Infinite-dimensional optimization
- 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 certain 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, if the metric defined on the plane is the Euclidean metric.
* 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. The intuition would suggest that the cup must have conical or cylindrical shape, which is false. The actual minimum surface is the catenoid.
* 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.
References
* David Luenberger (1997). "Optimization by Vector Space Methods." John Wiley & Sons. ISBN 0-471-18117-X.
Wikimedia Foundation.
2010.
Look at other dictionaries:
Optimization (mathematics) — In mathematics, the term optimization, or mathematical programming, refers to the study of problems in which one seeks to minimize or maximize a real function by systematically choosing the values of real or integer variables from within an… … Wikipedia
Mathematical optimization — For other uses, see Optimization (disambiguation). The maximum of a paraboloid (red dot) In mathematics, computational science, or management science, mathematical optimization (alternatively, optimization or mathematical programming) refers to… … Wikipedia
Shape optimization — is part of the field of optimal control theory. The typical problem is to find the shape which is optimal in that it minimizes a certain cost functional while satisfying given constraints. In many cases, the functional being solved depends on the … Wikipedia
Trajectory optimization — is the process of designing a trajectory that minimizes or maximizes some measure of performance. While not exactly the same, the goal of solving a trajectory optimization problem is essentially the same as solving an optimal control problem. The … Wikipedia
List of numerical analysis topics — This is a list of numerical analysis topics, by Wikipedia page. Contents 1 General 2 Error 3 Elementary and special functions 4 Numerical linear algebra … Wikipedia
Genetic algorithm — A genetic algorithm (GA) is a search heuristic that mimics the process of natural evolution. This heuristic is routinely used to generate useful solutions to optimization and search problems. Genetic algorithms belong to the larger class of… … Wikipedia
CMA-ES — stands for Covariance Matrix Adaptation Evolution Strategy. Evolution strategies (ES) are stochastic, derivative free methods for numerical optimization of non linear or non convex continuous optimization problems. They belong to the class of… … Wikipedia
Calculus of variations — is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite … Wikipedia
List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… … Wikipedia
Metaheuristic — In computer science, metaheuristic designates a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. Metaheuristics make few or no assumptions about the… … Wikipedia
