كتاب Introduction to Optimum Design
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Introduction to Optimum Design
Third Edition
Jasbir S. Arora
The University of Iowa
College of Engineering
Iowa City, Iowa  

كتاب Introduction to Optimum Design  I_t_o_11
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1 Introduction to Design Optimization
2 Optimum Design Problem Formulation
3 Graphical Optimization and Basic Concepts
4 Optimum Design Concepts
5 More on Optimum Design Concepts
6 Optimum Design with Excel Solver
7 Optimum Design with MATLAB
8 Linear Programming Methods for Optimum Design
9 More on Linear Programming Methods for Optimum Design
10 Numerical Methods for Unconstrained Optimum Design
11 More on Numerical Methods for Unconstrained Optimum Design
12 Numerical Methods for Constrained Optimum Design
13 More on Numerical Methods for Constrained Optimum Design
14 Practical Applications of Optimization
15 Discrete Variable Optimum Design Concepts and Methods
16 Genetic Algorithms for Optimum Design
17 Multi-objective Optimum Design Concepts and Methods
18 Global Optimization Concepts and Methods
19 Nature-Inspired Search Methods
20 Additional Topics on Optimum Design
Index
A
Acceptance criterion, 697
Acceptance rejection (A-R)
method, 697 698
Acceptance/rejection of trial
design, 717
ACO. See Ant Colony Optimization
Adaptive numerical method for
discrete variable
optimization, 636 641
continuous variable
optimization, 636 637
discrete variable optimization,
637 641
Advanced first-order second
moment method, 777 781
Agent, 727
Algebra, vector and matrix. See
Vector and matrix algebra
Algorithm, for traveling salesman
problem, 721 724
Algorithm does not converge, 217
Algorithms
attributes of good optimization,
588
conceptual local-global, 699 700
constrained problems, 417
constraint correction, 638
convergence of, 417
CSD, 526 527
Phase I, 337
Phase II, 339 345
robust, 587
selection of, 587
Simplex, 384 385
Algorithms, concepts related to
numerical. See Numerical
algorithms
Algorithms, SLP. See Sequential
Linear Programming
algorithms
Algorithms for step size
determination, ideas,
418 421
alternate equal interval search,
425
analytical method to compute
step size, 419 421
definition of one-dimensional
minimization subproblem,
419
equal interval search, 423 424
example—analytical step size
determination, 420
example—minimization of
function by golden section
search, 429
golden section search, 425 430
numerical methods and compute
step size, 421 430
Alternate equal interval
search, 425
Alternate quadratic interpolation,
447 448
American Association of State
Highway and Transportation
Officials (AASHTO),
231 232
Analyses
engineering, 4
operations, 702 705
Analysis, postoptimality. See
Postoptimality analysis
Analysis of means (ANOM), 749
Analytical method, 419 421
Ant behavior, 718 720
simple model/algorithm,
719 720
Ant Colony Optimization (ACO),
718 727
algorithm for design
optimization, 724 727
algorithm for traveling salesman
problem, 721 724
behavior, 718 720
Application to different
engineering fields, 52
example problem, 724 725
feasible solutions, finding, 725
pheromone deposit, 726 727
pheromone evaporation, 726
problem definition, 724
Array operation, 276
Artificial cost function, 336, 383
Artificial variables, 334 347,
382 383
cost function, 336
definition of Phase I problem,
336 337
degenerate basic feasible
solution, 345 347
example—feasible problem, 342
example—implications of
degenerate feasible solution,
346
example—unbounded problem,
344
example—use of artificial
variables, 344
example—use of artificial
variables for equality
constraints, 342
example—use of artificial
variables for $ type
constraints, 339
Phase I algorithm, 337
Phase II algorithm, 339 345
use for equality constraints, 342
Ascents, alternation of descents,
687
Asymmetric three-bar structure,
594 598
Augmented Lagrangian methods,
479 481
B
Basic feasible solution, degenerate,
346
BBM. See Branch and bound
method
Beam, design of rectangular,
174 187
Beam design problem, graphical
solution for, 82 94
Binary variable defined, 619
Binomial crossover, 717
Bound-constrained optimization,
549 553
861Bound-constrained optimization
(Continued)
optimality conditions, 549 550
projection methods, 550 552
step size calculation, 552 553
Bounded objective function
method, 675 676
Brackets
design of two-bar, 30 36
design of wall, 171 174
Branch and bound method (BBM),
623 628
basic, 623 624
example—BBM with local
minimizations, 626
example—BBM with only
discrete values allowed, 624
for general MV-OPT, 627 628
with local minimization,
625 627
British versus SI units. See U.S.
British versus SI units
Broyden-Fletcher-Goldfarb-Shanno
(BFGS) method, 470 472
C
Cabinet design, 37 40
Calculation of basic solution,
314 320
basic solutions to Ax 5 b,
317 320
pivot step, 316 317
tableau, 314 316
Calculus concepts, 103 115
example—calculation of gradient
vector, 105
example—evaluation of gradient
and Hessian of function, 106
example—linear Taylor’s
expansion of function, 109
example—Taylor’s expansion of
a function of one variable,
108
example—Taylor’s expansion of
a function of two variables,
108
gradient vector, 103 105
Hessian matrix, 105 106
necessary and sufficient
conditions, 115 116
quadratic forms and definite
matrices, 109 115
Taylor’s expansion, 106 109
Can design, 25 26
Canonical form/general solution of
Ax 5 b, 308 309
Changing constraint limits, effect
of, 153 156
Chromosome, 645, 715 716
Clustering methods, 691 694
Coefficient matrix, changes in,
361 375
Coefficient of variation, 773
Coefficients, ranging cost, 359 361
Coil springs, design of, 43 46
Column design
for minimum mass, 286 290
minimum weight tubular, 40 42
Column matrix, 787 820
Columns, graphical solutions for
minimum weight tubular,
80 81
Column vector, 787 820
Compression members, optimum
design of, 243 250, 244t
discussion, 250
example—elastic buckling
solution, 249
example—inelastic buckling
solution, 247
formulation of problem, 243 247
formulation of problem, for
elastic buckling, 249 250
formulation of problem, for
inelastic buckling, 247 248
Compromise solution, 665
Computer programs, sample, 823
equal interval search, 823 826
golden section search, 826 828
modified Newton’s method, 829
steepest descent method, 829
Concepts, optimum design. See also
Optimum design concepts
duality in NLP, 201 212
exercises, 178 180
necessary conditions, for
equality-constrained
problem, 130 137
necessary conditions, for general
constrained problem,
137 153
Concepts, solution. See Solution
concepts
Concepts and methods, multiobjective optimum design.
See Multi-objective optimum
design concepts and
methods
Conditions
descent, 416
second-ordered, 194 199
transformation of KKT, 403 404
Conditions, alternate form of KKT
necessary, 189 192
example—alternate form of KKT
conditions, 190
example—check for KKT
necessary conditions, 191
Conditions, concepts relating to
optimality, 116 117
Conjugate gradient method,
434 436, 484
example—use of conjugate
gradient algorithm, 435 436
Constrained design, numerical
methods for, 491 574
algorithms and constrained
problems, 492 495
basic concepts and ideas, 492 499
constrained quasi-Newton
methods, 573
constraint normalization, 496 498
constraint status at design point,
495 496
convergence of algorithms,
498 499
CSD method, 525 531
descent function, 498
example—constraint
normalization and status
at point, 497
inexact step size determination,
s, 0035
linearization of constrained
problem, 541
miscellaneous numerical
optimization methods,
564 569
potential constraint strategy,
534 537
QP problem, 513 514
QP subproblem, 514 520
SLP algorithm, 506 513
Constrained optimum design
problems, 281 282
example—constrained
minimization problem using
fmincon, 281
example—constrained optimum
point, 138
example—cylindrical tank
design, 127
862 INDEXexample—equality constrained
problem, 140
example—fmincon in
Optimization Toolbox, 281
example—inequality constrained
problem, 140
example—infeasible problem,
139
example—Lagrange multipliers
and their geometrical
meaning, 131
example—solution of KKT
necessary conditions, 145,
146, 150
example—use of Lagrange
multipliers, 136
example—use of necessary
conditions, 140
inequality constraints, 137 139
KKT, 139 152
necessary conditions, 137 153
necessary conditions: equality
constraints, 137 153
Constrained optimization, secondorder conditions for,
194 199
example—check for sufficient
conditions, 197
solution of KKT necessary
conditions using
Excel, 222
solution of KKT necessary
conditions using MATLAB,
149
Constrained optimum design,
numerical methods for,
533 574
bound-constrained optimization,
549 553
inexact step size calculation,
537 549
potential constraints trategy,
534 537
QP subproblem, 514 520
quasi-Newton Hessian
approximation, 557 558
search direction calculation,
514 520
SQP, 513 514, 553 563
step size calculation subproblem,
520 525
Constrained problems, concepts
related to algorithms for,
492 495
Constrained problems,
linearization of, 499 506
example—definition of linearized
subproblem, 500 506
example—linearization of
rectangular beam design
problem, 504
Constrained quasi-Newton
methods. See also Sequential
quadratic programming
descent functions, 563
deviation of QP subproblem,
554 557
example—use of constrained
quasi-Newton method, 560
observations on, 561 563
quasi-Newton Hessian
approximation, 557 558
Constrained steepest-descent (CSD)
method, 513, 525 527
algorithm, 526 527
algorithm, with inexact step size,
542 549
descent function, 538 542
example—calculation of descent
function, 540
example—golden section search,
429
example—use of CSD algorithm,
542
step size determination,
444 450
Constrained variable metric (CVM).
See Sequential quadratic
programming
Constraint correction (CC),
algorithm for, 638
Constraint limits, effect of
changing, 155
Constraint normalization, 496 498
Constraints, 300
linear, 23
notation for, 8 9
Constraints, formulation of, 22 25
equality and inequality
constraints, 23
feasibility design, 23
implicit constraints, 23 25
linear and nonlinear constraints,
23
Constraint status at design point,
495 496
Constraint strategy, potential,
534 537, 587
example—determination of
potential constraint set, 534
example—search direction and
potential constraint strategy,
536
Constraint tangent hyperplane, 194
Continuous variable optimization,
608 609, 636 637
Contours
plotting of function, 75 77
plotting of objective
function, 74
Control, optimal, 6
Control effort problem, minimum,
608 609
Controlled random search (CRS),
694 697
Control of systems by nonlinear
programming. See Nonlinear
programming, control of
systems by
Control problems
minimum time, 609 610
prototype optimal, 598 602
Conventional versus optimum
design, 4 5
Convergence of algorithms, 417
Convergence ratio, 482
Convex functions, 162 164
Convex programming problem,
164 170
Convex sets, 160 161
Correction algorithm, constraint,
638
Correlation coefficient, 773
Cost
algorithm for constraint
correction at constant, 638
algorithm for constraint
correction at specified
increase in, 638
constraint correction with
minimum increase in, 638
Cost coefficients, ranging, 359 361
Cost function, 300
Cost function, artificial, 336
Cost function scaling, effect on
Lagrange multipliers,
156 157
Covariance, 773
Covering methods, 684 685
Criterion, acceptance, 697
Criterion method, weighted global,
673 674
INDEX 863Criterion space and design space,
660 662
Crossover operation to generate
trial design, 716 717
CRS. See Controlled random search
CSD, 527 531, 572
constrained quasi-Newton
methods, 573
CSD method, 530 531
linearization of constrained
problem, 528 529
QP subproblem, 529
SLP algorithm, 529
CSD method. See Constrained
steepest-descent method
Cumulative distribution function,
770
Curve fitting, quadratic, 444 447
Cylindrical tank design, minimum
cost, 42 43
D
Davidon-Fletcher-Powell (DFP)
method, 467 469
DE. See Domain elimination (DE)
DE algorithm, 717 718
notation and terminology for,
715t
Definite matrices, quadratic forms
and, 109 115
Definitions, standard LP, 300 302
Degenerate basic feasible solution,
345 347
Derivative-based methods, 214
Derivative-free methods, 215
Derivatives of functions, 12 13
first partial derivatives, 12
partial derivatives, of vector
functions, 13
second partial derivatives, 13
Descent, methods of generalized,
686 688
Descent algorithm, 432 434
Descent condition, 417, 538 542
Descent direction, 415 417
descent step, 415 417
orthogonality of steepest,
454 455
rate of convergence, 417
Descent function, 498,
520 522, 563
example, 522
Descent method, steepest, 431 434,
451 455, 482 483
example—verification of
properties of gradient
vector, 453
properties of gradient vector,
451 454
Descents and ascents, alternation
of, 687
Descent search, steepest, 829
Descent step, 415 417
Design, 714
of cabinet, 37 40
of can, 25 26
of column, 286 290
of flywheel, 290 298
of insulated spherical tank,
26 28
of minimum cost cylindrical
tank, 42 43
of minimum weight tubular
column, 40 42
multiple optimum, 77
of rectangular beam, 547
of two-bar bracket, 30 36
of wall bracket, 171 178
Design, GA for optimum. See
Genetic algorithms (GA) for
optimum design
Design, global optimization
concepts and methods for.
See Global optimization
concepts and methods
Design, introduction to, 1 16
basic terminology and notation,
6 13
conventional versus optimum
design process, 4 5
design process, 2 4
engineering design versus
engineering analysis, 4
optimum design versus optimal
control, 6
Design, linear programming
methods for. See Linear
programming methods for
optimum design
Design, mathematical model for
optimum. See Mathematical
model for optimum design
Design, numerical methods for
constrained. See Constrained
design, numerical methods
for
Design, numerical methods for
constrained optimum. See
Constrained optimum
design, numerical methods
for
Design, numerical methods for
unconstrained optimum. See
Unconstrained optimum
design, numerical methods
for
Design concepts, optimum. See
Optimum design concepts
Design concepts and methods,
discrete variable. See
Discrete variable optimum
design concepts and
methods
Design concepts and methods,
multi-objective. See Multiobjective optimum design
concepts and methods
Design examples, engineering,
171 178
Design examples with MATLAB,
optimum, 284 298
Design of experiments for response
surface generation, 741 748
example—generation of a
response surface using an
orthogonal array, 744
example—optimization using
RSM, 746
Design optimization
applications with implicit
functions, 576 582
practical applications with
implicit functions, 575 618
Design optimization, issues in
practical. See Practical design
optimization, issues in
Design optimization applications
with implicit functions
adaptive numerical method for
discrete variable
optimization, 636 641
general-purpose software,
589 590
gradient evaluation for implicit
functions, 582 587
issues in practical design
optimization, 587 588
multiple performance
requirements, 592 598
optimal control of systems by
nonlinear programming,
598 612
864 INDEXoptimum design of three-bar
structure, 592 598
optimum design of two-member
frame, 590 591
out-of plane loads, 590 591
practical design optimization
problems, 576 582
Design point, 714
constraint status at, 578 582
Design problem formulation,
optimum, 17 64
design of cabinet, 37 40
design of can, 25 26
design of coil springs, 43 46
design of two-bar bracket,
30 36
general mathematical model for
optimum design, 50 64
insulated spherical tank design,
26 28
minimum cost cylindrical tank
design, 42 43
minimum weight design of
symmetric three-bar truss,
46 50
minimum weight tubular column
design, 40 42
problem formulation process,
18 25
saw mill operation, 28 30
Design problems
classification of mixed variable
optimum, 621 622
graphical solutions for rectanglar
beam, 82 94
with multiple solutions, 77 78
sufficiency check for rectangular
beam, 199 201
Design problems, constrained
optimum. See Constrained
optimum design problems
Design problems, unconstrained
optimum, 116 129
Design process, 2 4
Design representation, 645 646
Design space, 660 662
Design variables, scaling of,
456 459
example—effect of scaling of
design variables, 456
Design vector, 714
Desirable direction, 415
Determination, search direction,
431 436
Deterministic methods, 684 689
covering methods, 684 685
methods of generalized descent,
686 688
tunneling method, 688 689
zooming method, 685 686
Diagonal matrix, 791 821
Differential evolution algorithm,
714 718
A-R of trial design, 717
crossover operation to generate
trial design, 716 717
DE algorithm, 717 718
generation of donor design, 716
generation of initial population,
715 716
Digital human modeling, 614 617
Direct Hessian updating,
470 472
Directions
descent, 415 417
desirable, 415 417
method of feasibility, 564 565
orthogonality of steepest descent,
454 455
Direct search methods, 214 215,
412, 485 489, 713
Hooke-Jeeves method, 486 489
univariate search, 485 486
Discrete design with orthogonal
arrays, 749 753
example—discrete design with
an orthogonal array, 752
Discrete variable optimization,
609 610, 636 641
Discrete variable optimum design
concepts and methods,
619 642
adaptive numerical method for,
607 608
basic concepts and definitions,
620 623
BBM, 623 628
dynamic rounding-off method,
632 633
IP, 628 629
methods for linked discrete
variables, 633 635
neighborhood search method,
633
SA, 630 632
selection of methods, 635
sequential linearization methods,
629
Domain elimination (DE), 707 708
method, 700 702
Dominance, efficiency and, 664 665
Duality in nonlinear programming,
201 212
local duality, equality constraints
case, 201 206
local duality, inequality
constraints case, 206 212
Dynamic rounding-off method,
632 633
E
Efficiency and dominance, 664 665
Eigenvalues and eigenvectors,
816 818
example—calculation of
eigenvalues and
eigenvectors, 816 818
Eigenvectors, eigenvalues and,
816 818
Elements, off-diagonal, 791 821
Elimination, Gauss-Jordan,
800 803
Elimination domain, 700 702
Engine, optimization, 667
Engineering applications of
unconstrained methods,
472 477
Engineering design examples,
171 178
design of rectangular beam,
174 187
design of wall bracket,
171 174
Engineering design optimization
using Excel Solver, 231 238
data and information collection,
233 234
definition of design variables, 234
formulation of constraints,
234 235
identification of criterion to be
optimized, 234
project/problem statement,
231 233
solution, 238
Solver dialog box, 237 238
spreadsheet layout, 235 237
Engineering design versus
engineering analysis, 4
Equal interval search, 423 424,
823 826
alternate, 425
INDEX 865Equality-constrained problem,
necessary conditions,
130 137
Lagrange multipliers, 131 135
Lagrange multiplier theorem,
135 137
Equality constraints case, local
duality, 201 206
Equations
general solution of m 3 n linear,
792 803
solution of m linear, 804 809
Errors, minimization of,
602 608
Evaluation, gradient, 575 576
Excel Solver, 218 223
for LP problems, 225 227
for NLP, optimum design of
springs, 227 231
roots of a set of nonlinear
equations, 222 223
roots of a nonlinear equation,
219 221
for unconstrained optimization
problems, 224
Excel Solver, optimum design of
plate girders using. See also
Plate girders, optimum
design using Excel Solver
data and information collection,
233 234
identification/definition of
design variables, 234
identification of constraints,
234 235
identification of criterion to be
optimized, 234
project/problem statement,
231 233
solution, 235 237
Solver dialog box, 237 238
spreadsheet layout, 235 237
Excel Solver, optimum design with,
213 274. See also Optimum
design, with Excel Solver
for LP problems, 225 227
for NLP, optimum design of
springs, 227 231
numerical methods for optimum
design, 213 218
optimum design of compression
members, 243 250
optimum design of members for
flexure, 250 263
optimum design of plate girders
using excel solver, 231 238
optimum design of
telecommunication poles,
263 273
optimum design of tension
members, 238 243
for unconstrained optimization
problems, 224
Excel worksheet, 222 223
Expansion, Taylor’s. See Taylor’s
expansion
Expected value, 772 774
Expressions, variables and,
275 276
F
Feasible directions, method of,
564 565
Feasible points, finding, 216
Feasible region, identification
of, 73
Feasible solution, degenerate basic,
345 347
Feasible solutions, finding,
725 726
initial link, selection, 726
link from layer R, 726
solution for all ants, 726
Filters, Pareto-set, 670
First-order reliability method
(FORM), 781
Fitness functions, Pareto, 669
Fitting, quadratic curve, 444 447
Flywheel design for minimum
mass, 290 298
data and information collection,
290 292
definition of design variables,
292
formulation of constraints, 292
optimization criterion, 292
project/problem statement, 290
Formulation, design problem. See
Design problem formulation
Formulation process, problem. See
Problem formulation process
Formulations, comparison of three,
611 612
Function contours
plotting, 75 77
plotting of objective, 74
Functions
artificial cost, 336
descent, 498, 520 522
normalization of objective, 667
Pareto fitness, 669
plotting, 72 73
utility, 665 666
Functions, convex, 162 164
Functions, implicit, designing
practical applications with,
575 618
Functions, implicit, gradient
evaluation for, 582 587
example—gradient evaluation
for two-member
frame, 583
Functions of single variables,
optimality conditions for,
117 122
G
GA. See Genetic algorithms
Gaussian (normal) distribution,
773 774
Gaussian elimination procedure,
796 800
Gauss-Jordan elimination, 800 803
Gene, defined, 645
General concepts, gradient-based
methods. See Gradient-based
search methods
General constrained problem,
necessary conditions,
137 153
KKT necessary conditions,
139 152
role of inequalities, 137 139
summary of KKT solution
approach, 152 153
General iterative algorithm, 413 415
Generalized descent, methods of,
686 688
Generalized reduced gradient
(GRG) method, 567 569
General-purpose software, use of,
589 590
integration of application into,
589 590
Generation, 644, 714
Generation of donor design, 716
Generation of initial population,
715 716
Genetic algorithms (GA),
fundamentals of, 646 651
amount of crossover and
mutation, 649
866 INDEXcrossover, 648
elitist strategy, 670
immigration, 651
leader of population, 650
multi-objective, 667 671
multiple runs for problem, 651
mutation, 648 649
niche techniques, 671
number of crossovers and
mutations, 649
Pareto fitness function, 669
Pareto-set filter, 670
ranking, 669
reproduction procedure,
647 648
stopping criteria, 650
tournament selection, 670 671
VEGA, 668 669
Genetic algorithms (GA), for
optimum design, 643 656
applications, 653 655
basic concepts and definitions,
644 646
fundamentals of, 646 651
Genetic algorithms (GA), for
sequencing-type problems,
651 653
example—bolt insertion
sequence determination, 652
Global and local minima,
definitions of, 96 103
Global criterion method, weighted,
673 674
Global optimality, 159 170
convex functions, 162 164
convex programming problem,
164 168
convex sets, 160 161
example—checking for convexity
of function, 163, 164
example—checking for convexity
of problem, 166, 167, 168,
169
example—checking for convexity
of sets, 161
sufficient conditions for convex
programming problems,
169 170
transformation of constraint,
168 169
Global optimization concepts and
methods, 681 712
basic concepts of solution
methods, 682 684
deterministic methods, 684 689
numerical performance of
methods, 705 712
stochastic methods, 689 698
two local-global stochastic
methods, 699 705
Global optimization, of structural
design problems, 708 712
Goal programming, 676 677
Golden section search, 425 430,
523, 826 828
Golf methods, 688
Good optimization algorithm,
attributes of, 588
Gradient-based and direct search
methods, 411 412
nature-inspired search methods,
412
Gradient-based search methods,
411 412
basic concepts, 413
general algorithm, 415
general iterative algorithm,
413 415
Gradient evaluation for implicit
functions, 582 587
Gradient evaluation requires
special procedures,
575 576
Gradient method, conjugate,
434 436
Gradient projection method,
566 567
Gradient vectors, 103 105
properties of, 451 454
Graphical optimization, 65 94
design problem with multiple
solutions, 77 78
graphical solution for beam
design problem, 82 94
graphical solution for minimumweight tubular column,
80 81
graphical solution process,
65 71
infeasible problem, 79 80
problem with unbounded
solution, 79
use of Mathematica for graphical
optimization, 71 74
use of MATLAB for graphical
optimization, 75 77
Graphical optimization, use of
Mathematica for, 71 74
identification and shading of
infeasible region for
inequality, 73
identification of feasible region,
73 74
identification of optimum
solution, 74
plotting functions, 72 73
plotting of objective function
contours, 74
Graphical optimization, use of
MATLAB for, 75 77
editing graphs, 77
plotting of function contours, 75 77
Graphical solution, for beam
design problem, 82 94
Graphical solution, for minimumweight tubular column, 80 81
Graphical solution procedure,
step-by-step, 67 71
coordination of system set-up, 67
identification of feasible region
for inequality, 67 68
identification of optimum
solution, 69 71
inequality constraint boundary
plot, 67
plotting objective function
contours, 68 69
Graphical solution process, 65 71
profit maximization problem,
65 66
Graphs, editing, 77
H
Hessian approximation, quasiNewton, 557 558
Hessian matrix, 105 106
Hessian updating
direct, 470 472
inverse, 467 469
Hooke-Jeeves method, 486 489
algorithm, 486 489
exploratory search, 486
pattern search, 486
Hyperplane, constraint tangent, 194
I
Identity matrix, 791 821
Implicit functions, design
applications with, 575 618
adaptive numerical method for
discrete variable
optimization, 636 641
INDEX 867Implicit functions, design
applications with (Continued)
formulation of practical design
optimization problems,
576 582
general-purpose software,
589 590
gradient evaluation for implicit
functions, 582 587
issues in practical design
optimization, 587 588
multiple performance
requirements, 592 598
optimal control of systems by
NLP, 598 612
optimum design of three-bar
structure, 592 598
optimum design of two-member
frame, 590 591
out-of-plane loads, 590 591
Implicit functions, design practical
applications with, 575 618
Implicit functions, gradient
evaluation for, 582 587
example—gradient evaluation
for two-member frame, 583
Improving feasible direction,
564 565
Inaccurate line search, 448 449
Inequality, identification and
hatching of infeasible region
for, 73
Inequality constraints case, local
duality, 206 212
Inexact step-size calculation. See
Step-size calculation, inexact
Infeasible problem, 79 80
Infeasible region, identification and
shading of, 73
Insulated spherical tank design,
26 28
Integer programming (IP), 628 629
Integer variable, 619
Integration, stochastic, 698
Interpolation, alternate quadratic,
447 448
Interpolation, polynomial,
444 448
quadratic curve fitting, 444 447
Interval-reducing methods, 422 423
Interval search
alternate equal, 425
equal, 423 424, 823 826
Inverse Hessian updating, 467 469
IP. See Integer programming
Irregular points, 192 194
example—check for KKT
conditions at irregular
points, 192
K
Karush-Kuhn-Tucker (KKT), 189
conditions, transformation of,
404 405
conditions for LP problem,
400 402
optimality conditions, 400
solution, 400 402
necessary conditions, 139 152
necessary conditions, alternate
form of, 189 192
example—alternate form of
KKT conditions, 190
example—check for KKT
necessary conditions, 191
necessary conditions for QP
problem, 403 404
solution approach, 152 153
L
Lagrange multipliers, 131 135
effect of cost function scaling on,
156 157
physical meaning of, 153 159
constraint variation sensitivity
result, 159
effect of changing constraint
limit, 153 156
example—effect of scaling
constraint, 158
example—effect of scaling cost
function, 157
example—Lagrange
multipliers, 157, 158
example—optimum cost
function, 155
example—variations of
constraint limits, 155
scaling cost function on
Lagrange multipliers, 157
Lagrange multiplier theorem,
135 137
Lagrangian methods, augmented,
479 481
Length of vectors. See Norm/length
of vectors
Lexicographic method, 674 675
Limit state equation, 774 776
Linear constraints, 23
Linear convergence, 482
Linear equations, general solution
of m 3 n, 804 809
Linear equations in n unknowns,
solving n, 792 803
determinants, 793 796
example—determinant of matrix
by Gaussian
elimination, 799
example—Gauss-Jordan
reduction, 801
example—Gauss-Jordan
reduction process in tabular
form, 809
example—general solution by
Gauss-Jordan reduction, 806
example—inverse of matrix by
cofactors, 801
example—rank determination by
elementary operation, 804
example—solution of
equations by Gaussian
elimination, 798
Gaussian elimination procedure,
796 800
Gauss-Jordan elimination, 806
general solution of m 3 n linear
equations, 804 809
inverse of matrix, 800 803
linear systems, 792 793
rank of matrix, 803 804
Linear functions, 300
constraints, 300
cost function, 300
Linearization methods, sequential,
629
Linearization of constrained
problems, 499 506
example—definition of linearized
subproblem, 501
example—linearization of
rectangular beam design
problem, 504
Linear limit state equation, 776
Linear programming (LP), duality
in, 387 399
alternate treatment of equality
constraints, 391 392
determination of primal solution
from dual solution, 392 395
dual LP program, 388 389
dual variables as Lagrange
multipliers, 398 399
868 INDEXexample—dual of LP program,
389
example—dual of LP with
equality and $ type
constraints, 390
example—primal and dual
solutions, 394
example—recovery of primal
formulation from dual
formulation, 391
example—use of final primal
tableau to recover dual
solutions, 398
standard primal LP, 387 388
treatment of equality constraints,
389 390
use of dual tableau to recover
primal solution, 395 398
Linear programming methods, for
optimum design, 299 376,
377 410
artificial variables, 334 347
basic concepts related to LP
problems, 305 314
calculation of basic solution,
318 320
definition of standard LP
problem, 300 305
duality in LP, 387 399
example—structure of tableau,
318
KKT conditions for LP problem,
400 402
linear functions, 300
postoptimality analysis,
348 375
QP problem, 402 409
two-phase Simplex method,
334 347
Linear programming problem,
standard, 66, 300 305
example—conversion to standard
LP form, 304
linear constraints, 23
unrestricted variables, 303
Linear programming problems,
concepts related to, 299,
305 314
example—characterization of
solution for LP problems, 311
example—determination of basic
solutions, 311
example—profit maximization
problem, 306
LP terminology, 310 313
optimum solutions to LP
problems, 313 314
Linear programs (LPs), 299
Linear systems, 792 793
Line search, 522 525
Linked discrete variable, 619
Linked discrete variables, methods
for, 633 635
Loads, out-of-plane, 590 591
Local duality, equality constraints
case, 201 206
Local duality, inequality
constraints case, 206 212
Local-global algorithm, conceptual,
699 705
Local minima, definition, 96 103
Lower triangle matrix, 791 821
M
Marquardt modification, 465 466
Mass
column design for minimum, 286
flywheel design for minimum,
290 298
Mathematica, use of, for graphical
optimization. See Graphical
optimization, use of
Mathematica for
Mathematical model for optimum
design, 50 64
active/inactive/violated
constraints, 53 54
application to different
engineering fields, 52
discrete integer design variables,
54
feasibility set, 53
important observations about
standard model, 52 53
maximization problem treatment,
51
optimization problems, types of,
55 64
standard design optimization
model, 50 51
treatment of greater than type
constraints, 51 52
MATLAB, optimum design
examples with, 284 298
column design for minimum
mass, 286 290
flywheel design for minimum
mass, 290 298
location of maximum shear
stress, 284 285
two spherical bodies in contact,
284 285
MATLAB, optimum design with,
275 298
constrained optimum design
problems, 281 282
Optimization Toolbox, 275 277
unconstrained optimum design
problems, 278 280
MATLAB, use of for graphical
optimization, 75 77
editing graphs, 77
plotting of function contours,
75 77
Matrices, 785 787
addition of, 787
column, 790
condition numbers of, 819 822
definition of, 785 787
diagonal, 791 821
equivalence of, 790
identity, 791 821
inverse of, 800 803
lower triangle, 791 821
multiplication of, 788 789
null, 787
partitioning of, 791 792
quadratic forms and definite,
109 110
rank of, 803 804
row, 790
scalar, 790 791
square, 791
transpose of, 790
upper triangle, 791 821
vector, 787
Matrices, norms and condition
numbers of, 818 822
condition number of matrix,
819 822
norm of vectors and matrices,
818 819
Matrices, types of, 787 792
addition of matrices, 790
elementary row—column
operations, 790
multiplication of matrices, 788 789
partitioning of matrices, 791 792
scalar product dot product of
vectors, 790 791
square matrices, 791
vectors, 787
INDEX 869Matrix, changes in coefficient,
361 375
Matrix, Hessian, 105 106
Matrix algebra, vector and, 785
concepts related to set of vectors,
810 816
definition of matrices, 785 787
eigenvalues and eigenvectors,
816 818
norm and condition number of
matrix, 818 822
solution of m linear equations in
n unknowns, 792 803
types of matrices and their
operations, 787 792
Matrix operation, 276
Mechanical and structural design
problems, 614
Members for flexure, optimum
design of. See Optimum
design of members for
flexure
Meta-Model, 731 732
normalization of variables,
737 739
RSM, 733
Method of feasible directions,
564 565
Methods See also individual method
entries
alternate Simplex, 385 386
A-R, 707
augmented Lagrangian,
479 481
BFGS, 469
bounded objective function,
675 676
clustering, 691 694
conjugate gradient, 434 437
constrained quasi-Newton, 573
constrained steepest descent,
525 527
covering, 684 685
deterministic, 684 689
DFP, 467 469
domain elimination, 700 702
dynamic rounding-off, 632 633
of generalized descent,
686 688
golf, 687
gradient projection, 566 567
GRG method, 567 569
interval reducing, 423
lexicographic, 674 675
linear programming, 299 410
modified Newton’s, 829
multiplier, 479 481
multistart, 691
neighborhood search, 633
operations analysis of, 702 705
performance, 706 707
performance of stochastic
zooming, 707 708
scalarization, 666
sequential linearization, 629
Simplex, 321 334
stochastic zooming, 702
tunneling, 688 689
two-phase Simplex, 334 347
unconstrained, 472 481
vector, 666
weighted global criterion,
673 674
weighted min-max, 672 673
weighted sum, 671 672
zooming, 685 686
Methods, for linked discrete
variables, 633 635
Methods, miscellaneous numerical
optimization, 564 569
gradient projection method,
566 567
GRG method, 567 569
method of feasibility directions,
564 565
Methods, multi-objective optimum
design concepts and. See
Multi-objective optimum
design concepts and
methods
Methods, Newton’s. See Newton’s
methods
Methods, numerical performance
of, 705 712
DE methods, 707 708
global optimization of structural
design problems, 708 712
performance of methods using
unconstrained problems,
706 707
stochastic zooming method,
707 708
summary, 705 706
Methods, for optimum design,
global concepts and,
681 712
Methods, quasi-Newton. See QuasiNewton methods
Methods, sequential quadratic
programming (SQP). See also
Sequential quadratic
programming
observations on constrained,
561 563
Methods, two local-global
stochastic. See Stochastic
methods, local-global
Methods, unconstrained
optimization. See
Unconstrained optimization
methods
Minima, definitions of global and
local, 96 103
example—constrained minimum,
100
example—constrained problem,
99
example—existence of a global
minimum, 102
example—use of the definition of
maximum point, 101
example—using Weierstrass
theorem, 102
existence of minimum, 102 103
Minimization techniques,
sequential unconstrained,
479
Minimum, existence of, 102 103
Minimum control effort problem,
608 609
Minimum mass
column design for, 286 290
flywheel design for, 290 298
Minimum-weight tubular column,
graphical solution for, 80 81
Min-max method, weighted,
672 673
Mixed variable optimum design
problems (MV-OPT), 620
classification of, 621 622
definition of, 620
Modifications, Marquardt,
465 466
Monte Carlo simulation
(MCS), 781
Motion, optimal control of system,
611 612
Multi-objective optimum design
concepts and methods,
657 680
bounded objective function
method, 675 676
870 INDEXcriterion space and design space,
660 662
example—single-objective
optimization problem, 658
example—two-objective
optimization problem, 659
generation of Pareto optimal set,
666 667
goal programming, 676 677
lexicographic method,
674 675
multi-objective GA, 667 671
normalization of objective
functions, 667
optimization engine, 667
preferences and utility functions,
665 666
problem definition, 657 659
scalarization methods, 666
selection of methods, 677 679
solution concepts, 662 665
terminology and basic concepts,
660 667
vector methods, 666
weighted global criterion
method, 673 674
weighted min-max method,
672 673
weighted sum method,
671 672
Multi-objective GA, 667 671
elitist strategy, 670
niche techniques, 671
Pareto fitness function, 669
Pareto-set filter, 670
ranking, 669
tournament selection, 670 671
VEGA, 668 669
Multiple optimum designs, 77
Multiple performance
requirements, 592 598
asymmetric three-bar structure,
594 598
comparison of solutions, 598
symmetric three-bar structure,
592 594
Multiple solutions, design problem
with, 77 78
Multiplier methods, 479 481
Multipliers, physical meaning of
Lagrange. See Lagrange
multipliers, physical
meaning of
Multistart method, 691
N
Nature-inspired search methods,
215, 412, 713 730
Ant Colony Optimization,
718 727
differential evolution algorithm,
714 718
Particle Swarm Optimization,
727 729
Necessary conditions, for equalityconstrained problem,
130 137
Lagrange multipliers, 131 135
Lagrange multiplier theorem,
135 137
Necessary conditions, for general
constrained problem,
137 153
Karush-Kuhn-Tucker necessary
conditions, 139 152
role of inequalities, 137 139
summary of KKT solution
approach, 152 153
Neighborhood search method, 633
Newton’s methods. See also QuasiNewton methods
classical, 460
example—conjugate gradient
and modified Newton’s
methods, 465
example—use of modified
Newton’s method, 462, 463
Marquardt modification, 465 466
modified, 461 465, 829
Niche techniques, 671
Nonlinear equations, solution of,
475 477
Nonlinear limit state equation,
776 777
Nonlinear programming (NLP), 411
Nonlinear programming, control of
systems by, 598 612
comparison of three
formulations, 611 612
minimization of errors in state
variables, 602 608
minimum control effort problem,
608 609
minimum time control problem,
609 610
optimal control of system
motion, 611 612
prototype optimal control
problem, 598 602
Nonlinear programming, duality
in. See Duality in nonlinear
programming
Nonquadratic case, 483
Normalization, constraint, 496 498
Normalization of variables,
737 739
example—response surface using
normalization procedure,
740
example—response surface using
the normalization procedure,
738
procedure, 737 741
Norm/length of vectors, 10 11
Notation
basic terminology and, 6 13
for constraints, 8 9
summation, 9 10
Null matrix, 787
Numerical algorithms, 415 417
convergence, 417
descent direction and descent
step, 415 417
example—checking for descent
condition, 417
general algorithm, 415
Numerical methods, to compute
step size, 421 430
alternate equal-interval search, 425
equal-interval search, 423 424
general concepts, 421 423
golden section search, 425 430
Numerical methods, for
constrained design. See
Constrained design,
numerical methods for
Numerical methods for constrained
optimum design. See
Constrained optimum
design, numerical methods
for
Numerical methods for optimum
design, 213 218
search methods, classification of,
214 215
simple scaling of variables,
217 218
solution process, 215 217
Numerical methods for
unconstrained optimum
design. See Unconstrained
optimum design, numerical
methods for
INDEX 871Numerical optimization methods,
564 569
gradient projection method,
566 567
GRG method, 567 569
method of feasibility directions,
564 565
Numerical performance of
methods. See Methods,
numerical performance of
O
Objective function contours,
plotting of, 74
Objective functions, normalization
of, 667
Off-diagonal elements, 791 821
Operations analysis of methods,
702 705
Optimal control, versus optimum
design, 6
Optimal control of system motion,
611 612
Optimal control problem,
prototype, 598 602
Optimality, global. See Global
optimality
Optimality, Pareto, 663 664
Optimality conditions
for bound constrained
optimization, 549 550
concepts relating to, 116 117
for functions of single variables,
117 122
Optimality, weak Pareto, 664
Optimal set, generation of Pareto.
See Pareto optimal set,
generation of
Optimization
continuous variable, 636 637
discrete variable, 637 641
engines, 667
Optimization, bound constrained,
549 553
Optimization, graphical. See
Graphical optimization
Optimization, issues in practical
design, 587 588
attributes of good optimization
algorithm, 588
potential constraint
strategy, 587
robustness, 587
selection of algorithm, 587
Optimization, practical applications
of, 575 618
Optimization, practical applications
of, 575 618
discrete variable optimum
design, 636 641
formulation of practical design
optimization problems,
576 582
general-purpose software, use of,
589 590
gradient evaluation for
implicit functions,
582 587
issues in practical design
optimization, 587 588
multiple performance
requirements, 592 598
optimal control of systems by
NLP, 598 612
optimum design of three-bar
structure, 592 598
optimum design of two-member
frame, 590 591
out-of-plane loads, 590 591
structural optimization problems,
alternative formulations for,
612 613
time-dependent problems,
alternative formulations for,
613 617
Optimization, second-order
conditions for constrained.
See Constrained
optimization, second-order
conditions for
Optimization, use of Mathematica
for graphical. See Graphical
optimization, use of
Mathematica for
Optimization, use of MATLAB for
graphical. See Graphical
optimization, use of
MATLAB for
Optimization algorithm, attributes
of good. See Good
optimization algorithm,
attributes of
Optimization algorithms, by
nature-inspired search
methods, 713 730
Optimization methods,
miscellaneous numerical,
564 569
gradient projection method,
566 567
GRG method, 567 569
method of feasibility directions,
564 565
Optimization methods,
unconstrained, 477 481
augmented Lagrangian,
479 481
multiplier, 479 481
sequential unconstrained
minimization techniques,
478 479
Optimization problems, practical
design. See Practical design
problems, formulation of
Optimization problems, types of,
55 64
Optimization Toolbox, 275 277
array operation, 276
matrix operation, 276
scalar operation, 276
variables and expressions,
275 276
Optimum design, 731 784
conventional versus, 4 5
design of experiments for
response surface generation,
741 748
discrete design with orthogonal
arrays, 749 753
example  application of Taguchi
method, 764, 766
example  calculation of
reliability index, 782
example—discrete design with
an orthogonal array, 752
example—generation of a
response surface using an
orthogonal array, 744
example—generation of
quadratic response surface,
735
example—optimization using
RSM, 746
example reliability-based design
optimization, 784
example—response surface using
normalization procedure,
738 739, 740 741
example  robust optimization,
759
general mathematical model for,
50 64
872 INDEXmeta-models for design
optimization, 731 741
RBDO, design under uncertainty,
767 784
robust design approach,
754 766
Optimum design, discrete variable.
See Discrete variable
optimum design concepts
and methods
Optimum design, GA for. See
Genetic algorithms (GA) for
optimum design
Optimum design, global concepts
and methods for, 681 712
basic concepts of solution
methods, 682 684
deterministic methods, 684 689
numerical performance of
methods, 705 712
stochastic methods, 689 698
two local-global stochastic
methods, 699 705
Optimum design, LP methods for.
See Linear programming
methods, for optimum
design
Optimum design, mathematical
model for. See Mathematical
model for optimum design
Optimum design, numerical
methods for constrained.
See also Constrained design,
numerical methods for
approximate step-size
determination, 572
bound-constrained optimization,
549 553
examples—constraint
normalization and status at
point, 497
inexact step size calculation,
537 549
linearization of constrained
problem, 499 506
miscellaneous numerical
optimization methods,
564 569
plate girders optimum design
using Excel Solver, 231 238
potential constraints strategy,
534 537, 587
QP problem, 402 409
QP subproblem, 514 520
quasi-Newton Hessian
approximation, 557 558
search direction calculation,
514 520
SQP, 513 514, 553 563
sequential quadratic
programming methods,
553 563
SLP algorithm, 506 513
step-size calculation subproblem,
520 525
Optimum design, numerical
methods for unconstrained.
See Unconstrained optimum
design, numerical methods
for
Optimum design, with Excel
Solver, 213 274
example—design of a shape for
inelastic LTB, 259
example—design of a shape for
elastic LTB, 261
example—design of noncompact
shape, 262
example—elastic buckling
solution, 249
example—inelastic buckling
solution, 247
example—optimum design of
pole, 268
example—optimum design with
the local buckling constraint,
270
example—optimum design with
the tip rotation constraint,
269
example—selection of W10
shape, 241
example—selection of W8 shape,
242
Excel Solver for LP problems,
225 227
Excel Solver for NLP, optimum
design of springs, 227 231
Excel Solver for unconstrained
optimization problems, 224
numerical methods for optimum
design, 213 218
optimum design of compression
members, 243 250
optimum design of members for
flexure, 250 263
optimum design of plate girders
using Excel Solver, 231 238
optimum design of
telecommunication poles,
263 273
optimum design of tension
members, 238 243
Optimum design concepts, 95 212
alternate form of KKT necessary
conditions, 189 192
basic calculus concepts, 103 115
constrained optimum design
problems, 281 282
engineering design examples,
171 178
exercises, 208 212
global optimality, 159 170
irregular points, 192 194
necessary conditions, for
equality-constrained
problem, 130 137
necessary conditions, for general
unconstrained problem,
137 153
physical meaning of Lagrange
multipliers, 153 159
postoptimality analysis, 153 159
second-order conditions for
constrained optimization,
194 199
sufficiency check for rectangular
beam design problem,
199 201
unconstrained optimum design
problems, 278 280
Optimum design concepts and
methods, discrete variable.
See Discrete variable
optimum design concepts
and methods
Optimum design concepts and
methods, multi-objective.
See Multi-objective optimum
design concepts and
methods
Optimum design examples with
MATLAB. See MATLAB,
optimum design examples
with
Optimum design of compression
members, 243 250, 244t
discussion, 250
example—elastic buckling
solution, 249
example—inelastic buckling
solution, 247
INDEX 873Optimum design of compression
members (Continued)
formulation of problem, 243 247
formulation of problem, for
elastic buckling, 249 250
formulation of problem, for
inelastic buckling, 247 248
Optimum design of members for
flexure, 250 263
data and information collection,
250 254
definition of design variables,
258
deflection requirement, 258 262
example—design of a compact
shape for elastic LTB, 261
example—design of a compact
shape for inelastic LTB, 259
example—design of noncompact
shape, 262
formulation of constraints,
258 262
moment strength requirement,
254 255
nominal bending strength of
compact shapes, 255 256
nominal bending strength of
noncompact shapes,
256 257
optimization criterion, 258
project/problem description, 250
shear strength requirement,
257 258
Optimum design of plate girders
using Excel Solver. See Plate
girders, optimum design
using Excel Solver
Optimum design of
telecommunication poles. See
Telecommunication poles,
optimum design of
Optimum design of tension
members. See Tension
members, optimum design of
Optimum design of three-bar
structure. See Three-bar
structure, optimum design of
Optimum design of two-member
frame. See Two-member
frame, optimum design of
Optimum design problem
formulation, 17 64
design of cabinet, 37 40
design of can, 25 26
design of coil springs, 43 46
design of two-bar bracket, 30 36
general mathematical model for
optimum design, 50 64
insulated spherical tank design,
26 28
minimum cost cylindrical tank
design, 42 43
minimum weight design of
symmetric three-bar truss,
46 50
minimum weight tubular column
design, 40 42
problem formulation process,
18 25
saw mill operation, 28 30
Optimum design problems,
constrained. See Constrained
optimum design problems
Optimum design problems,
unconstrained. See
Unconstrained optimum
design problems
Optimum designs, multiple, 77
Optimum design versus optimal
control, 6
Optimum design with MATLAB.
See MATLAB, optimum
design with
Optimum solution, identification
of, 74
Optimum solutions to LP
problems, 313 314
Order of convergence, 482
Out-of-plane loads, 590 591
P
Parameters, ranging right side,
354 358
Pareto fitness function, 669
Pareto optimality, 663 664
weak, 664
Pareto optimal set, generation of,
666 667
Pareto-set filter, 670
Particle position, 728
Particle Swarm Optimization
(PSO), 727 729
algorithm, 728 729
behavior and terminology, 727 728
Particle velocity, 728
Performance of methods using
unconstrained problems,
706 707
Performance requirements,
multiple, 592 598
Phase I algorithm, 337
Phase II algorithm, 339 345
Phase I problem, definition of,
336 337
Pheromone deposit, 726 727
Pheromone evaporation, 726
Physical programming, 665 666
Pivot step, 316 317
Plate girders, optimum design
using Excel Solver, 231 238
data and information collection,
233 234
definition of design variables,
234
formulation of constraints,
234 235
optimization criterion, 234
project/problem description,
231 233
Solver Parameters dialog box,
237 238
spreadsheet layout, 235 237
Plotting
of function contours, 75 77
functions, 72 73
of objective function contours, 74
Points
constraint status at design,
495 496
sets and, 6 8
utopia, 665
Points, irregular, 192 194
example—check for KKT
conditions at irregular
points, 192
Polynomial interpolation,
444 448
alternate quadratic interpolation,
447 448
quadratic curve fitting, 444 447
Postoptimality analysis, 153 159,
348 375
changes in coefficient matrix,
361 375
changes in resource limits,
348 349
constraint variation sensitivity
result, 159
effect of scaling constraint on
Lagrange multiplier, 158
effect of scaling cost function on
Lagrange multipliers, 157
874 INDEXexample— 5 and $ type
constraints, 352
example— # type constraints,
350, 360
example—effect of scaling
constraint, 158
example—effect of scaling cost
function, 156 157
example—equality and $ type
constraints, 357, 361
example—Lagrange multipliers,
156 157, 158
example—optimum cost
function, 155
example—ranges for cost
coefficients, 360, 361
example—ranges for resource
limits, 356, 357
example—recovery of Lagrange
multipliers for $ type
constraint, 352
example—variations of
constraint limits, 155
ranging cost coefficients,
359 361
ranging right-side parameters,
354 358
recovery of Lagrange multipliers
for $ type constraints, 352
Potential constraint strategy, 587
Practical applications, design
optimization, 575 618
alternative formulations for timedependent problems,
613 617
Practical design optimization,
issues in, 587 588
attributes of good optimization
algorithm, 588
potential constraint strategy, 587
robustness, 587
selection of algorithm, 587
Practical design problems,
formulation of, 576 582
example of practical design
optimization problem,
577 582
example—design of two-member
frame, 612 613
general guidelines, 576 577
Preferences and utility functions,
665 666
Probability density function (PDF),
769 770
Probability of failure, 770 771
Problem formulation, optimum
design. See Optimum design
problem formulation
Problem formulation process,
18 25
data and information collection,
19 20
definition of design variables,
20 21
formulation of constraints,
22 25
optimization criterion, 21 22
project/problem description, 18
Problems. See also Subproblems
classification of mixed variable
optimum design problems,
621 622
concepts related to algorithms
for constrained problems,
492 495
definition of Phase I, 336 337
example of practical design,
577 582
formulation of spring design, 46
graphical solutions for beam
design, 82 94
infeasible, 79 80
integer programming, 40
linear programming, 66,
299, 377
minimum control effort,
608 609
minimum time control, 609 610
MV-OPT, 620
optimum solutions to LP
problems, 313 314
profit maximization, 65 66
prototype optimal control,
598 602
solution to constrained problems,
477 481
sufficiency check for rectangular
beam design, 199 201
with unbounded solutions, 79
Problems, concepts related to linear
programming. See Linear
programming problems,
concepts related to
Problems, constrained optimum
design. See Constrained
optimum design problems
Problems, convex programming,
164 170
Problems, definition of standard
linear programming. See
Linear programming
problem, standard
Problems, formulation of practical
design optimization. See
Practical design problems,
formulation of
Problems, GA for sequencing-type.
See Genetic algorithms (GA),
for sequencing-type
problems
Problems, global optimization of
structural design. See Global
optimization, of structural
design problems
Problems, linearization of
constrained. See Linearization
of constrained problems
Problems, performance of methods
using unconstrained. See
Performance of methods
using unconstrained problems
Problems, QP. See Quadratic
programming (QP) problems
Problems, time-dependent. See
Time-dependent problems
Problems, unconstrained design.
See also Unconstrained
optimum design problems
concepts relating to optimality
conditions, 116 117
example—adding constant to
function, 124
example—cylindrical tank
design, 127
example—effects of scaling, 124
example—local minima for
function of two variables,
125, 129
example—local minimum points
using necessary conditions,
119, 120, 121
example—minimum cost
spherical tank using
necessary conditions, 122
example—multivariable
unconstrained minimization,
279
example—numerical solution of
necessary conditions, 128
example—single-variable
unconstrained minimization,
278
INDEX 875Problems, unconstrained design
(Continued)
example—using necessary
conditions, 119, 127
example—using optimality
conditions, 125, 129
optimality conditions for
functions of several
variables, 122 129
optimality conditions for
functions of single variables,
117 122
Procedures, Gaussian elimination,
796 800
Procedures, gradient evaluation
requires special, 575 576
Process, design, 2 4
Process, problem formulation. See
Problem formulation process
Profit maximization problem,
65 66
Programming
duality in linear, 387 399
goal of, 676 677
physical, 665 666
Programming, control of systems
by nonlinear. See Nonlinear
programming, control of
systems by
Programming problems
convex, 164 170
linear, 56, 299, 305 314
Programs, sample computer, 823
equal interval search, 823 826
golden section search, 826 828
modified Newton’s method, 829
steepest-descent search, 829
Projection method, gradient,
566 567
Prototype optimal control problem,
598 602
Pure random search, 690 691
Q
QP. See Quadratic programming
problems
Quadratic convergence, 482
Quadratic curve fitting, 444 447
Quadratic forms and definite
matrices, 109 115
example—calculations for gradient
of quadratic form, 114
example—calculations for Hessian
of quadratic form, 114
example—determination of form
of matrix, 112, 113
example—matrix of quadratic
form, 110
Quadratic function, 482 483
Quadratic interpolation, alternate,
447 448
Quadratic programming (QP)
problems, 402 409, 514 520
definition of, 402 403, 514 518
derivation of, 554 557
example—solution to QP
subproblem, 519
example—definition of QP
subproblem, 515
example—solution of QP
problem, 406
KKT necessary conditions for,
403 404
Simplex method for solving,
405 409
solution to, 518 520, 569 573
transformation of KKT
conditions, 404 405
Quasi-Newton Hessian
approximation, 557 558
Quasi-Newton methods, 466 472,
484 485
BFGS method, 470 472
DFP method, 467 469
direct Hessian updating,
470 472
example—application of BFGS
method, 471
example—application of DFP
method, 468
inverse Hessian updating,
467 469
observations on constrained,
561 563
Quasi-Newton methods,
constrained. See Sequential
quadratic programming
R
Random search, pure, 690 691
Ranging cost coefficients, 359 361
Ranging right-side parameters,
354 358
Rate of convergence, 417
Rate of convergence of algorithms,
481 485
conjugate gradient method, 484
definitions, 481 482
Newton’s method, 483
quasi-Newton methods, 484 485
steepest-descent method,
482 483
Rectangular beam, design of,
174 187
Rectangular beam design problem,
sufficiency check for,
199 201
Recursive quadratic programming
(RQP), 554. See also
Sequential quadratic
programming
Reducing methods, interval, 422 423
Regions
identification and shading of
infeasible, 73
Reliability-based design
optimization (RBDO), under
uncertainty, 767 784
calculation of reliability index,
774 781
example  calculation of
reliability index, 782
example—reliability-based
design optimization, 784
review of background material
for, 768 774
Reliability index, 773
Representation, design, 645 646
Reproduction, defined, 647 648
Requirements, multiple
performance, 592 598
Response surface method (RSM),
733
example—generation of
quadratic response surface,
735
quadratic response surface
generation, 733 735
Right-side parameters, ranging,
354 358
Robust algorithms, 587
Robust design approach, 754 766
Taguchi method, 761 766
Robust optimization, 754 760
example—robust optimization,
759
mean, 754 755
PDF, 755 756
problem definition,
756 759
standard deviation, 755
variance, 755
876 INDEXRole of inequalities, 137 139
Roots of a set of nonlinear
equations, 222 223
Excel worksheet, 222 223
solution to KKT cases with
Solver, 223
Solver Parameters dialog box,
223
Roots of nonlinear equation,
219 221
Solver Parameters dialog box,
220 221
Rounding-off method, dynamic,
632 633
Row matrix, vector, 787 820
S
SA. See Simulated annealing
Saw mill operation, 28 30
Scalarization methods, 666
Scalar matrix, 791 821
Scalar operation, 276
Scaling of design variables,
456 459
example—effect of scaling of
design variables, 456
Search direction calculation,
514 520
definition of QP subproblem,
514 518
example—definition of QP
subproblem, 515
example—solution to QP
subproblem, 519
solution to QP subproblem,
518 520
Search direction determination,
431 436, 459 466
Searches
alternate equal interval, 425
equal interval, 423 424, 823 826
golden section, 425 430,
826 828
inexact line, 448 449
line search, 522 525
pure random, 690 691
steepest descent, 829
Search method, neighborhood, 633
Search methods, classification of,
214 215
derivative-based, 214
derivative-free, 215
direct search, 214 215
nature-inspired, 215
Second-order conditions for
constrained optimization,
194 199
Second-order information, 194
Sequencing-type problems, GA for,
651 653
Sequential linearization methods,
629
Sequential linear programming
(SLP) algorithm, 506 513
algorithm observations, 512 513
example—sequential linear
programming
algorithm, 509
example—use of sequential
linear programming, 510
move limits in, 506 508
SLP algorithm, 508 512
Sequential quadratic programming
(SQP), 513 514, 553 563,
707
algorithm, 558 561
derivation of QP subproblem,
554 557
descent functions, 563
example—solving spring design
problem using SQP method,
560
example—use of SQP method,
558
observations on, 561 563
option, 590 591
quasi-Newton Hessian
approximation, 557 558
Sequential unconstrained
minimization techniques,
478 479
Set, generation of Pareto optimal,
666 667
Sets, convex, 160 161
Sets and points, 6 8
Simple scaling of variables,
217 218
Simplex algorithms, 384 385
Simplex in two-dimensional space,
321
Simplex method
alternate, 385 386
artificial cost function, 382 383
canonical form/general solution
of Ax 5 b, 308 309
example—Big-M method for
equality and $ type
constraints, 386
example—identification of
unbounded problem with
Simplex method, 333
example—LP problem with
multiple solutions, 331
example—pivot step, 316
example—solution by Simplex
method, 328
example—solution of profit
maximization problem, 329
general solution to Ax 5 b,
377 379
interchange of basic and
nonbasic variables, 316
pivot step, 316, 384
Simplex algorithms, 384 385
steps of, 322
tableau, 378 379
two-phase, 334 347
Simplex method, derivation of,  


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