- Optimization (mathematics)
mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integervariables from within an allowed set.
An optimization problem can be represented in the following way:"Given:" a function "f" : "A" R from some set "A" to the
real numbers :"Sought:" an element "x"0 in "A" such that "f"("x"0) ≤ "f"("x") for all "x" in "A" ("minimization") or such that "f"("x"0) ≥ "f"("x") for all "x" in "A" ("maximization").
Such a formulation is called an
optimization problemor a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming- see History below). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physicsand computer visionmay refer to the technique as energy minimization, speaking of the value of the function "f" as representing the energy of the systembeing modeled.
Typically, "A" is some
subsetof the Euclidean spaceR"n", often specified by a set of "constraints", equalities or inequalities that the members of "A" have to satisfy. The domain "A" of "f" is called the "search space",while the elements of "A" are called " candidate solutions" or "feasible solutions".
The function "f" is called an "objective function", "energy function", or "cost function" [ [http://ic.arc.nasa.gov/people/dhw/papers/78.pdf No Free Lunch Theorems for Optimization] ] . A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an "optimal solution".
Generally, when the feasible region or the objective function of the problem does not present convexity, there may be several local minima and maxima, where a "local minimum" x* is defined as a point for which there exists some δ > 0 so that for all x such that
: holds; that is to say, on some region around x* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.
A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem. The branch of
applied mathematicsand numerical analysisthat is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization.
The first optimization technique, which is known as
steepest descent, goes back to Gauss. Historically, the first term to be introduced was linear programming, which was invented by George Dantzigin the 1940s. The term "programming" in this context does not refer to computer programming(although computers are nowadays used extensively to solve mathematical problems). Instead, the term comes from the use of "program" by the United States military to refer to proposed training and logisticsschedules, which were the problems that Dantzig was studying at the time. (Additionally, later on, the use of the term "programming" was apparently important for receiving government funding, as it was associated with high-technology research areas that were considered important.)
Other important mathematicians in the optimization field include:
John von Neumann
Leonid Vitalyevich Kantorovich
Naum Z. Shor
* Leonid Khachian
* Boris Polyak
* Yurii Nesterov
* Arkadii Nemirovskii
* Michael J. Todd
Optimization problems are often expressed with special notation. Here are some examples:
This asks for the minimum value for the objective function "x2" + 1, where "x" ranges over the
real numbers R. The minimum value in this case is 1, occurring at "x" = 0.
This asks for the maximum value for the objective function 2"x", where "x" ranges over the reals. In this case, there is no such maximum as the objective function is unbounded, so the answer is "
infinity" or "undefined".
This asks for the value (or values) of "x" in the interval (−∞, −1] that minimizes (or minimize) the objective function "x"2 + 1 (the actual minimum value of that function does not matter). In this case, the answer is "x" = −1.
This asks for the ("x", "y") pair (or pairs) that maximizes (or maximize) the value of the objective function "x"·cos("y"), with the added constraint that "x" lies in the interval [−5, 5] (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form (5, 2π"k") and (−5, (2"k" + 1)π), where "k" ranges over all
Linear programmingstudies the case in which the objective function "f" is linear and the set A is specified using only linear equalities and inequalities. Such a set is called a polyhedronor a polytopeif it is bounded.
Integer programmingstudies linear programs in which some or all variables are constrained to take on integervalues.
Quadratic programmingallows the objective function to have quadratic terms, while the set A must be specified with linear equalities and inequalities.
Nonlinear programmingstudies the general case in which the objective function or the constraints or both contain nonlinear parts.
Convex programmingstudies the case when the objective function is convex and the constraints, if any, form a convex set. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
Second order cone programming(SOCP).
Semidefinite programming(SDP) is a subfield of convex optimization where the underlying variables are semidefinitematrices. It is generalization of linear and convex quadratic programming.
Stochastic programmingstudies the case in which some of the constraints or parameters depend on random variables.
* Robust programming is, as stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. This is not done through the use of random variables, but instead, the problem is solved taking into account inaccuracies in the input data.
Combinatorial optimizationis concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
Infinite-dimensional optimizationstudies the case when the set of feasible solutions is a subset of an infinite- dimensional space, such as a space of functions.
Constraint satisfactionstudies the case in which the objective function "f" is constant (this is used in artificial intelligence, particularly in automated reasoning).
* Disjunctive programming used where at least one constraint must be satisfied but not all. Of particular use in scheduling.
Trajectory optimizationis the speciality of optimizing trajectories for air and space vehicles.In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):
Calculus of variationsseeks to optimize an objective defined over many points in time, by considering how the objective function changes if there is a small change in the choice path.
Optimal controltheory is a generalization of the calculus of variations.
Dynamic programmingstudies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that relates these subproblems is called the Bellman equation.
Mathematical programming with equilibrium constraintsis where the constraints include variational inequalitiesor complementarities.
Crudely all the methods are divided according to variables called:-
Single Variable Optimization
Multi Variable Optimization
For twice-differentiable functions, unconstrained problems can be solved by finding the points where the
gradientof the objective function is zero (that is, the stationary points) and using the Hessian matrixto classify the type of each point. If the Hessian is positive definite, the point is a local minimum, if negative definite, a local maximum, and if indefinite it is some kind of saddle point.
However, existence of derivatives is not always assumed and many methods were devised for specific situations. The basic classes of methods, based on smoothness of the objective function, are:
First order methods
Actual methods falling somewhere among the categories above include:
Gradient descentaka steepest descent or steepest ascent
Nelder-Mead methodaka the Amoeba method
Subgradient method- similar to gradient method in case there are no gradients
* Newton's method
Interior point methods
Conjugate gradient method
Line search- a technique for one dimensional optimization, usually used as a subroutine for other, more general techniques.
Should the objective function be convex over the region of interest, then any local minimum will also be a global minimum. There exist robust, fast numerical techniques for optimizing twice differentiable convex functions.
Constrained problems can often be transformed into unconstrained problems with the help of
Here are a few other popular methods:
Ant colony optimization
Particle swarm optimization
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 equationon 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 recent and growing subset of this field is
multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineeringproblems. Mainstream economicsalso relies heavily on mathematical programming. An often studied problem in microeconomics, the utility maximization problem, and its dual problemthe Expenditure minimization problem, are economic optimization problems. Consumersand firms are assumed to maximize their utility/ profit. Also, agents are most frequently assumed to be risk-aversethereby wishing to minimize whatever risk they might be exposed to. Asset prices are also explained using optimization though the underlying theory is more complicated than simple utility or profit optimization. Tradetheory also uses optimization to explain trade patterns between nations.
Another field that uses optimization techniques extensively is
*Important publications in optimization
Interior point methods
Radial basis function
NAG Numerical Libraries-The NAG Library contains a comprehensive collection of Optimization routines, which cover a diverse set of problems and circumstances.http://www.nag.co.uk/optimization/index.asp
NPSOL- a Fortran package designed to solve the nonlinear programming problem: the minimization of a smooth nonlinear function subject to a set of constraints on the variables.http://www.sbsi-sol-optimize.com/asp/sol_product_npsol.htm
OpenOpt- a free toolbox with connections to lots of solvers, for Python language programmers
IPOPT- an open-source primal-dual interior point method NLP solver which handles sparse matrices
KNITRO- solver for nonlinear optimization problems
Mathematica- handles linear programming, integer programming and constrained non-linear optimization problems
* Mordecai Avriel (2003)." Nonlinear Programming: Analysis and Methods." Dover Publishing. ISBN 0-486-43227-0.
* Stephen Boyd and Lieven Vandenberghe (2004). [http://www.stanford.edu/~boyd/cvxbook/ Convex Optimization] , Cambridge University Press. ISBN 0-521-83378-7.
* Elster K.-H. (1993), Modern Mathematical Methods of Optimization, Vch Pub. ISBN 3055014529.
* Jorge Nocedal and Stephen J. Wright (2006). [http://www.ece.northwestern.edu/~nocedal/book/num-opt.html Numerical Optimization] , Springer. ISBN 0-387-30303-0.
* Panos Y. Papalambros and Douglass J. Wilde (2000). [http://www.optimaldesign.org/ Principles of Optimal Design : Modeling and Computation] , Cambridge University Press. ISBN 0-521-62727-3.
* Yang X.-S. (2008), Introduction to Mathematical Optimization: From Linear Programming to Metaheuristics, Cambridge Int. Science Publishing. ISBN 1-904602827.
* [http://www-fp.mcs.anl.gov/otc/Guide/index.html NEOS Guide] currently being replaced by the [http://wiki.mcs.anl.gov/neos NEOS Wiki]
* [http://www.mathprog.org/ Mathematical Programming Society]
* [http://www.coin-or.org/ COIN-OR] — Computational Infrastructure for Operations Research
* [http://glossary.computing.society.informs.org/ Mathematical Programming Glossary]
* [http://www.ipp.mpg.de/de/for/bereiche/stellarator/Comp_sci/CompScience/csep/csep1.phy.ornl.gov/mo/mo.html Mathematical optimization]
* [http://www.mat.univie.ac.at/%7Eneum/glopt.html Global optimization]
* [http://www2.arnes.si/%7Eljc3m2/igor/links.html Optimization Related Links]
* Jon Dattorro, [http://www.convexoptimization.com Convex Optimization & Euclidean Distance Geometry]
* [http://plato.asu.edu/guide.html Decision Tree for Optimization Software] Links to optimization source codes
* [http://www.optimization-online.org Optimization Online] A repository for optimization e-prints
* [http://www.bowdoin.edu/faculty/a/alevy/book.shtml The Basics of Practical Optimization] An undergraduate optimization text
* [http://www.aimms.com AIMMS]
* [http://www.ampl.com/ AMPL]
* [http://www.gams.com/ GAMS] — General Algebraic Modeling System
* [http://www.maximal-usa.com/mpl/ MPL]
* [http://www.ilog.com/products/oplstudio/ OPL]
* [http://www.ateji.com/optimj.html OPTIMJ]
* [http://www.conopt.com/ CONOPT]
* [http://www.ilog.com/products/cplex/ CPLEX] - linear, quadratic, and mixed-integer programming solver
* [http://www.dna-evolutions.com/dnaappletsample.html JOpt]
* [http://trilinos.sandia.gov/packages/moocho/ Moocho] - a very flexible open-source NLP solver
* [http://www.mosek.com/ Mosek]
* [http://www.sas.com/technologies/analytics/optimization/ SAS/OR]
* [http://www.stanford.edu/group/SOL/software.html Free Optimization Software by Systems Optimization Laboratory, Stanford University]
* [http://www.ime.usp.br/~egbirgin/tango/ TANGO Project] - Trustable Algorithms for Nonlinear General Optimization
* [http://www.fea-optimization.com/optm/index_e.htm SmartDO] - Engineering global optimization (commercial) software
* [http://ool.sourceforge.net/ OOL (Open Optimization library)] - a set of optimization routines in C.
* [http://www.ilog.com/products/cplex/product/interfaces.cfm CPLEX Component Libraries]
* [http://www2.arnes.si/~ljc3m2/igor/ioptlib/ IOptLib (Investigative Optimization Library)] - a free open source library for development of optimization algorithms (ANSI C).
* [http://www.alglib.net/optimization/ ALGLIB] Optimization sources. C++, C#, Delphi, Visual Basic.
* [http://optalgtoolkit.sourceforge.net/ OAT (Optimization Algorithm Toolkit)] - a set of standard optimization algorithms and problems in Java.
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