Users:Structural Optimization/Optimization Algorithms/Conjugate Gradient
Contents |
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 βk such that
sk = -∇ fk + β * sk-1.
Thus, the actual search search direction sk incorporates a specified amount of the previous search direction sk-1. The parameter βk can be computed by the current and the previous objective gradients
βk = (|∇ fk|2) / (|∇ fk-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 βk and the storage of the old search direction sk-1 and the old gradient ∇ fk-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 FILTFUNC=OPT-LINKFUNC 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
- ↑ G. Vanderplaats, Numerical Oüptimization Techniques for Engineering Design: With Applications, McGraw Hill, 1984
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