Mathematical optimization

In mathematics, the term optimization refers to the study of systems that have the form

maximize f(x) such that x in A

or

minimize f(x) such that x in A

where x = (x₁, x₂, ...,x_n) is a vector of variables, f is a function from this vector to the real numbers, and A is a subset of some space.

Such a formulation is called a mathematical program. Values of x that are in A are called feasible solutions. 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.

A great many real-world and theoretical problems may be modeled in this general framework.

In general there will be several local maxima and minima, where a local minimum x^* is defined as 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.

The space used is generally Rⁿ, so f(x) takes the form of a function of a real-valued vector.

Constraints are typically also added to the problem, so the set A is defined explicitly. The constraints are grnerally formulated as a set, C = { c₀(x), c₁(x), ... ,c_m(x) }. Constraints of the form c_i(x) = 0 are termed equality constraints, and constraints of the form c_j(x) <= 0 are termed inequality constraints.

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, though computers are used extensively to solve mathematical programs. Instead, programming comes from the use of the term 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 the term "programming" was associated with important high-technology research areas that were deserving of funding.)

In fact, some mathematical programming work had been done previously...(anyone? - Gauss did some stuff here)

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

Convex Optimization

For linear constraints, these problems can often be solved simply by finding the places where the gradient of the function is 0 (i.e. the stationary points) and using the Hessian 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 the region of interest (as defined by the 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.

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 ODE 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 LCP (linear complementarity problem), which can also be viewed as a QP (quadratic programming problem).

(Are there other fields with strong connections?)