In certain optimization

Optimization (mathematics)

In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

 problems the unknown optimal solution might not be a number or a vector, but rather a continuous quantity, for example a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 or the shape of a body. Such a problem is an infinite-dimensional optimization problem, because, a continuous quantity cannot be determined by a finite number of certain degrees of freedom

Degrees of freedom (physics and chemistry)

A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

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Examples

• Find the shortest path between two points in a plane. The variables in this problem are the curves connecting the two points. The optimal solution is of course the line segment joining the points, if the metric defined on the plane is the Euclidean metric.

• Given two cities in a country with lots of hills and valleys, find the shortest road going from one city to the other. This problem is a generalization of the above, and the solution is not as obvious.

• Given two circles which will serve as top and bottom for a cup of given height, find the shape of the side wall of the cup so that the side wall has minimal area. The intuition would suggest that the cup must have conical or cylindrical shape, which is false. The actual minimum surface is the catenoid
  Catenoid
  A catenoid is a three-dimensional surface made by rotating a catenary curve about its directrix. Not counting the plane, it is the first minimal surface to be discovered. It was found and proved to be minimal by Leonhard Euler in 1744. Early work on the subject was published also by Jean Baptiste...
  
  .

• Find the shape of a bridge capable of sustaining given amount of traffic using the smallest amount of material.

• Find the shape of an airplane which bounces away most of the radio waves from an enemy radar.

Infinite-dimensional optimization problems can be more challenging than finite-dimensional ones. Typically one needs to employ methods from partial differential equation

Partial differential equation

In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s to solve such problems.

Several disciplines which study infinite-dimensional optimization problems are calculus of variations

Calculus of variations

Calculus of variations is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite integrals involving unknown...

, optimal control

Optimal control

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. The method is largely due to the work of Lev Pontryagin and his collaborators in the Soviet Union and Richard Bellman in the United States.-General method:Optimal...

 and shape optimization

Shape optimization

Shape optimization is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints...

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