Optimization (mathematics) |
In mathematics, the term optimization refers to the study
of problems that have the form
- Given: a function
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). A great many real-world and
theoretical problems may be modeled 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 feasible 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 greater than the value at that point. Local
maxima are defined similarly. In general, it is easy 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.
Notation
Optimization problems are often expressed with special notation. Here are some examples:
- minx in R x2+1
This asks for the minimum value for the expression x2+1, where x ranges over the real numbers R. The minimum value in this case is 1, occurring at
x=0.
- maxx in R 2x
This asks for the maximum value for the expression 2x, where x ranges over the reals. In this case, there is
no such maximum as the expression is unbounded, so the answer is "infinity" or
"undefined".
- arg minx 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 maxx 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.
Techniques
Techniques for solving mathematical programs depend on the nature of the objective function and constraint set. The following
major subfields exist:
- linear programming studies the case in which the objective
function f is linear and the set A is specified using only linear equalities and inequalities
- integer programming studies linear 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 linear equalities and inequalities
- nonlinear programming studies the general case in
which the objective or constraints or both contain nonlinear parts
- stochastic programming studies the case in which
some of the constraints depend on random variables
- 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 ideal 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 nonlinear optimization problems with many poor local minima:
Uses
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 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 engineering
problems.
Another field that uses optimization techniques extensively is operations research.
History
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). Instead, 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 research areas that were considered important.)
- In fact, some mathematical programming work had been done previously...(anyone? - Gauss did some stuff here)
See also
External links
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