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Positive de¬nite matrix, 4
Singular values, 26
Positive semide¬nite matrix, 4
Singular vectors, 26
Preconditioned CG
Small residual problem, 22
algorithm, 8
spd, 4
Preconditioning
Spendley“Hext“Himsworth algorithm, 159
conjugate gradient, 8
SR1 update, 81
Predicted reduction
Stagnation, 19
line search, 40
Nelder“Mead
trust region, 51
example, 139
Projected BFGS method, 104
Standard assumptions, 14
Projected Newton method, 100
State equation, 10
algorithm, 100
State variable, 10
Projection
Stationary point, 5, 43, 100
onto „¦, 89
bound constraints, 88
PSB update, 81
nondegenerate, 90
Steepest descent
Quadratic model, 6, 40
direction, 39
Quadratic objective functions, 6
method, 39
Quasi-Newton methods, 71
Stencil failure, 117, 123
Steplength, 39
Reduced Hessian
Stiff initial value problems, 11
de¬nition, 89
Strict complementarity, 90
Response surfaces, 148
Suf¬cient conditions
for optimality, 6
Safeguarding, 44
Suf¬cient decrease, 39, 41
Scaled gradient projection, 99
gradient projection, 91
Scaling, 46
projected Newton, 97
Scaling matrix, 99
Secant equation, 71 simplex form, 138


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180 INDEX

Surrogate models, 148

Taylor™s theorem, 5
Termination
bound constrained problems, 91
small function differences, 21
small gradients, 21
small steps, 21
unconstrained problems, 21
Tosca, Floria, xiv
TR-CG
algorithm, 64
Truncated Newton methods, 28
Trust region, 50
actual reduction, 51
algorithm
adjustment, 51
paradigm, 52
dogleg, 58
classical, 58
convergence theorem, 52
Levenberg“Marquardt
algorithm, 58
parameter, 57
methods, 50
Newton point, 58
predicted reduction, 51
problem, 50
radius, 50
trial solution, 50
trial step, 50
unidirectional, 54
Trust region CG
algorithm, 64

Unconstrained optimization problem, 3

Weber™s problem, 118, 127, 152
Wolfe conditions, 49
strong, 49

Youngman
Henny, 159

Zero residual problem, 22




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