Optimization (mathematics)

In Mathematical, the optimization is the study of the problems which are form:

being given: a function $f:\; With\; \backslash \; mapsto\; \backslash \; mathbb\; R$ of a $A$ unit in the whole of the Real number S

To seek: an element $x\_0$ of $A$ such as $f\; (x\_0)\; \backslash \; Ge\; F\; (X)$ for all $x$ in $A$ (“maximization”) or such as $f\; (x\_0)\; \backslash \; the\; F\; (X)$ for all $x$ in $A$ (“minimization”).

Such a formulation is sometimes called mathematical program (term not directly related to the programming data-processing, but used for example for the linear Programming - to see the history below). Several theoretical and practical problems can be studied in this general framing.

Since the maximization of $f$ is equivalent to the minimization of $-f$, a method to find the minimum (or the maximum) is enough to solve the problem of optimization.

It frequently happens that $A$ is a Sous-ensemble given Euclidean Espace $\backslash \; mathbb\; R^n$, often specified by a whole of Contrainte S, equalities or inequalities which the elements of $A$ must satisfy. The elements of $A$ are called the acceptable solutions and the function $f$ is called the function objective . A possible solution which maximizes (or minimizes, if it is the goal) the function objective is called a optimal solution . In the particular case where $A$ is a subset of $\backslash \; mathbb\; N^n$ or $\backslash \; mathbb\; N^p\; \backslash \; times\; \backslash \; mathbb\; R^q$, one speaks about combinative Optimization.

A local minimum $x^*$ is defined as a point such as there exists a vicinity V of $x^*$ such as for all $x\; \backslash \; in\; V$, $f\; (X)\; \backslash \; Ge\; F\; (x^*)$; i.e. in a vicinity of $x^*$ all the values of the function are larger than the value in this point. When $A$ is a subset of $\backslash \; mathbb\; R^n$, or more generally a normalized vector space, that is written: for a $\backslash \; delta\; >\; 0$ given and all $x$ such as $||X\; -\; x^*||\; \backslash \; the\; \backslash \; delta$ one has $f\; (X)\; \backslash \; Ge\; F\; (x^*)$. The local maxima are similarly defined. In general, it is easy to find the minima (maxima) local, which are sometimes numerous. To check that the found solution is a minimum (maximum) total, it is necessary to resort to additional knowledge on the problem (for example the convexity of the function objective).

There does not exist known method ensuring whatever the type of function which one will find total a extremum .

Notation

The problems of optimization are often expressed with a special notation. Here some examples:

$\backslash \; min\_\; \{X\; \backslash \; in\; \backslash \; mathbb\; \{R\}\}\; x^2\; +\; 1$

One seeks the minimal value for the expression $x^2\; +\; 1$, where $x$ extends on the real numbers $\backslash \; mathbb\; R$. The minimal value in this case is 1, being caused with $x\; =\; 0$.

$\backslash \; max\_\; \{X\; \backslash \; in\; \backslash \; mathbb\; \{R\}\}\; 2x$

One seeks the maximum value for the expression $2x$, where $x$ extends on realities. In this case, there is not such a maximum since the expression is not limited, therefore the answer is “the Infini” or “indefinite”.

$\backslash \; arg\; \backslash \; min\_\; \{X\; \backslash \; in]\; -\; \backslash \; infty,\; -1]\}\; x^2\; +\; 1$

One seeks the values of $x$ in the interval $]\; -\; \backslash \; infty,\; -1]$ which minimizes the expression $x^2\; +\; 1$. (The true minimal value of this expression is not important.) In this case, the answer is $x\; =\; -1$.

$\backslash \; arg\; \backslash \; max\_\; \{X\; \backslash \; in\; 5,\; there\; \backslash \; in\; \backslash \; mathbb\; \{R\}\}\; X\; \backslash \; cos\; (there)$

One seeks the pairs $(X,\; there)$ which maximize the value of the expression $x\; \backslash \; cos\; (there)$, with the added constraint that the absolute value of $x$ cannot exceed 5. (Again, the true maximum value of the expression is not important.) In this case, the solutions are the pairs of the form $(5,\; 2k\; \backslash \; pi)$ and $(-\; 5,\; (2k\; +\; 1)\; \backslash \; pi)$, where $k$ extends on all the entireties.

Techniques

The techniques to solve the mathematical problems depend on the nature of the function objective of the constrained unit. The following major under-fields exist:

• the linear Programming studies the cases where the border of the unit has and the function objective are linear. It is a method very employed to establish the programs of the oil refineries, but also to determine the most profitable composition of a salted mixture, under constraints, starting from contract the prices of the moment.
• the linear Programming of integers studies the linear programs in which some or all the variables are forced to take whole values. These problems can be solved by various methods: Secant Separation and evaluation, Plane.
• the quadratic Programmation makes it possible the function objective to have quadratic terms, while preserving a description of the unit starting from linear equalities/inequalities
• has the non-linear Programmation studies the general case in which the objective or the constraints (or both) contain non-linear parts
• the stochastic Programmation studies the case in which some of the constraints depend on random variable
• the dynamic Programming uses the property that an optimal solution is necessarily composed of optimal under-solutions (attention: the opposite is not true in general) to break up the problem by avoiding the combinative Explosion. It is usable only when the function objective is monotonous increasing. It is the dynamic programming which allows for example
• the airframe manufacturers to find the plans of takeoff optimal their machines,
• with the engineers of basin to distribute the mining production between their various wells
• with the media planners to effectively distribute an advertizing budget between various supports

For the derivable functions twice, problems without constraints can be solved by finding the places where the gradient of the function is 0 (for example stationary points) and by using the Matrice hessienne to classify the type of point. If the hessien is definite positive, the point is a local minimum; if it is definite negative, a local maximum and if it is indefinite it is one point-collar .

If the function is convex on the whole of the acceptable solutions (defined by the constraints) then any local minimum is also a total minimum. Robust and fast digital techniques exist to optimize convex functions doubling derivable. Apart from these functions, less effective techniques must be employed.

The problems with constraints can often be transformed into problems without constraints using the Multiplicateur of Lagrange: this method indeed amounts introducing increasing penalties as one approaches the constraints. An algorithm due to '' Hugh Everett '' makes it possible to update in a coherent way the values of the multipliers at each iteration to guarantee convergence. This one also generalized the interpretation of these multipliers to apply them to functions which are neither continuous, nor derivable. The lambda becomes right then a coefficient of penalty .

Many techniques exist to find a good minimum local in the non-linear problems of optimization with several poor local minima, they are generally regarded as Métaheuristique S.

Uses

The problems of the dynamic with rigid Corps S (especially the dynamics of the rigid bodies articulated) often need techniques of mathematical programming, since one can see the dynamics of the rigid bodies like resolution of a ordinary differential equation on a forced variety; the constraints are various non-linear geometrical constraints such as “these two points must always coincide”, or “this point must always be on this curve”. Also, the problem of calculate the forces of contact can be completed by solving a linear Problème of complementarity, which can also be seen like a PPQ (quadratic problem of programming).

Several problems of design can also be expressed in the form of programs of optimization. This application is called the optimization of form. A recent subset and crescent of this field are called the multidisciplinary Optimization which, although useful in several problems, was particularly applied to the problems of the aerospace Génie.

Another field which uses the techniques of optimization is the Operations research.

Optimization is one of the central tools of the Microéconomie which are based on the principle of the Rationalité and of the optimization of the behaviors, the Profit for the undertaken, and the Utilité for the consuming .

History

Historically, the first introduced term was that of linear Programming, invented by George Dantzig in the Années 1940. The term programming in this context does not refer to the data-processing Programmation (although the computers are largely used nowadays to solve mathematical programs). It comes from the use of the word program by the American armed forces to establish schedules of formation and choices Logistique S, that Dantzig studied at the time. The use of the term “programming” also had an interest to release appropriations in one time when the Planification became a priority of the governments.

See too

• Descent of gradient
• Algorithm of the simplex
• Game theory,
• the Compiler S for the Computer programming languages,
• Operations research,
• Logical fuzzy,
• random Optimization,
• Hugh Everett
• Métaheuristique,
• Inequality of variation,
• mixed Complementarity,
• simple Algorithm
• Microéconomie.

External bonds

• Guide NEOS

More

• Michel Minoux, mathematical Programming - theory and algorithms , Dunod editions, 1983

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