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 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.
- 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 equations to solve such problems.
Several disciplines which study infinite-dimensional optimization problems are calculus of variations, optimal control and shape optimization.
External links and references
- David Luenberger (1997). Optimization by Vector Space Methods. John Wiley & Sons. ISBN 047118117X.