Infinite-dimensional optimization

In certain optimization problems the unknown optimal solution might not be 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 certain degrees of freedom.

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.