Optimization (mathematics)

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In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an allowed set. This problem can be represented in the following way

Given: a function f : A <math>\to</math> 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 ("minimization") or such that f(x₀) ≥ f(x) for all x in A ("maximization").

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example for linear programming - see history below). Many real-world and theoretical problems may be modeled in this general framework.

Typically, A is some subset of the 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 feasible solutions. The function f is called an objective function, or cost function. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

The domain A of f is called the search space, while the elements of A are called candidate solutions or feasible solutions.

Generally, when the feasible region or the objective function of the problem does not present convexity, there may be several local minima and maxima, where a local minimum x^* is defined as a point for which there exists some δ > 0 so that for and all x such that

<math>\|\mathbf{x}-\mathbf{x}^*\|\leq\delta</math>;

the expression

<math>f(\mathbf{x}^*)\leq f(\mathbf{x})</math>

holds; that is to say, on some region around x^* all of the function values are greater than or equal to the value at that point. Local maxima are defined similarly.

A large number of algorithms proposed for solving non-convex problems – including the majority of commercially available solvers – are not capable of making a distinction between local optimal solutions and rigorous optimal solutions, and will treat the former as actual solutions to the original problem. The branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a non-convex problem is called global optimization.

1 Notation
2 Major subfields
3 Techniques
4 Uses
5 History
6 See also
7 References
8 External links

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Notation

Optimization problems are often expressed with special notation. Here are some examples:

<math>\min_{x\in\mathbb R}\; x^2 + 1</math>

This asks for the minimum value for the objective function x² + 1, where x ranges over the real numbers R. The minimum value in this case is 1, occurring at x = 0.

<math>\max_{x\in\mathbb R}\; 2x</math>

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

<math>\operatorname{argmin}_{x\in[-\infty,-1]}\; x^2 + 1</math>

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

<math>\operatorname{argmax}_{x\in[-5,5],\;y\in\mathbb R}\; x\cdot\cos(y)</math>

This asks for the (x, y) pair(s) that maximize the value of the objective functionx·cos(y), with the added constraint that x lies in the interval [−5, 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.

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Major subfields

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 function or the 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 in which the optimization strategy is based on splitting the problem into smaller subproblems.
Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
Constraint satisfaction studies the case in which the objective function f is constant (this is used in AI, particularly in Automated reasoning).

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Techniques

For twice-differentiable functions, unconstrained problems can be solved by finding the points where the gradient of the objective function is zero (that is, the stationary points) and using the Hessian matrix to classify the type of each 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.

One can find the stationary points by starting with a guess for a stationary point, and then iterate towards it by using methods such as

Should the objective function be convex over the region of interest, then any local minimum will also be a global minimum. There exist robust, fast numerical techniques for optimizing doubly differentiable convex functions.

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

Here are a few other popular methods:

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Uses

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.

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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.)

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