Users:Structural Optimization/Optimization Algorithms/Conjugate Gradient

Contents 1 Motivation 2 Input Parameters 2.1 Input Example 3 References

Motivation

Application of the Conjugate Gradient (CG) algorithm^[1] is a method to improve the convergence rate of the steepest descent approach. The search direction of the CG method incorporates the curvature of the objective by the parameter β^k such that

s^k = -∇ f^k + β * s^k-1.

Thus, the actual search search direction s^k incorporates a specified amount of the previous search direction s^k-1. The parameter β^k can be computed by the current and the previous objective gradients

β^k = (|∇ f^k|²) / (|∇ f^k-1|²).

In the most optimization problems the CG method shows a faster convergence than the steepest descent method. In general the faster convergence preponderates the increased numerical effort of computation of β^k and the storage of the old search direction s^k-1 and the old gradient ∇ f^k-1.

Input Parameters

Block headline
Parameter	Values, Default(*)	Description
OPT-CTR	int : CG_NAND	ID and identifier of optimization algorithm.
Common Compulsory Parameters, valid for all optimization algorithms
FILTER	OPT-FILTER int, int, ....	One or more filter function IDs.
OBJ	OPT-RESPONSE_FCT int, int, ...	One or more response functions that are considered as objective.
OUTPUT	PC-OUT int	The Output object.
DOMAIN	EL-DOMAIN int	The respective domain on which the optimization problem is defined.
REGULARIZATION	EL-REGULARIZATION int	The regularization object.
LINE_SEARCH	OPT-LINE_SEARCH int	The line search object.
CONVERGENCE_CONTROL	OPT-CONVERGENCE int	The convergence checker.
Common Optional Parameters, valid for all optimization algorithms
DESIGN_SPACE_BOUNDS	ND-SET int, int, ...	One or more node set IDs that define the boundary of the design space.
RESTART_DATA_FREQ	int	Frequency of restart output,

Input Example

Example of a complete input block:

OPT-CTR 1 : CG_NAND
  ! compulsory parameter
  FILTER=OPT-FILTER 1
  OBJ=OPT-RESPONSE_FCT 1
  OUTPUT=PC-OUT 1
  DOMAIN = EL-DOMAIN 1
  REGULARIZATION = EL-REGULARIZATION 1
  LINE_SEARCH = OPT-LINE_SEARCH 1
  CONVERGENCE_CONTROL = OPT-CONVERGENCE 1
  ! optional parameter
  DESIGN_SPACE_BOUNDS = ND-SET 6

References

1. ↑ G. Vanderplaats, Numerical Oüptimization Techniques for Engineering Design: With Applications, McGraw Hill, 1984

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