Documentation/Optimization - Versionsgeschichte

Admin am 19. März 2014 um 22:09 Uhr

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Admin am 12. September 2013 um 20:16 Uhr

2013-09-12T20:16:44Z

Dirk: /* The Objective Function */

2013-02-15T20:55:51Z

‎The Objective Function

Dirk: Created page with "=Optimization= == Introduction == Optimization searches for the best solution '''x'''* to a problem and can be defined as '''x'''* = argmin (''f''('''x''')) where ''f'' ..."

2013-02-15T20:55:25Z

Created page with "=Optimization= == Introduction == Optimization searches for the best solution '''x'''* to a problem and can be defined as '''x'''* = argmin (''f''('''x''')) where ''f'' ..."

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=Optimization=

== Introduction ==

Optimization searches for the best solution '''x'''* to a problem and can be defined as

'''x'''* = argmin (''f''('''x'''))

where ''f'' is the function to be optimized (the objective function) and '''x''' are the design variables for which optimal settings have to be found.

This definition can be extended to multiple objectives '''f''' and by constraints.

== Some Important Expressions ==
=== Manual vs. Automated Optimization ===

Optimization is an everyday task. For example, searching the fastest way to work or home is an optimization problem. We speak about ''automated'' optimization, if a computer algorithm solves the problem in an automated fashion, i.e. without user interaction.

=== The Objective Function ===
Optimization typically searches for the optimal solution to a problem. For automated optimization, the problem must be encoded in a mathematical function ''f'', which has to be either minimized or maximized. In OpenOpal, only minimization is considered as maximization can be expressed as:

max(f) = - min(-f)

=== Design Variables ===
The optimal solution is searched by modifying the design variables '''x'''. The objective function depends on the design variables: f = f('''x''').

=== Multiple Objectives ===

If multiple objectives '''f''' should be optimized, then a multi-objective optimization problem has to be solved. Some optimization algorithms search concurrently for multiple compromise solutions for the objectives (Pareto optimization) or a single compromise solution, defined as a weighed sum of all objectives.

=== Constraints ===

While the optimization algorithm tries to minimize all objectives, constraints simply have to be fulfilled.

For example, the maximal stress in a truss should not exceed a certain limit. If the stress is below the limit, no advantage is gained. If the stress is above the limit, the solution is typically constrained by a penalty value, which increases with increasing constraint violation.

=== Direct vs. Indirect Search ===

In OpenOpal, we implement optimization algorithms that search a problem in an iterative fashion, i.e. by computing several different solutions to the problem. The best solution is returned. This iterative search can be either direct or indirect:

'''Direct''' search uses only direct information (i.e. the objective and constraint value(s).
'''Indirect''' algorithms use indirect information (i.e. gradient and/or higher order derivative information of the objective(s) and constraint(s)).

=== Stochastic vs. Deterministic Algorithms ===

While '''stochastic''' algorithms such as Evolutionary Algorithms and Particle Swarm use random values in their search method, '''deterministic''' algorithms like the Simplex Method or gradient based search do not.

@@ Zeile 18: / Zeile 18: @@
 === Objective Function ===
-Optimization typically searches for the optimal solution to a problem. For automated optimization, the problem must be encoded in a mathematical function ''f'', which has to be either minimized or maximized. In OpenDINO, only minimization is considered as maximization can be expressed as:
+Optimization typically searches for the optimal solution to a problem. For automated optimization, the problem must be encoded in a mathematical function ''f'', which has to be either minimized or maximized. In OpenDino, only minimization is considered as maximization can be expressed as:
      max(f) = - min(-f)
@@ Zeile 37: / Zeile 37: @@
 === Direct vs. Indirect Search ===
-In OpenDINO, we implement optimization algorithms that search a problem in an iterative fashion, i.e. by computing several different solutions to the problem. The best solution is returned. This iterative search can be either direct or indirect:
+In OpenDino, we implement optimization algorithms that search a problem in an iterative fashion, i.e. by computing several different solutions to the problem. The best solution is returned. This iterative search can be either direct or indirect:
 '''Direct''' search uses only direct information (i.e. the objective and constraint value(s).

@@ Zeile 17: / Zeile 17: @@
 Optimization is an everyday task. For example, searching the fastest way to work or home is an optimization problem. We speak about ''automated'' optimization, if a computer algorithm solves the problem in an automated fashion, i.e. without user interaction.
-=== The Objective Function ===
+=== Objective Function ===
 Optimization typically searches for the optimal solution to a problem. For automated optimization, the problem must be encoded in a mathematical function ''f'', which has to be either minimized or maximized. In OpenOpal, only minimization is considered as maximization can be expressed as: