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 variables in this problem are the curves connecting the two points. The optimal solution is of course the line segment joining the points.

• Given two cities in a land with lots of hills an 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, find the shape of the side wall of the cup so that it has minimal area.

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

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.