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 the points which has the shortest length. This problem is a generalization of the above, and the answer to it is clearly non-trivial. One could guess that this is an infinite dimensional problem by noticing that the set of roads joining these two points is extraordinarly huge.

• Find the shape of a bridge capable of sustaining given amount of traffic. This is a finite-dimensional problem if we decide in advance what type of bridge we want. For example, if a suspension bridge is desired, one thing one needs to decide is how many cables to use for suspension, then how strong to make each. Another thing to determine is how thick the bridge should be. There are other things to determine, but you got the idea, ultimately, there is a finite list of parameters, and once the value of those parameters is known, the bridge is completely determined. The bridge problem is infinite dimensional however, 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 sofisticated methods from partial differential equations to solve such problems.