.
** Mirror descent
Heuristics
Besides (finitely terminating) algorithms and (convergent) iterative methods, there are heuristic algorithm, heuristics. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:
* Differential evolution
* Dynamic relaxation
* Evolutionary algorithms
* Genetic algorithms
* Hill climbing with random restart
* Memetic algorithm
* Nelder–Mead method, Nelder–Mead simplicial heuristic: A popular heuristic for approximate minimization (without calling gradients)
* Particle swarm optimization
* Simulated annealing
* Stochastic tunneling
* Tabu search
Applications
Mechanics
Problems in rigid body dynamics (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differential equation on a constraint manifold; the constraints are various nonlinear geometric constraints such as "these two points must always coincide", "this surface must not penetrate any other", or "this point must always lie somewhere on this curve". Also, the problem of computing contact forces can be done by solving a linear complementarity problem, which can also be viewed as a QP (quadratic programming) problem.
Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the engineering optimization, and another recent and growing subset of this field is multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineering problems.
This approach may be applied in cosmology and astrophysics.
Economics and finance
Economics is closely enough linked to optimization of agent (economics), agents that an influential definition relatedly describes economics ''qua'' science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses. Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibrium (economics), equilibria. The ''Journal of Economic Literature'' JEL classification codes, codes classify mathematical programming, optimization techniques, and related topics under JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories, JEL:C61-C63.
In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem, are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize their utility, while firms are usually assumed to maximize their Profit (economics), profit. Also, agents are often modeled as being Risk aversion, risk-averse, thereby preferring to avoid risk. Asset pricing, Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. International trade theory also uses optimization to explain trade patterns between nations. The optimization of Portfolio (finance), portfolios is an example of multi-objective optimization in economics.
Since the 1970s, economists have modeled dynamic decisions over time using control theory. For example, dynamic search theory, search models are used to study labor economics, labor-market behavior. A crucial distinction is between deterministic and stochastic models. Macroeconomics, Macroeconomists build dynamic stochastic general equilibrium, dynamic stochastic general equilibrium (DSGE) models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.
Electrical engineering
Some common applications of optimization techniques in electrical engineering include active filter design, stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures, handset antennas, electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical
surrogate model A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so a model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design ...
and space mapping methodologies since the discovery of space mapping in 1993.
Civil engineering
Optimization has been widely used in civil engineering. Construction management and transportation engineering are among the main branches of civil engineering that heavily rely on optimization. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures, resource leveling,
Hydrological optimization, water resource allocation, traffic management and schedule optimization.
Operations research
Another field that uses optimization techniques extensively is
operations research
Operations research ( en-GB, operational research) (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a discipline that deals with the development and application of analytical methods to improve deci ...
. Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses stochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and
stochastic optimization
Stochastic optimization (SO) methods are optimization methods that generate and use random variables. For stochastic problems, the random variables appear in the formulation of the optimization problem itself, which involves random objective funct ...
methods.
Control engineering
Mathematical optimization is used in much modern controller design. High-level controllers such as model predictive control (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.
Geophysics
Optimization techniques are regularly used in geophysics, geophysical parameter estimation problems. Given a set of geophysical measurements, e.g. seismology, seismic recordings, it is common to solve for the mineral physics, physical properties and structure of the earth, geometrical shapes of the underlying rocks and fluids. The majority of problems in geophysics are nonlinear with both deterministic and stochastic methods being widely used.
Molecular modeling
Nonlinear optimization methods are widely used in conformational analysis.
Computational systems biology
Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.
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 function#As a polynomial function, li ...
has been applied to calculate the maximal possible yields of fermentation products,
and to infer gene regulatory networks from multiple microarray datasets as well as transcriptional regulatory networks from high-throughput data.
Nonlinear programming
In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem where some of the constraints or the objective function are nonlinear. An optimization problem is one of calculation of the extrema (maxima, minima or sta ...
has been used to analyze energy metabolism and has been applied to metabolic engineering and parameter estimation in biochemical pathways.
Machine learning
Solvers
See also
* Brachistochrone
* Curve fitting
* Deterministic global optimization
* Goal programming
* List of publications in mathematics#Optimization, Important publications in optimization
* Least squares
* Mathematical Optimization Society (formerly Mathematical Programming Society)
* :Optimization algorithms and methods, Mathematical optimization algorithms
* :Mathematical optimization software, Mathematical optimization software
* Process optimization
* Simulation-based optimization
* Test functions for optimization
* Variational calculus
* Vehicle routing problem
Notes
Further reading
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External links
* Links to optimization source codes
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{{Authority control
Mathematical optimization,
Operations research
Mathematical and quantitative methods (economics), Optimization