home | alphabetical index | |||||||

## Optimization (mathematics)In mathematics, the termoptimization 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*x*_{0}in*A*such that*f*(*x*_{0}) ≥*f*(*x*) for all*x*in*A*("maximization") or such that*f*(*x*_{0}) ≤*f*(*x*) for all*x*in*A*("minimization").
mathematical program. A great many real-world and theoretical problems may be modeled in this general framework.
Typically,
In general there will be several local maxima and minima, where a local minimum x
## NotationOptimization problems are often expressed with special notation. Here are some examples: This asks for the minimum value for the expressionx+1, where ^{2}x ranges over the real numbers R. The answer in this case is 1.
- max
_{x in R}2x
x, 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 min
_{x in ]-∞,-1]}*x*^{2}+1
x in the interval ]-∞,-1] which minimizes the expression x^{2}+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*)
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 (5,2&pik) and (-5,(2k+1)π), where k ranges over all integers.## Techniques- 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.
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: ## UsesAdditionally, 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). (Are there other fields with strong connections? Economics) ## History
Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s. The term
*In fact, some mathematical programming work had been done previously...(anyone? - Gauss did some stuff here)*
- game theory
- compilers for programming languages
- operations research
- fuzzy logic
- random optimization
- genetic algorithm
- variational inequality
- mixed complementarity
External links:
| |||||||

copyright © 2004 FactsAbout.com |