Optimisasi (matematik)

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Dina matematika, watesan optimisasi nujul kana pangajaran ngeunaan masalah nu mibanda bentuk

Given: a fungsi f : A -> R from some set A to the real numbers

Sought: an element x₀ in A such that f(x₀) ≥ f(x) for all x in A ("maximization") or such that f(x₀) ≤ f(x) for all x in A ("minimization").

Such a formulation is sometimes called a mathematical program (a term not directly related to computer programming, but still in use for example for linear programming - see history below). Many réal-world and théoretical problems may be modéled in this general framework.

Typically, A is some subset of Euclidean space Rⁿ, often specified by a set of constraints, equalities or inequalities that the members of A have to satisfy. The elements of A are called the feasible solutions and the function f is called the objective function. A féasible solution that maximizes (or minimizes, if that is the goal) the objective function is called an optimal solution.

In general there will be several local maxima and minima, where a local minimum x^* is defined as a point such that for some δ > 0 and all x such that ||x - x^* || ≤ δ the formula f(x) ≥ f(x^*) holds; that is to say on some ball around x^* all of the function values are gréater than the value at that point. Local maxima are defined similarly. In general, it is éasy to find local minima, however additional facts about the problem (e.g. the function being convex) are required to ensure that the solution found is a global minimum.

Notasi

Masalah optimization biasa dilambangkeun ku notasi husus. Here are some examples:

min_{x in R} x²+1

Keur nyebutkeun nilai minimum dina rumus x²+1, nu mana x aya dina "rentang" real numbers R. Nilai minimum dina kasus ieu nyaéta 1, dina waktu x=0.

max_{x in R} 2x

This asks for the maximum value for the expression 2x, where x ranges over the réals. In this case, there is no such maximum as the expression is unbounded, so the answer is "infinity" or "undefined".

arg min_{x in [-∞,-1]} x²+1

This asks for the value(s) of x in the interval [-∞,-1] which minimizes the expression x²+1. (The actual minimum value of that expression does not matter.) In this case, the answer is x = -1.

arg max_{x in [-∞,5], y in R} x · cos(y)

This asks for the (x,y) pair(s) that maximize the value of the expression x·cos(y), with the added constraint that x cannot exceed 5. (Again, the actual maximum value of the expression does not matter.) In this case, the solutions are the pairs of the form (5,2πk) and (-5,(2k+1)π), where k ranges over all integers.

Teknik

tehnik keur ngarengsekeun program matematik gumantung kana kondisi alami fungsi obyektif fungsi jeung constraint set. The following major subfields exist:

linear programming studies the case in which the objective function f is linéar and the set A is specified using only linéar equalities and inequalities
integer programming studies linéar programs in which some or all variables are constrained to take on integer values
quadratic programming allows the objective function to have quadratic terms, while the set A must be specified with linéar equalities and inequalities
nonlinear programming studies the general case in which the objective or constraints or both contain nonlinéar parts
program stokastik nalungtik kasus nu gumantung kana variabel acak
dynamic programming studies the case which has optimal substructure and overlapping subproblems.

For twice-differentiable functions, unconstrained problems can be solved by finding the places where the gradient of the function is 0 (i.e. the stationary points) and using the Hessian matrix to classify the type of 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.

Should a function be convex over a region of interest (as defined by constraints) then any local minimum will also be a global minimum. Robust, fast numerical techniques exist for optimizing doubly differentiable convex functions. Outside of these functions, less idéal techniques must be used.

Constrained problems can often be transformed into unconstrained problems with the help of the Lagrange multiplier.

Several techniques exist for find a good local minimum in nonlinéar optimization problems with many poor local minima:

Pamakean

Additionally, 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 nonlinéar géometric 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 engineering problems.

Another field that uses optimization techniques extensively is operations research.

Sajarah

Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s. The term programming in this context does not refer to computer programming (although computers are nowadays used extensively to solve mathematical programs). Instéad, the term comes from the use of program by the United States military to refer to proposed training and logistics schedules, 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 reséarch aréas that were considered important.)

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