Infinite-dimensional optimization

In certain optimization problems the unknown optimal solution can be not a number or a vector, but rather a continuous quantity, for example a function 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 degrees of freedom.

Examples

• Find the shortest path between two points in a plane. The answer to this is of course the line segment joining the points.

• Given two points in a landscape with lots of hills an valleys, find a road joining them which has the shortest length. This problem is a generalization of the above, and the answer to it is clearly non-trivial.