Infinite-dimensional optimization

In certain optimization problems the unknown optimal solution might 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 the points which has the shortest length. This problem is a generalization of the above, and the answer to it is clearly non-trivial.

• Find the shape of a bridge capable of sustaining given amount of traffic. This an infinite dimensional optimization problem if nothing is known in advance of the optimal bridge, so all possible configurations must be searched.

Infinite dimensional optimization problems are much more challenging than finite dimensional ones. Typically one needs to employ methods from partial differential equations to solve such problems.

Several disciplines which study infinite dimensional optimization problems are calculus of variations, optimal control and shape optimization.