Users:Structural Optimization/Optimization Algorithms/Steepest Descent

Contents 1 Motivation 2 Input Parameters 2.1 Input Example 3 References

Motivation

The most simple unconstrained optimization algorithm is the method of Steepest Descent (SD). It computes the search direction at step k by

s^k = -∇ f^k.

Thus, it is always ensured that the objective can be improved. The basic drawback of the SD method is bad convergence if the magnitudes of design variables differ seriously. This ends up in the well known zigzagging behavior and slow convergence rate. There exist two efficient approaches to improve the convergence behavior.

• Variable scaling by the Variable Metric Method
• Use of conjugated search directions by the Conjugate Gradient Method.

The method of steepest descent can be extended to the Constraint Steepest Descent (CSD) method which is applicable to constrained optimization problems ^[1].

Input Parameters

Block headline
Parameter	Values, Default(*)	Description
OPT-CTR	int : STEEP_DESC_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 : STEEP_DESC_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. ↑ J. Arora, Introduction to Optimum Design, Elsevier, second edition, 2004

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