Optimization (mathematics)

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.

Overview

An optimization problem can be represented in the following way

Given: a function f : A ${\displaystyle \to }$ 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 in linear programming - see History below). Many real-world and theoretical problems may be modeled in this general framework. Problems formulated using this technique in the fields of physics and computer vision may refer to the technique as energy minimization, speaking of the value of the function f as representing the energy of the system being modeled.

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

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 all x such that

${\displaystyle \|\mathbf {x} -\mathbf {x} ^{*}\|\leq \delta ;\,}$

the expression

${\displaystyle f(\mathbf {x} ^{*})\leq f(\mathbf {x} )}$

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.

Notation

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

${\displaystyle \min _{x\in \mathbb {R} }\;x^{2}+1.\,}$

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.

${\displaystyle \max _{x\in \mathbb {R} }\;2x.}$

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".

${\displaystyle \operatorname {argmin} _{x\in (-\infty ,-1]}\;x^{2}+1.\,}$

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

${\displaystyle \operatorname {argmax} _{x\in [-5,5],\;y\in \mathbb {R} }\;x\cdot \cos(y).\,}$

This asks for the (xy) pair (or pairs) that maximizes (or maximize) the value of the objective function x·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.

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. Such a set is called a polyhedron or a polytope if it is bounded.
• 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.
• Convex programming studies the case when the objective function is convex and the constraints, if any, form a convex set. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
• Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinite matrices. It is generalization of linear and convex quadratic programming.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

• Calculus of variations seeks to optimize an objective defined over many points in time, by considering how the objective function changes if there is a small change in the choice path.
• Optimal control theory is a generalization of the calculus of variations.
• Dynamic programming studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that relates these subproblems is called the Bellman equation.

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.

However, existence of derivatives is not always assumed and many methods were devised for specific situations. The basic classes of methods, based on smoothness of the objective function, are:

Actual methods falling somewhere among the categories above include:

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 twice 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:

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.

Mainstream economics also relies heavily on mathematical programming. An often studied problem in microeconomics, the utility maximization problem, and its dual problem the Expenditure minimization problem, are economic optimization problems. Consumers and firms are assumed to maximize their utility/profit. Also, agents are most frequently assumed to be risk-averse thereby wishing to minimize whatever risk they might be exposed to. Asset prices are also explained using optimization though the underlying theory is more complicated than simple utility or profit optimization. Trade theory also uses optimization to explain trade patterns between nations.

Another field that uses optimization techniques extensively is operations research.

History

The first optimization technique, which is known as steepest descent, goes back to Gauss. 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 problems). 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.)

Other important mathematicians in the optimization field include: