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 |  موضوع: كتاب Introduction to Optimum Design 4th Edition الخميس 19 أبريل 2018, 11:51 pm | |
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Introduction to Optimum Design 4th Edition
Jasbir singh arora
The University of Iowa,
College of Engineering,
Iowa City, Iowa
ويتناول الموضوعات الأتية :
Subject Index
A
Absolute guarantee, 710
Absolute minimum, 108
Absolute-value constraint, 37
Acceptance criterion, 722
Acceptance–rejection (A–R) methods, 710, 721
ACO. See Ant colony optimization (ACO)
Active constraint, 515
Active inequality, 152
Active/tight constraint, 155
Adaptive numerical optimization procedure, 58
Additive model, 814
one-way table, 814
orthogonal array, 813, 815
Advanced frst order second-moment method
(AFOSM), 841
Algorithm, failure of, 516
Allowable strength design (ASD) approach,
640, 647
Allowable stress, 35
American Association of State Highway and
Transportation Offcials (AASHTO), 268
American Institute of Steel Construction
(AISC), 639, 644, 652, 656
manual, 700
Analysis of means (ANOM), 813
additive model for function, 816
Analyze designs, 6
Annealing process, 693
Answer Report from Solver, for linear programming
problem, 265
Ant colony optimization (ACO), 755
algorithm, 759–760
fnding feasible solutions, 762–763
pheromone deposit, 763
pheromone evaporation, 763
problem defnition, 760–761
algorithm for traveling salesman problem, 757
ant behavior, 755
simple model/algorithm, 756
path fnding capability, 756
probabilities, 759, 763
virtual ant changes, defnition of, 758
Ant Colony Optimization and Swarm Intelligence
(ANTS), 740
Antenna, geometrical view, 671
Approximate Pareto set, 783
Approximation, linear, 728
Armijo-like procedure, 460, 576
Artifcial cost function, 345, 394
Artifcial variables, 343
equality constraints, 351
two-phase simplex method, 343
artifcial cost function, 345
degenerate basic feasible solution, 355
phase I algorithm, 346
infeasible problem, 346
phase II algorithm, 348
phase I problem, defnition of, 345
unbounded solution, 353
Aspiration point, 781
Associative law, 855
Asymmetric matrix, 857
Asymptotic error constant, 495
Attainability, 776
Attainable set, 776
Augmented Lagrangian methods, 220, 490, 492
Augmented matrix, 859
B
Banded matrix, 857
Bar cross-sectional shapes, 36
Barrier function methods, 490
advantages and disadvantages of, 492
Basic calculus concepts, review of, 113
Basic feasible solutions, 321, 336, 390, 412
Basic infeasible solutions, 330
Basis functions, 60
Bayesian approach, 717
BBM. See Branch-and-bound methods (BBM)
Beam design problem, graphical solution, 90
using MATLAB, 91
Bending stress, partial derivatives of, 612
BFGS method. See Broyden-Fletcher-Goldfarb-Shanno
(BFGS) method
Binary variable, 683
Binomial crossover, 752
Block shear, 644
Bolt insertion sequence, 748
Boltzmann–Gibbs distribution, 694930 Subject Index
Bound-constrained optimization, 572
algorithm, 597
optimality conditions, 573
projection methods, 574
step-by-step algorithm, 575–576
step size calculation, 576
variable close to upper/lower bound, 577
Bounded objective function method, 788–789
Branch-and-bound methods (BBM), 687, 711
linear problems, 687
local minimizations, 690
nonlinear continuous problems, 691
solution of continuous subproblems, 691
Branching, 689
Broyden-Fletcher-Goldfarb-Shanno (BFGS) method,
482, 574
quasi-Newton methods, 283
limited-memory, 576
updating procedure, 597
B-splines, 630
Buckling constraints, 53
C
Cabinet design, 40–43
data and information collection, 40
formulation of constraints, 41
mathematical formulation, 41
optimization criterion, 41
project/problem description, 40
Can, design of, 28
project/problem description, 28
Candidate optimum points, 127
Candidate points, 784
Canonical representation, 872
Cantilever beam, 20
design problem, 21
design variables, 23
formulation for optimum design, 28
of hollow square cross-section, 21
notation and data for, 22
Cantilever column, 43
Cantilever structures, subjected to shock
input, 626
Center of gravity (CG), 98
Chain rule of differentiation, 468
Cholesky factors, 582
Chromosome, 750, 751
Classical Newton method, 472
Clustering methods, 718
Coeffcient matrix, 859, 871
Cofactor expansion for |A|, 861
Coil springs, 47, 48
design of, 47–49
Column matrix, 853
Compression members, optimum design of
constraints, formulation of, 648
data/information collection, 645
design variables, defnition of, 648
formulation of problem, 644
optimization criterion, 648
problem formulation
for elastic buckling, 651
discussion, 652
for inelastic buckling, 649
project/problem description, 645
Compromise-programming methods, 787
Compromise solution, 779
Computational algorithm, 548
Computational approximations, 796
Conjugate gradient directions, 445
Conjugate gradient method, 448
Constrained optimizations
algorithms, conceptual steps, 513, 514
second-order conditions, 212
general constrained problems, 213
insights for, 214
strong suffcient condition, 215
suffcient conditions, for general constrained
problems, 214
Constrained optimum design, 555
numerical methods for, 511
algorithms for constrained problems
implementation of iterations, 513–514
iterative process, 512
basic concepts related, 512
algorithm, convergence of, 516
algorithms for constrained problems, 512
constraint status, at design point, 515
descent function, 516
constrained steepest–descent (CSD) method, 512
potential constraint strategy, 556–558
Constrained problem, linearization of, 517
linearized subproblem
defnition of, 518–519
notation, 517
Constrained problems, penalty function for, 694
Constrained steepest descent (CSD)
algorithm, 542, 548, 558, 560
geometrical interpretation, 562
inexact step size, 565
with inexact step size, 565
observations, 548–549
step size determination, 568
direction, 531, 547, 591
method, 512, 547
Constraint boundary, for inequality, 73
Constraint correction (CC) algorithm, 701
Constraint function cells, 274Subject Index 931
Constraint normalization, 242, 244
equality constraint, 244
inequality constraint, 245
Constraints, 11
inequality, 11
for LP problem, 352
sensitivity theorem, 196
set, 107
variation sensitivity theorem, 172
Continuous functions, 14
Continuous-variable optimization problem, 59
ContourPlot command, 79, 80
Contraction operation, 502
Control force constraint, 629
Controlled random search (CRS), 720, 731
basic idea of, 720
global phase, 720
local phase, 721
Conventional design method, 7
vs. optimum design process, 6–7
Convex functions, 184
characterization of, 181
Convex interval, 179
Convexity, 180
Convex polyhedral set, 313
Convex polyhedron, 322
Convex programming problem, 190
Convex sets, 184
Coordinate system set-up, 73
Cost coeffcients, 370
Cost function, 37
contours, 89
value, 517
Cost space. See Criterion space
CPU time, 760
Cramer’s rule, 860
Criterion space, 773
two-objective optimization problem, graphical
representation of, 775
Critically important, 247
Crossover operation with one-cut point, 745
Cross-section, of plate girder, 270
CRS. See Controlled random search (CRS)
CSD. See Constrained steepest descent (CSD)
Cumulative distribution function (CDF), 834
Curvature condition, 462
D
Data optimization, 646
Davidon–Fletcher–Powell (DFP) method, 479
DEA. See Differential evolution algorithm (DEA)
Defnition of minima, graphical interpretation, 109
De?ection constraint, 49
Dependent variable cells, 274
Derivative-based optimization method, 609
Derivative-based search methods, 239, 424
Descent function, 541
calculation of, 545
golden section search, 543
second trial point, 546
value, 545
Descent method, 427
Design activities, 6
Design change vector, 581
Designing engineering systems, 4
Design of experiments, for response surface
generation, 805
Design optimization
formulated as problem, 3
iterative process, 4
overview of, 3
Design problem
with multiple solutions, 85–86
with unbounded solutions, 86–87
Design space, 773
Design under uncertainty, 833
Design variable bounds
constraints, 623
for global optimization problem, 729
Design variables, 22, 42, 248
vector, 641
Determinant of matrix, 860
Determinants, 859–862
leading principal minor, 862
properties of, 861–862
singular matrix, 862
Deterministic methods, 709, 710
DFP quasi-Newton method, 497
Diameter constraint, 49
Differential elastic line equation, 668
Differential equation (DE), 626
Differential evolution algorithm (DEA), 750, 752–753
application, 753
crossover operation to generate the trial design, 752
donor design, generation of, 751
initial population, generation of, 750
main steps, 753
notation and terminology, 751
trial design, acceptance/rejection of, 752
Dirac delta function, 630
Direction-fnding subproblem, 589
Directions of descent, 427
Direct search methods, 239, 498, 739
Discontinuous functions, 14
Discrete design
with orthogonal arrays, 813–816
variables, 684
Discrete/integer-variable optimization problem, 59932 Subject Index
Discrete variable design, 58, 683, 813
Discrete variable optimum design concepts/methods,
683, 701
adaptive numerical method, 699–701
continuous variable optimization, 701
basic concepts/defnitions, 684
mixed variable optimum design problem
(MV-OPT)
classifcation of, 685
defnition of, 684
branch-and-bound methods, 687
basic, 687–688
general MV-OPT, 691–692
with local minimization, 689–690
dynamic rounding-off method, 696
algorithm, 696
integer programming, 692–693
linked discrete variables, 698, 699
neighborhood search method, 697
selection of, 699
sequential linearization methods, 693
simulated annealing, 693–695
algorithm, 694
Displacement constraint, 629, 634
Displacement response
optimum with minimization
of control effort, 636
of error, 632
time as performance index, 637
Distributive law, 855
Domain elimination (DE) method, 731
?ow diagram, 725
Domination pressure, 784
Double-subscript notation, for
variables, 798
D–string, 742, 744
Duality, in linear programming, 399
dual LP problem, 399
dual tableau to recover
primal solution, 407
dual variables as Lagrange multipliers, 410
proof, 410
equality constraints
alternate treatment, 402
treatment of, 401
primal solution, determination of, 403
standard primal LP problem, 399
Duality, in nonlinear programming, 220
equality/inequality-constrained problem, 226
gradient matrix of equality constraints, 221
gradient of, 223
Hessian of Lagrangian function, 221
Lagrangian function, 221
local duality
equality constraints case, 220
inequality constraints case, 226
theorem, 224
lower bound for primal cost function, 228
problem solving, 222
saddle points, 228
theorem, 228
strong duality theorem, 227
weak duality theorem, 227
Dual problem, 220
Dual tableau, 408
Dual variables, 359
Dummy variables, 851, 852
Dynamic displacement constraint, 631
Dynamic rounding-off algorithm, 696
E
e-Constraint, 789
?-Constraint approach, 789
Eigenvalues, 882, 883
Eigenvectors, 882, 883
Elastic buckling, 647, 651
hot-rolled I–shapes, 659
W–shapes, 659
Elastic line equation, 670
Elements of the matrix, 852
Elimination process, 859
Elite points, 783
Energy expenditure, 24
Engineering analysis, 6
Engineering design
with analysis, 6
examples, 189
rectangular beam, 193
convexity, 193
KKT necessary conditions, 194–197
sensitivity analysis, 197
vs. engineering analysis, 6
wall bracket, 189
convexity, 191
convex programming problem, 190
KKT necessary conditions, 191
problem formulation, 191
sensitivity analysis, 193
Equal interval search, 438, 891
computer program, 892
Equality constraint, 58
function value, 517
gradient conditions, 150
Lagrange multiplier, 144–145
theorem, 149
problem, 143, 844
Equality/inequality-constrained problem,
226, 493Subject Index 933
Equivalence class sharing, 785
Equivalent single-degree-of-freedom system
displacement response of, 628
Errors, meta-model, 796
Euclidean space, 884
Euler stress, 647
Evtushenko’s method, 711
Exact penalty function, 539
Excel, 152, 307
Excel Solver, 237, 253, 260
for linear programming problems, 260
Answer Report from Solver, 265
Sensitivity Report from Solver, 266
Solver Parameters dialog box, 263
Solver Results dialog box, 264
worksheet, 262
nonlinear equation, roots of, 253
Solver Answer Report, 257
Solver Output, 256
Solver Parameters dialog box, 254, 255
Solver Results dialog box, 256
nonlinear equations, roots of set, 257
Solution to KKT Cases with Solver, 260
Solver Parameters dialog box, 259
worksheet, 257, 258
nonlinear programming
optimum design of springs, 266–268
of plate girders, 268
data and information collection, 270–271
design variables, defnition of, 271
formulation of constraints, 272
optimization criterion, 271
project/problem description, 268–269
solution, 274–276
Solver Parameters Dialog Box, 273
spreadsheet layout, 272–273
for unconstrained optimization problems, 260
worksheet and Solver Parameters dialog
box, 261
Excel worksheet, 254, 261
for linear programming problem, 262
for spring design problem, 267
Exhaustive search, 178
Experimental errors, 796
Exterior penalty methods, 491
F
Feasibility tolerance, 242
Feasible criterion space, 774
Feasible design space, 107, 772
Feasible directions (FD) method, 531
Feedback loops, 5
Feedback mechanism, 755
Fibonacci sequence, 437
Finite–element application, 609
Finite exact methods, 710
Finite number, 709
First-order necessary condition, 129
First-order necessary conditions, 131–134
First-order reliability method
(FORM), 845
Fitness sharing, 784
Flagpole, 100
Fletcher–Reeves formula, 449
Flexural members, 653
Flywheel-shaft system, 295
Free-body diagram, 34
Function gradients, 520
Function of variable, 13
Fundamental natural frequency, 53
G
Gaussian (normal) distribution, 821
Gaussian elimination method, 314, 858, 862, 865
elimination process, 862
inverse, 868
procedure, 862–865, 867
Gauss-Jordan elimination method, 317, 326, 330, 392,
395, 397, 866, 869, 872
general solution of linear system, 876
inverse, 868
in tabular form, 875
General design optimization model, 144
Generalized descent methods, 712, 713
Generalized reduced gradient (GRG) method,
531, 593
nonlinear, 260
General-purpose software, use of, 615
integration of application, 616
software selection, 616
Genetic algorithms (GAs), 778
Genetic and Evolutionary Computation Conference
(GECCO), 740
Geometrical representation, 10
Global convergence, 428
properties, 494
Global/local minima, defnitions of, 106
minimum, existence of, 112
minimum/maximum, 107, 108
Global minimum, 184, 708
point. See Absolute minimum
Global optimality, 178
constraint, transformation of, 187
convex functions, 181
convex programming problems, 183
suffcient conditions, 188
convex sets, 179
Hessian condition, 182934 Subject Index
Global optimization concepts/methods, 707
basic solution concepts, 708
global minimum
characterization of, 708
searching for, 709
controlled random search
basic idea of, 720
deterministic methods, overview of, 710
covering methods, 711–712
generalized descent, 712–714
descents/ascents, alternation of, 713
Golf methods, 714
tunneling method, 714–715
zooming method, 712
deterministic/stochastic methods, 709
local-global stochastic methods, 723
numerical performance methods, 729
features of methods, 730
stochastic zooming and domain elimination
methods, 731–732
structural design problems, 732–733
unconstrained problems, performance, 731
Global optimization methods, 708
characteristics of, 730
Global optimum design, 188
Global phase, 720
Goal programming, 789–790
Golden section search, 439, 441, 894
subroutine GOLD, 895
Golden section search procedure, 437
Goldstein test, 462, 463
Golf methods, 714
Gradient-based search methods, 238, 424, 425. See also
Derivative-based search methods
Gradient conditions, 150
Gradient evaluation, 534, 543
implicit functions, 609–610
of implicit functions, 609–610
Gradient method, 442
Gradient projection (GP) method, 531, 591
Gradient vector, 114, 115, 464
partial derivatives of function, 114
Graph-editing capability, 82
Graphical optimization
feasible region, identifcation of, 80
inequality, infeasible region
identifcation and shading of, 79–80
MATLAB uses, 81
editing of graph, 85
function contours, plotting of, 82
proft maximization problem, 83
objective function contours, plotting of,
80–81
optimum solution, identifcation of, 81
plotting functions, 78–79
use of mathematica, 77
Graphical representation, 153, 185
Graphical solution process, 71, 72
for beam design problem, 90
for minimum-weight tubular column, 88–89
proft maximization problem–formulation, 72–73
step-by-step procedure, 73–77
Grid points, stopping criterion, 634
H
Hessian modifcation, 478
Hessian matrix, 15, 114, 115, 139, 143, 187, 469, 472,
475, 498
of Lagrange function, 579
for quadratic form, 126
second-order partial derivatives, 115
Hessian updating procedure, limited-memory, 595
Heuristic methods, 710
Hooke–Jeeves method, 498
I
Identity matrix, 857
IDESIGN program, 701
IF THEN ELSE condition, 648, 661
Implicit constraints, 59
Implicit enumeration, 686
Implicit functions, gradient evaluation, 609–610
Improving feasible direction, 589
Inactive constraint, 515
Inactive inequality, 153
Including inequality constraints, 580
Independent variables, 316
Independent variable transformation, 631
Inelastic buckling, 647
constraint, 649
Inequality
constrained problem, 155
constraint boundary, 73
plot, 73
constraint functions
for optimum design problem, 184
constraint function value, 517
feasible/infeasible side, 74
Inexact line search, 542, 560
basic concept of, 460
Inexact step size calculation
basic concept, 560
descent condition, 560–563
Infeasible design, 26
optimization problem, 88
problem, 87
Infeasible problem, 154, 248
Infnite solutions, 316Subject Index 935
Inner array, 827
Insulated spherical tank design, 29
constraints, formulation of, 30
data and information collection, 30
optimization criterion, 30
project/problem description, 29
Integer programming problems, 33, 43
Integer variable, 58, 683, 684
Inter-disciplinary environment, 4
Interval arithmetic, 711
Interval-reducing methods, 434, 539
Inverse barrier function, 491
Inverse using cofactors, 866
Irregular optimum point, graphical solution, 211
Isocost curves, 139
Iterative process, 6
J
Jacobian matrix, 593
K
Karush-Kuhn-Tucker (KKT), 159, 207, 532, 708
alternate form, 208
cases with four inequalities, 171
conditions, for LP problem, 411
optimality conditions, 412
solution of, 412–414
frst-order necessary conditions, 157, 257
graphical solution, 164
important observations, 160
irregular points, 211
Lagrange function, 194
limitation of, 170
necessary conditions, 161, 163, 207, 208, 210,
218, 410
alternate form, 207–208
graphical solution, 211
irregular points, 210
quadratic programming (QP) problems, 415
second-order conditions, for constrained
optimization, 212
general constrained problems, 213
insights for, 214
strong suffcient condition, 215
suffcient conditions, for general constrained
problems, 214
solver results for, 259
worksheet and Solver Parameters dialog
box, 258
Kilopascal (KPa), 243
KKT. See Karush-Kuhn-Tucker (KKT)
L
Lagrange function, 228, 577, 581
Lagrange multipliers, 57, 144, 145, 148, 149, 151, 155,
170, 172, 173, 175, 268, 359, 362, 540, 578, 581, 594,
620, 623, 635, 845
constraint, scaling, 176, 177
equality constraint, 149, 844
for lower-and upper-bound constraints, 573
of primal constraints, 399
vectors, 221
for inequality constraints, 415
Lagrangian function, 573, 588
Lagrangian methods, 708
Length of vector, 12
Lexicographic method, 788
Life-cycle cooling cost, 30
LINDO, 307
Linear approximations, 522, 728
Linear combination, 12
of vectors, 876
Linear equations
m linear equations, in n unknowns, 869
rank of matrix, 869
m × n linear equations, 870–872
basic solutions, 874
set of vectors, 876
linear independence, 876
vector spaces, 880
Linear functions
constraints, 308
cost function, 308
Linear independence, 145
of vectors, 877
Linearized equality constraints, 517, 518
Linearized feasible region
graphical representation of, 521
Linearized subproblem, 520, 535, 544
Linear least squares problem, 485
Linearly independent, 877
Linear optimization problems, 851
Linear problems, BBM for, 687
Linear programming (LP). See also Linear optimization
problems
artifcial variables, two-phase simplex method, 343
basic concepts
boundary of feasible set, 314
convexity of, 314
infnite roots, 314
related to problems, 313
basic solutions, calculation of, 323
basic solutions, 326
pivot step, 324
tableau, 323
basic theorems, 335
multiple solutions, 336
graphical solution, 360936 Subject Index
optimum solution, 356
number of basic solutions, 322
to problems, 321
postoptimality analysis, 356
coeffcient matrix, changes, 372
constraint limits, changes, 358
cost function, change, 359
Lagrange multipliers, recovery of, 358
Lagrange multiplier values, 358
limits on changes in resources, 365
new values of basic variables, 366
ranging cost coeffcients, 369
basic variables, 370
nonbasic variables, 369
ranging right-side parameters, 365
problems, 26, 33, 57, 307
Answer Report from Solver, 265
Excel solver, 260
with multiple solutions, 340
Sensitivity Report from Solver, 266
Solver Parameters dialog box, 263
Solver Results dialog box, 264
solving, 307
worksheet, 262
simplex method, 329
basic idea, 330
basic steps, 330
cost function in terms of nonbasic
variables, 330
2D/3D space, 329
KKT conditions, problem, 411
standard problem, defnition of, 308
expanded form, 309
matrix form, 309
summation form, 309
terminology, 319
basic feasible solution, 319
basic solution, 319
basic variables, 319
basis, 319
convex polyhedron, 319
degenerate basic solution, 319
feasible solution, 319
feasible solution, 319
nonbasic variables, 319
optimum basic solution, 319
optimum solution, 319
vertex/extreme point, 319
transcription to standard, 310
maximization of function, 311
nonnegative constraint limits, 310
treatment of inequalities, 310
“? type” constraints, treatment of, 310
“? type” constraints, treatment of, 311
unrestricted variables, 311
Linear programming methods, for optimum design,
307, 389
alternate simplex method, 397
simplex algorithm, 396–397
simplex method, derivation, 389
artifcial cost function, 394–395
basic variable, 393
canonical form, 389–390
nonbasic variable, 391, 392
cost function, 391
optimum cost function, 392
pivot step, 395–396
reduced cost coeffcients, 392
unbounded problem, 392
Linear simultaneous equations, 457
Linear systems, 858–859
Line search function, 430
Line search problem, 429
Line search termination criterion, 431, 456
Linked discrete variable, 683
Lipschitz condition, 548
Lipschitz constant, 710, 711
Load and resistance factors design (LRFD)
approach, 640
Local duality theorem, 220, 224, 225
Local-global stochastic methods, 723
conceptual, algorithm, 723, 724
domain elimination method, 724–726
stopping criteria, 726
operations analysis of methods, 727
checking, proximity of point, 727
design variable constraints, 729
point and trajectory distance, 728
trajectory approximation, 728
stochastic zooming method, 726
Local minimum point. See Relative minimum
Local optimality conditions, 128
Log barrier function, 491
LP. See Linear programming (LP)
M
Marquardt’s algorithm, 478
Marquardt’s method, 478
Mathematica, 152
Mathematical approximations, 796
Mathematica Optimization Tool Box, 237
Mating strings, 743
MATLAB, 152, 165, 279, 307
constrained optimum design problems, 285
fle, proft maximization problem, 83
Linear programming (LP). See also Linear optimization
problems (cont.)Subject Index 937
graphical solution method and basic optimization
concepts, 279
optimization toolbox, 279–281
output from, explanation of, 281
scalar/array/matrix operations, 280
variables/expressions, 279
optimum design examples, 288
column design for minimum mass, 290
data/information collection, 291
design variables, defnition of, 292
formulation of constraints, 292
optimization criterion, 292
project/problem statement, 290
?ywheel design for minimum mass, 293
constraints, formulation of, 297
data/information collection, 296
design variables, defnition of, 297
optimization criterion, 297
project/problem statement, 293
maximum shear stress location, 288–289
constraints, formulation of, 289
criterion, 289
data and information collection, 288
design variables, defnition of, 288
program, 81, 82
unconstrained optimum design
problems, 282
MATLAB Optimization Tool Box, 237
Matrices
addition of matrices, 853
condition number, 885
defnition of, 851
elementary row–column operations, 856
equivalence of matrices, 856
inverse of matrix
Gaussian elimination, 867
Gauss–Jordan elimination, 866
inverse by cofactors, 866
multiplication of matrices, 853
norm of, 884
condition number, 884
vectors, 884
null matrix, 853
partitioning of matrices, 857–858
scalar product/dot product of vectors, 856
square matrices, 857
transpose of a matrix, 855
types of, 853
vector, 853
Matrix of order, 852
Maximin ftness function, 783
Maximum constraint violation, 540
Maximum point, defnition of, 111
m-Digit binary string, 742
Mean value frst-order second-moment method
(MVFOSM), 841
Megapascal (MPa), 243
Members for ?exure, optimum design of, 652
constraints, formulation of, 660
data/information collection, 652
de?ection requirement, 660
design of ?exural members, 653
data for optimizing, 653
moment strength requirement, 656
nominal bending strength
of compact shapes, 657
of noncompact shapes, 658
project/problem description, 652
shear strength requirement, 659
Member stresses, 623
Metaheuristics methods, 764
Meta-models
for design optimization, 771, 773
mathematical model, 772
response surface method (RSM), 773
normalization of variables, 776, 778
normalization of variables, procedure, 777–778
quadratic response surface generation,
773–777
errors, 796
examples, 796
m-File, for objective function, 283
Minimum-area beam design problem, 91
Minimum-cost cylindrical tank design, 46–47
Minimum-weight system, 700
Minimum-weight tubular column design, 43, 89
formulation 1 for, 44–45
formulation 2 for, 45–46
graphical solution, 88–89
Mixed variable optimum design problem (MV-OPT)
classifcation of, 685
combinatorial problems, 685–686
design variables, to other parameters, 685
discrete variables, nondiscrete values, 685
functions continuous/differentiable, 685
functions nondifferentiable, 685
solution concepts, overview of, 686–687
defnition of, 684
Modifed Newton method, 473–477, 713, 897
computer program, 904
Monte Carlo simulation (MCS), 845
Most probable failure point (MPFP), 842
reliability index, 845, 846
geometric representation, 843
Most probable point (MPP), 842
Move limits, 525
Multilayered graphical representation
of discrete variable problem, 762938 Subject Index
Multi-objective genetic algorithms (GAs), 781
elitist strategy, 783
niche techniques, 784–785
Pareto ftness function, 783
Pareto-set flter, 783
ranking, 782–783
Tournament selection, 784
vector-evaluated, 782
Multi-objective optimization methods, 780, 790, 791
Multi-objective optimization problems, 771
Multi-objective optimum design concepts/methods,
771, 773
criterion space/design space, 773–776
objective functions, normalization of, 780–781
optimization engine, 781
pareto optimal set, generation of, 780
preferences/utility functions, 779–780
problem defnition, 771–772
selection of methods, 790
solution concepts, 776
compromise solution, 779
effciency/dominance, 778
pareto optimality, 777
utopia point, 778
weak pareto optimality, 777
vector methods/scalarization methods, 780
Multi-objective problems, 241
Multiple optimum designs, 86
Multiple performance requirements, three-bar
structure, 619
asymmetric three–bar structure, 621–625
comparison of solutions, 625
symmetric three–bar structure, 619–621
Multiplier methods, 490
Multivariable unconstrained minimization, 283
MV-OPT problems, 685, 687
N
Natural frequency, 51
Nature-inspired algorithms, 739
Nature-inspired methods, 739
Nature-inspired search methods, 739
algorithms, 739
drawbacks of, 740
ant colony optimization, 755
algorithm, 759–760
fnding feasible solutions, 762–763
pheromone deposit, 763
pheromone evaporation, 763
problem defnition, 760–761
algorithm for traveling salesman problem, 757
ant behavior, 755
simple model/algorithm, 756
virtual ant changes, defnition of, 758
differential evolution algorithm, 750, 752–753
crossover operation to generate the trial
design, 752
donor design, generation of, 751
initial population, generation of, 750
notation and terminology, 751
trial design, acceptance/rejection of, 752
genetic algorithms (GA), 747
applications of, 749
basic concepts/defnitions, 741
design representation, 741–742
ftness function, 743
initial generation/starting design set, 742–743
crossover
number of, 746
fundamentals of, 743
crossover, 744
mutation, 745
population, leader of, 746
reproduction procedure, 744
stopping criteria, 747
mutation, number of, 745
for optimum design, 741
immigration, 747
multiple runs, for problem, 747
particle swarm optimization (PSO), 764
algorithm, 765–766
swarm behavior/terminology, 764
sequencing-type problems, genetic algorithm, 748
relocation, 749
NBR-6123 code, 666
n-Dimensional column vector, 859
Necessary conditions, 136
concepts of, 127
for constrained problem, 152
gradient condition, geometrical meaning of, 158
Karush–Kuhn–Tucker necessary conditions, 154
KKT conditions, 159
frst-order necessary conditions, 170
important observations, 160
limitation of, 170
role of inequalities, 152
switching conditions, 158
Nelder–Mead algorithm, 503–505, 721
Nelder–Mead simplex method, 498
Newton methods, 428, 455, 473, 496
Newton–Raphson iterative procedure, 579
Newton–Raphson method, 142, 578, 579, 592
Newton search direction, 474
Niche techniques, 784
Noise coeffcient, 722
Noncompact shape, 664
Nondifferentiable problems, 694
Nondominated check, 783Subject Index 939
Nongradient-based method, 283
Nonhomogeneous system, 859
Nonlinear discrete problems, 692
Nonlinear equations, 489
Excel solver
roots of set, 257
Excel solver roots, 253
roots of set
Solution to KKT Cases with Solver, 260
Solver Parameters dialog box, 259
worksheet, 257, 258
Solver Answer Report, 257
Solver Output, 256
Solver Parameters dialog box, 254, 255
Solver Results dialog box, 256
Nonlinear optimization problems, 519
Nonlinear programming algorithms, 537
Nonlinear programming, Excel solver
optimum design of springs, 266–268
Nonlinear programming, optimal control of systems, 625
minimum time control problem, 637–639
prototype optimal control problem, 625–629
state variable, minimization of error, 629–635
minimum control effort problem, 635–637
numerical results, 631
numerical solution, formulation for, 630–631
problem normalization, 631–634
results discussion, 634–635
system motion, formulations, 639
Nonlinear programming problem (NLP) methods, 26,
54, 307, 423, 577, 688
Nonquadratic case, 497
Nonunimodal function, 433
?-Norm, 885
Normalization procedure, 803, 805
Normalized shear stress, 290
Norm of matrices, 884
Norm of vectors, 12, 884
Notebook, 78
Null/zero matrix, 853
Numerical algorithms, 556
Numerical aspects, of problem formulation. See
Problem formulation, numerical aspects of
Numerical optimization methods
feasible directions, method of, 588–590
generalized reduced gradient method, 592–593
gradient projection method, 591–592
Numerical performance methods, 729
Numerical search methods, 238, 432, 512
derivative-based methods, 238–239
derivative-free methods, 240
direct search methods, 239–240
nature-inspired search methods, 240
selection of method, 241
Numerical solution process, for optimum design, 250
algorithm, 252
feasible points, 251–252
general purpose software, integration of application,
250–251
O
Off-diagonal elements, 121, 123, 857
One-cut-point, 744
One-dimensional minimization, 459
Optimal Bayesian estimate, 717
Optimal control problems, 625, 627
Optimality conditions, 128
basic concept, 128
functions of several variables, 135
for functions of single variable, 129
for unconstrained variable problems, 131
Optimality criteria methods, 105
Optimization engine, 781
Optimization methods, 4
classifcation of, 106
Optimization problem, 4
Optimization toolbox functions, 281
Optimization variable, 23
Optimum control forces
to minimize control effort, 636
to minimize error, 633
to minimize time, 638
Optimum design process
formulation, of complex engineering systems, 603
general mathematical model
active/inactive/violated constraints, 57
application to different engineering felds, 56
discrete/integer design variables, 58
feasible set, 57
“greater than type” constraints, treatment of, 56
maximization problem treatment, 55
standard design optimization model, 54–55
standard model, important observations, 56–57
types of problems, 59
continuous/discrete-variable, 59
design variables as functions, 60
dynamic-response, 60
implicit constraints, 59
network optimization problems, 59
smooth/nonsmooth, 59
problem formulation, numerical aspects of, 241
vs. conventional, 6–7
vs. optimal control, 8
Optimum Lagrange multipliers, 174
Optimum points, representation of, 107
Orthogonal arrays method, 807
Orthogonal steepest–descent paths, 468
Outer array, 827940 Subject Index
Out-of-plane loads, two-member frame, 617
Output from optimization function, 281
P
Pareto optimality, 777
Pareto optimal set, 777, 785
illustration of, 775
Pareto optimal solution, 787
Pareto-set flter approach, 783
Partial derivatives, 114
of functions, 14
of vector functions, 15
Partial pivoting, 865
Particle swarm optimization (PSO), 764
algorithm, 765–766
swarm behavior/terminology, 764
Pattern search methods, 499
PDF. See Probability density function (PDF)
Penalty function method, 490
Penalty methods, 713
Performance requirements, 25
Pheromone density, 760
Pheromone values, 757
Physical programming, 779
Pitting, 288
Plate girders
design problem, spreadsheet for, 274
Excel solver, 268
data and information collection, 270–271
design variables, defnition of, 271
formulation of constraints, 272
optimization criterion, 271
project/problem description, 268–269
solution, 274–276
Solver Parameters Dialog Box, 273
spreadsheet layout, 272–273
Point maximizing, 55
Points, 8
Poisson’s ratio, 288
Polak–Ribiére formulas, 449
Pole structure, section, 668
Polyhedron, 322
Positive defnite quadratic function, 497
Postoptimality analysis, 171, 356
changing constraint limits, 171–172
constraint, effect of scaling, 176
constraint variation sensitivity result, generalization
of, 177
cost function scaling, effect of, 175
frst-order changes, in cost function, 172
Lagrange multipliers, 171
nonnegativity of, 173
practical use of, 174
Potential constraint index set, 556
Potential constraint strategy, 548, 556, 615
Potential cost functions, 603
Potential energy function, 487
Potentially active, 556
Practical applications
development of problem formulation, 60–61
of optimization, 601
Practical design optimization, issues, 614
algorithm, selection of, 614
potential constraint strategy, 614
robustness, 614
good optimization algorithm, attributes
of, 614–615
Practical design optimization problems, formulation
of, 602
example of, 603–604
general guidelines, 602–603
Predictive dynamics, 675
Preliminary design, 5
Primal cost function, 228
Primal problem, 408
Principle of stationary potential energy, 486
Probability density function (PDF), 819,
833, 834
limit state, 839
Probability mass function, 833
Problem formulation process, 20
cantilever beam, optimization criterion, 25
constraints, formulation of
equality and inequality, 26
feasible design, 26
linear/nonlinear, 26
restrictions, 25
data/information collection, 21
design variables
for cantilever beam, 23
design variables, defnition of, 22
numerical aspects of, 241
general guidelines, 241–242
iterative process for development, 248–250
scaling of constraints, 242–243
scaling of design variables, 246–247
optimization criterion, 24
project goals, 20
Problem parameter vector, 820
Proft maximization problem, 83, 315, 328
ABCDE, feasible region, 75
graphical representation, 85
graphical solution, 77
objective function contour, 76
Projection operator, 574
Pshenichny’s descent function, 539, 540
PSO. See Particle swarm optimization (PSO)
Push-off factor, 590Subject Index 941
Q
QP. See Quadratic programming (QP)
Quadratic approximation, 457, 800
Quadratic form, 122
Quadratic interpolation, 456
Quadratic loss function, 491, 825
Quadratic programming (QP), 531
defnition of, 414
KKT conditions
necessary, 415
transformation of, 415–416
problems, 414, 555
methods to solve, 555
simplex method for problem solving, 416–417
subproblem, 533, 537, 538, 594
graphical representation, 533
Quadratic programming subproblem, 536
direct solution, of QP subproblem, 596–597
KKT necessary conditions, 594–596
solution, 593
Quadratic response surface, generation of,
773–777, 799
Quasi-Newton methods, 428, 479, 497, 580, 587
R
Random tunneling, 722
Random variable, with mean value, 819
Realistic practical bounds, 247
Real telecommunication steel pole, 666
Recall, step size calculation problem, 456
Rectangular beam design problem, second-order
conditions, 218
Hessian of Lagrangian, 219
Hessians of cost function, 219
KKT necessary conditions, 218, 220
Rectangular matrix, 857
Rectangular m × n system, 858
Reduced cost coeffcients, 395
Reduced gradient, 592
Reduced sample points, 718
Regression analysis, 485
Relative minimum, 108
Reliability-based design optimization (RBDO), design
under uncertainty, 795, 833
formulation of, 848
reliability index, 838
advanced frst order second-moment method
(AFOSM), 841–845
limit state equation, 838–839
linear limit state equation, 840
nonlinear limit state equation, 840
review of background material, 833
coeffcient of variation, 837
correlation coeffcient, 837
covariance, 837
cumulative distribution function (CDF), 834
expected value, 835–836
Gaussian (normal) distribution, 837
inverse, 838
mean/variance, 836
probability density function (PDF), 833
probability of failure, 835
reliability index, 837
standard deviation, 836
Reliability index, geometric representation, 843
Reliable algorithms, 516
Resource limits, 308, 358
Response surface generation (RSG)
design of experiments, 805–807
Response surface method (RSM), 773,
797, 805
approximate a function, 782
bending stress constraint, 772
least squares method, 797
mean values, one-way table
graphical representation of, 815
normalization of variables, 776, 778
procedure, 777–778
optimization, 810
optimum values, 812
quadratic response surface generation,
773–777
shear stress constraint, 773
Response surface methods, 240
Robust design approach
defned, 817
effect of uncertainties in problem parameters, 818
robust optimization, 818
mean, 818
probability density function, 819
problem defnition, 820–823
standard deviation, 819
variance, 819
Taguchi method, 825–827
Robust design method, 818
Robust design optimization, 818
Robustness, 817
index, 823
Robust optimization solution, 821, 824
Root-fnding process, 488
Roulette wheel process, 744
Row matrix, 853
RSM. See Response surface method (RSM)
Rupture limit state constraint, 642
S
Saddle point theorem, 228
Sample computer programs, 891942 Subject Index
Sawmill operation, 32
data and information collection, 32
defnition of design variables, 32
formulation of constraints, 33
mathematical formulation, 33
optimization criterion, 33
project/problem description, 32
Scalar function, 884
Scalarization, 780
Scalar matrix, 857
Scalar multiplication, 881
Scalar quantity, 12
Scaling procedure, 247
Schaffer’s method, 782
Schema, 741
Search direction defnition, 575
Search methods, 106
Second-order conditions, for constrained
optimization, 212
general constrained problems, 213
insights for, 214
strong suffcient condition, 215
suffcient conditions, for general constrained
problems, 214
Second-order necessary conditions, 130, 136, 213
Selection process, 782
Selection step, 752
Self-explanatory ?owchart, 6
Sensitivity analysis, 171
Sensitivity Report from Solver
for linear programming problem, 266
Sequential linear programming (SLP)
algorithm, 524–526, 535
method, 530
move limits, 524
observations, 530–531
positive/negative design changes, 526
selection of proper move limits, 525
Sequential quadratic programming (SQP), 531, 547, 617
algorithm, 582, 587
descent functions, 587–588
methods, 577, 582, 631, 731
QP subproblem, 578
observations, 587
quadratic programming subproblem
derivation of, 578–580
quasi-Newton Hessian approximation, 580–582
subproblem, search direction calculation, 532
defnition of, 532
solving, 537
Serviceability requirement, 660
Sets, 9
Shear-stress constraint, 49
Show Formulas command, 262
Sign support column, 100–102
Simplest stochastic method
for global optimization, 717
Simplex method, 338, 389, 417
for LP problem, 411
Simulated annealing (SA) method, 687, 693
Single-objective optimization problem, 772
graphical representation of, 773
Singular matrix, 866
Slack variables, 154, 191, 310, 367
Slenderness ratios, 647
constraint, 642
SLP. See Sequential linear programming (SLP)
Small feasible region, 108
Smooth optimization problems, 238
Solve constrained design problems, 489
Solver Answer Report, for spring design problem, 269
Solver Parameters dialog box, 255, 260
for linear programming problem, 263
for plate girder design problem, 275
Solver Results dialog box, 256
for linear programming problem, 264
Solving n linear equations
in n unknowns, 858
Spherical tank, with intermediate variables, 31
Spring constant, 628
Spring design problem
with design variables, 49
material data, 697
SQP. See Sequential quadratic programming (SQP)
Square matrix, 852, 857
Standard design optimization model, 54
defned, 707
Standard LP problem, 308
Stationary points, frst-order necessary condition, 129
Steepest ascent, 465
Steepest–descent algorithm, 443
Steepest-descent direction, 158, 531
Steepest descent method, 442, 444, 469, 470, 479
computer program, 898
gradient-based methods for unconstrained
optimization, 897
Steepest–descent steps, 719
Step-by-step algorithm, 499
Step size calculation subproblem
descent function, 539–541
line search, 542
Step size determination, 429, 562
Stochastic integration, 722–723
Stochastic methods, overview of, 710, 716
clustering methods, 718, 719
density clustering, 719
mode analysis clustering, 720
reduced sample points, 718Subject Index 943
single linkage clustering, 719
vector quantization, 720
multistart method, 717
stopping criterion, 717
pure random search method, 717
Stochastic zooming method (ZOOM), 724, 731
Stress constraints, 52
Strong duality theorem, 227
Strong Wolfe conditions, 462
Structural optimization problems, alternative
formulations, 672
two–member frame design, alternate
formulation, 673
Submatrices, 857
Subprogram calls, 616
Subroutine EQUAL, 891
Subroutine FUNCT, 891, 894
Subroutine SYSEQ, 897
Subvectors, 857
Suffcient condition, 130
concepts of, 127
Suffcient-decrease condition, 461
Summation notation, 11, 118
Superscript notation, 11
Surplus variable, 310
Swarm intelligence methods, 764
Symmetric matrix, 857
Symmetric three–bar structure, 621
Symmetric three-bar truss, minimum-weight
design of, 50–54
System evolution model, 5
T
Tableau form, 324
Taguchi method, 825, 827, 828, 830
Tangential vector, 464
Taylor expansion, of constraint, 844
Taylor series, 840
Taylor series expansion, 822
Taylor’s expansion, 117–119, 129, 464
function of two variables, 119
of Lagrange function, 213
linear function, 120
quadratic form, differentiation of, 126
quadratic forms/defnite matrices, 120
Telecommunication poles, 667
optimum design of, 664
constraints, formulation of, 670
data/information collection, 665
design variables, defnition of, 669
optimization criterion, 669
project/problem description, 664
Ten-bar cantilever truss, 700, 702
Tension members, 641
constraints, formulation of, 642–643
data and information collection, 640
design variables, defnition of, 641
discussion, 644
optimization criterion, 642
optimum design of, 639
project/problem description, 640
Terminal displacement constraint, 629
Terminal velocity constraint, 629
Three-bar truss, 51
Three elementary row–column operations, 862
Time-dependent optimization problems, 674
Time-dependent problems
alternative formulations, 674
digital human modeling, 675
mechanical/structural design problems, 674
Total pivoting procedure, 865
Trade-off, 789
Trajectory methods, 712
Transformation function, 490
Transformation of variables, 801
Transportation problems, 33
Traveling salesman (TS), 758
problem, 748, 758
Tripod, 103
Tubular column, 44
Tunneling method, 714
basic concept of, 715
global descent property, 715
Twice-continuous differentiability, 13
Two-bar bracket design, 33, 34
data and information collection, 34
defnition of design variables, 35
formulation of constraints, 37
optimization criterion, 37
project/problem description, 33
Two-bar bracket problem, with intermediate
variables, 38
Two-bar truss, 486
minimization of total potential energy, 487
Two-cut point
crossover operation with, 745
methods, 744
Two–member frame design, 605
Two-objective optimization problem, 772
U
Unconstrained methods, engineering applications, 484
data interpolation, 485
nonlinear equations, solution of, 488–489
total potential energy, minimization of, 486
Unconstrained minimization, 489
problem, 455. See also Unconstrained optimum
design, numerical methods944 Subject Index
Unconstrained minimum, representation
of, 110, 111
Unconstrained optimization methods, 488
algorithms, rate of convergence, 494
convergence ratio, 495
defnitions, 494
linear convergence, 495
order of convergence, 494
quadratic convergence, 495
superlinear convergence, 495
conjugate Gradient method, 497
constrained problem using, solution of, 489
direct search methods, 498
contraction, 502
expansion, 502
Hooke–Jeeves algorithm, 499–500
Hooke–Jeeves methods, 499
exploratory search, 499
pattern search, 499
initial simplex, 503
Nelder–Mead algorithm, 503–504
Nelder–Mead simplex method, 500–501
re?ection, 501
shrinking operation, 503
termination criterion, 503
univariate search, 498
Newton method, 496
quasi-Newton methods, 497
BFGS quasi-newton method, 498
DFP method, nonquadratic case, 497
DFP method, quadratic case, 497
sequential unconstrained minimization
techniques, 490
augmented Lagrangian algorithm,
493–494
augmented Lagrangian (multiplier) methods,
492–494
barrier function methods, 491–492
penalty function method, 491
steepest-descent method, 495
nonquadratic case, 496
quadratic function, 495
Unconstrained optimization problems, 813
algorithms, descent direction/convergence
of, 426
convergence of, 428
descent direction/descent step,
427–428
rate of convergence, 428
classifcation of, 424
conjugate gradient methods, 448
optimum solution, 448
equal–interval search
alternate equal–interval search, 436
golden section search, 437
algorithm for 1D search by golden sections, 440
initial bracketing of minimum-phase I, 437
initial bracketing of minimum point, 438
interval of uncertainty-phase II, 439
initial bracketing of minimum-phase I, 434
process, 435
reducing interval of uncertainty-phase II, 435
Excel solver, 260
worksheet and Solver Parameters dialog box, 261
general algorithm, 426
general iterative algorithm, 425
numerical methods, 423
to compute step size, 432
1D search methods, 433
general concepts, 432
interval-reducing methods, 434
unimodal functions, 432
search direction determination, 445
conjugate gradient algorithm, 446
convergence of, 447
steepest–descent method, 442–443
step size determination, basic ideas, 429
analytical method to compute step size, 431
1D minimization problem, 430
reduction to function, 429
subproblem, defnition of, 429
Unconstrained optimum design, numerical
methods, 455
design variables, scaling of, 469
Newton method, search direction determination, 472
classical Newton method, 472–473
Marquardt modifcation, 478–479
modifed Newton method, 473–477
polynomial interpolation, 456
alternate quadratic interpolation, 459
Armijo’s rule, inexact line search, 460–461
Goldstein test, inexact line search, 462–463
quadratic curve ftting, 456–458
Wolfe conditions, inexact line search, 462
quasi-Newton methods, search direction
determination, 479
BFGS method, direct Hessian updating, 482–483
DFP method, inverse Hessian updating, 479–481
steepest-descent directions, orthogonality of, 468
steepest–descent method, 463
gradient vector, properties of, 463–465
step size determination, 456
Unconstrained points, 109
Unconstrained problem, 128
Uniform prior distribution, 717
Upper triangular matrix, 857
US–British, SI Units, 15, 16
conversion factors, 16Subject Index 945
User-defned constant, 821
Utility function, 780
Utopia point, 787
V
Variable-interval search method, 437
Variance represents dispersion, 819
Vector-evaluated genetic algorithm (VEGA), 782
Vector functions, partial derivatives of, 15
Vector optimization methods, 720, 771, 780
Vector quantization multistart, 719
Vector representation, 9
Vectors, 8
norm of, 884
Vector space, 880
Velocity responses
at optimum with minimization of time, 638
Vertical column, with eccentric load, 291
Violated constraint, 515
V–strings, 746
W
Water tower support column, 98, 99
Weak duality theorem, 227
Weakly Pareto optimal points, 792
Weierstrass theorem, 112, 113
Weighted global criterion method,
786–788
Weighted min–max method, 785
additional constraints, 786
advantages of, 786
disadvantages of, 786
Weighted sum method, 785
multi-objective optimization, 785
Weighted Tchebycheff method, 785
Welded plate girders, 268
Wiener process, 722
Wolfe conditions, 462
W–shape
for the ?exural member, 653
for member, 640
Y
Young’s modulus, 44
Z
Zero reduced cost coeffcient, 334
Zooming method, 712
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