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Gradient Methods in Structural Optimization
Jegan Mohan C.Ferienakadamie – Sarntal18th September 2006
Gradient Methods - Contents
IntroductionTypes of AlgorithmsGradient Methods
Steepest Descent MethodConjugate Gradient MethodMethod of feasible Directions…..
Introduction
Optimization problem is basically a minimization problem.
Introduction
Structural optimization – parameters to be optimized can be weight, stress distribution, loads etc.The constrain parameters can be displacements, stress etc.One step solution generally not possibleIterative methods are best suited
Types of Algorithms
One of the ways of classifying optimizationalgorithms are based on the kind of datathey use.
Zero order methods – function values ( direct search)First order methods – gradient ( Steepest descent, conjugate gradient)Second order methods (Newton type)
Descent Methods
Start point inside feasible domainChoose a descent search directionFollow this direction until function value is sufficiently decreased, however keep inside feasible domainChoose new search direction
Quadratic form
Descent Method
Steepest Descent
Gradient of the function is the direction of steepest ascentNegative of the function gradient would be the direction of the steepest descentLength of search path determined by line search algorithmSubsequent directions orthogonal to each other
Steepest Descent
Steepest Descentf(x) is a quadratic functionSearch direction sk is negative of gradient
Steepest Descent
Steepest Descent
Line search is performed to determine the minimum of function along a lineNon quadratic functions require numerical techniques like Equal interval search, Polynomial curve fitting etc., for line searchAnalytical methods for Quadratic functions
Steepest DescentFor badly conditioned problem, the steepest descent exhibit “zigzag” behaviorCondition number of a matrix ( w.r.t L2 Norm) is the ratio of biggest and smallest eigen-values.Bad condition = large condition numberEigen values are the axes of the hyper surfaceBadly conditioned systems have narrow function valleySolution oscillates and takes more iterations to reach solution
Steepest Descent
Steepest Descent
To avoid zigzaging, one may consider artificial widening of function by variable transformation
Optimization is done for the transformed variable and the results are then transformed back to the orginalvariablesHessian of the Matrix has to be computed – time consuming
Conjugate Gradient
Takes into account the curvature of the problemGenerates improved search directions and converges faster than Steepest DescentSearch directions are conjugate to each otherQuadratic problems – CG converges in the maximum of n steps ( n – no. of variables)
Conjugate Gradient
W.r.t steepest descent, the search directions are modified by
Criterion for β is such that all search directions are conjugate w.r.t. Q
Conjugate Gradient
Method of feasible directions
Extension of steepest descent or conjugate gradientWidely used in structural optimizationQuite robustMethod starts inside feasible domainAs long as inside feasible domain, no difference between this method and CG/Steepest descentIf in one of the iterations, the boundary is hit, the next search direction should bring back to inside domain
Method of feasible directions
Search direction must satisfy the following criteria
1. Feasible ( i.e. keeping inside feasible domain)
2. Usable ( i.e. must reduce objective )
Method of feasible directions
Other gradient methods
Generalized reduced gradient method (GRG)Modified feasible direction method
The End
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