COPYRIGHTED MATERIAL. Index

Transcription

1 Rao bindex.tex V3-05/21/ :46am Page 803 Index A Absolute minimum, 63 Active constraint, 8, 94 Addition of constraints, 218 Addition of new variables, 214 Additive algorithm, 605 Adjoint equations, 682 Adjoint variable, 682 Admissible variations, 78 All-integer problem, 588 Analytical methods, 253 Answers to selected problems, 795 Ant colony optimization, 3, 693, 714 algorithm, 717 ant searching behavior, 715 basic concept, 714 evaporation, 716 path retracing, 715 pheromone trail, 715 pheromone updating, 715 Applications of geometric programming, 525 Approximate mean, 642 Approximate variance, 642 Arithmetic-geometric inequality, 500 Artificial variables, 139 Augmented Lagrange multiplier method, 459 equality-constrained problems, 459 inequality-constrained problems, 462 mixed equality-inequality-constrained problems, 463 Augmented Lagrangian function, 460 Availability of computer programs, 786 Average, 635 B Balas algorithm, 604 Balas method, 589, 604 Barrier methods, 433 Basic feasible solution, 131, 136 Basic set operations, 724 Basic solution, 130, 136 Basic variables, 136 Basis, 130 Basis vector approach, 743 Beale s function, 365 Beam-column, 55 Bearing, 531 Behavior constraints, 7 BFGS formula, 353 BFGS method, 360 Bias of random directions, 312 Binary numbers, 607 Binary programming, 624 Binary variables, 607 Bivariate distribution, 639 Boltzmann s constant, 703 Boltzmann s probability distribution, 703 Boundary value problem, 549 Bounded objective function method, 764 Bound point, 8 Brachistochrone problem, 671 Bracket function, 443, 696 Branch and bound method, 609 Branching, 610 Brown s badly scaled function, 365 Broydon-Fletcher-Goldfarb-Shanno method, 304, 360 COPYRIGHTED MATERIAL C Calculus methods, 3 Calculus of variations, 3, 668 Canonical form, 133 Cantilever beam, 527 Cauchy method, 304, 339 Cauchy s inequality, 500 Central limit theorem, 647 Chance constrained programming,

2 Rao bindex.tex V3-05/21/ :46am Page Index Change in constraint coefficients, 215 Change in cost coefficients, 212 Change in right hand side constants, 208 Characteristics of constrained problem, 380 Choice of method, 784 Circular annular plate, 690 Classical optimization techniques, 63 Classification: of optimization problems, 14 of unconstrained minimization methods, 304 Classification of optimization problems based on: deterministic nature of variables, 29 existence of constraints, 14 nature of design variables, 15 nature of equations involved, 19 number of objective functions, 32 permissible values of design variables, 28 physical structure of the problem, 16 separability of the functions, 30 Closed half space, 128 Cluster analysis, 3 Coefficient of variation, 636 Collapse mechanism, 121 Comparison: of constrained methods, 785 of elimination methods, 272 of methods, 294 of unconstrained methods, 784 Complementary geometric programming, 520 degree of difficulty, 523 solution procedure, 522 Complement of a fuzzy set, 724 Complex method, 384 Composite constraint surface, 8 Computational aspects of optimization, 784 Computer programs, availability of, 786, 788 Computer program for: ant colony optimization, 789 fuzzy logic toolbox, 788 genetic algorithm and direct search toolbox, 788 modern optimization methods, 788 multiobjective optimization, 789 neural network toolbox, 788 particle swarm optimization, 788 simulated annealing algorithm, 788 Concave function, 779 Concept of cutting plane, 591 Concept of suboptimization, 549 Concrete beam, 29 Condition number of a matrix, 306 Cone clutch, 49, 528 Conjugate directions, 319 Conjugate gradient method, 341, 355, 361 Consistency condition, 220 Constrained minimization (GMP), 508 Constrained optimization problem, 6, 380 characteristics, 380 Constrained optimization techniques, 380 Constrained variation, 77, 79 Constraint qualification, 98 Constraint surface, 8 Contact stress between cylinders, 297 Contact stress between spheres, 250 Continuous beams, 576 Continuous dynamic programming, 573 Continuous feasible solution, 609 Continuous random variable, 634 Contours of objective function, 10 Contraction, 332 Contraction coefficient, 332 Control variables, 16 Control vector, 678 Convergence of constrained problems, 464 Convergence of order p, 305 Conversion of final to initial value problem, 566 Conversion of nonserial to serial system, 548 Convex: function, 779 polygon, 127 polyhedron, 127, 129 polytope, 129 programming problem, 98, 104, 442 set, 129

3 Rao bindex.tex V3-05/21/ :46am Page 805 Index 805 Cooling fin, 675 Correlation, 640 Correlation coefficient, 640 Correlation matrix, 646 Covariance, 640 Covariance matrix, 648 CPM and PERT, 3 Crane hook, 60 Crisp set theory, 723 Criterion function, 9 Critical points, 750 Crossover, 699 Cubic interpolation method, 253, 280 Cumulative distribution function, 634 Curse of dimensionality, 573 Curve of minimum time of descent, 672 Cutting plane method, 390, 589 algorithm, 387 geometric interpretation, 388 Cyclic process, 331 Cylinders in contact, 297 D Darcy-Weisbach equation, 663 Darwin s theory, 694 Davidon-Fletcher-Powell method, 304, 354 DC motor, 53 Decision variables, 6 Decomposition principle, 177, 200 Degenerate solution, 142 Degree of difficulty, 496 Derivatives of eigenvalues and eigenvectors, 747 of static displacements and stresses, 745 of transient response, 749 Descent direction, 336 Descent methods, 304, 335 Design constraints, 7 Design equations, 547 Design of: cantilever beam, 527 column, 11 cone clutch, 528 continuous beams, 576 drainage system, 579 four bar mechanism, 535 gear train, 579 helical spring, 529, 659 hydraulic cylinder, 527 lightly loaded bearing, 531 planar truss, 249 two bar truss, 533 Design of experiments, 3 Design point, 7 Design space, 7 Design variable linking, 738 Design variables, 6 Design vector, 6 DFP formula, 352 DFP method, 354 Dichotomous search, 253, 257 Differential calculus methods, 253, 493 Differential of f, 68 Direction finding problem, 395 Direct methods, 380, 383 Direct root method, 253, 286 Direct search methods, 309 Direct substitution, 76 Discrete programming problem, 588 Discrete random variable, 634 Discriminate analysis, 3 Drainage system, 579 Dual function, 501 Duality in linear programming, 192 Duality theorems, 195 Dual problem, 192, 509 Dual simplex method, 195 Dynamic optimization problem, 16 Dynamic programming, 3, 544 applications, 576 calculus method of solution, 555 computational procedure, 553 continuous, 573 conversion of final to initial value problem, 566 problem of dimensionality, 572 recurrence relation, 551 tabular method of solution, 560 E Electrical bridge network, 52 Elementary operations, 133 Elimination methods, 253, 254

4 Rao bindex.tex V3-05/21/ :46am Page Index Elimination methods comparison, 271 Engineering applications of optimization, 5 Engineering optimization literature, 35 Equality constraints, 6 Euler equation, 671 Euler-Lagrange equation, 671 Evaluation of gradient, 337 Event, 633 Exhaustive search, 253, 256 Expansion, 331 Expansion coefficient, 332 Expected value, 635 Experiment, 633 Extended interior penalty function, 451 Exterior penalty function method, 443, 455 algorithm, 445 convergence proof, 447 mixed equality-inequality constraints, 455 parametric constraints, 459 Extrapolation of design vector, 448 Extrapolation of objective function, 450 Extrapolation technique, 447 Extreme point, 130 F Factor analysis, 3 Failure mechanisms of portal frame, 246 Fast reanalysis techniques, 740 Fathomed, 610 Feasible direction, 95, 393 Feasible direction methods, 393, 394 Feasible solution, 130 Feasible space, 8 Fibonacci method, 253, 263 Fibonacci numbers, 263 Final interval of uncertainty, 263, 272 Final value problem, 548 First level problem, 757 First order methods, 305 Fitness, 696 Fletcher and Powell s helical valley function, 364 Fletcher-Reeves method, 304, 341 algorithm, 343 Floor design, 542 Flow chart: for augmented Lagrange multiplier method, 461 for cubic interpolation method, 284 for Fibonacci search method, 266 for linear extended penalty function method, 453 for parallel simulated annealing, 762 for Powell s method, 325 for simplex algorithm, 143 for simplex method, 153 for simulated annealing, 706 for two-phase simplex method, 153 Flywheel design, 482 Forced boundary conditions, 671 Four bar mechanism, 535 Four bar truss, 60, 557 Free boundary conditions, 671 Free point, 8 Freudenstein and Roth function, 364 Functional, 669 Functional constraints, 7 Function of a random variable, 638 Function of several random variables, 640 mean, 641 variance, 641 Fuzzy decision, 726 Fuzzy feasible region, 822 Fuzzy optimization, 3, 693, 722 computational procedure, 726 fuzzy set theory, 722 Fuzzy systems, 722, 725 G Game theory, 3 Gaussian distribution, 643 Gear train, 579 General iterative scheme of optimization, 305 Generalized penalty function method, 619 Generalized reduced gradient, 415 Generalized reduced gradient method, 412 algorithm, 416 General primal dual relations, 193

5 Rao bindex.tex V3-05/21/ :46am Page 807 Index 807 Genetic algorithms, 3, 693, 694, 701 Genetic operators, 697 Geometric boundary conditions, 671 Geometric constraints, 7 Geometric programming, 3, 22, 492 applications, 525 arithmetic-geometric inequality, 500 complementary geometric programming, 520 constrained problem, 508, 509 degree of difficulty, 496 mixed inequality constraints, 518 normality condition, 495 orthogonality conditions, 495 primal dual relations, 501 unconstrained problem, 493 Geometry of linear programming problems, 124 Global criterion method, 764 Global minimum, 63 Goal programming method, 765 Golden mean, 270 Golden section, 270 Golden section method, 253, 267 Gomory s constraint, 592 Gomory s cutting plane method, 591 for all integer problem, 592 graphical representation, 589 for mixed integer problem, 599 Gradient, 95, 335 Gradient evaluation, 337 Gradient of a function, 335 Gradient methods, 335 Gradient projection method, 404 algorithm, 409 Graphical optimization, 10 Graphical representation, 589 Grid search method, 304, 314 H Hamiltonian, 682 Helical spring, 22, 529 Helical torsional spring, 541 Hessian matrix, 71, 302 Heuristic search methods, 381 Historical development, 3 Hitchcock-Koopman s problem, 221 Hollow circular shaft, 51 Hopfield network, 729 Huang s family of updates, 353 Hydraulic cylinder design, 527 Hyperplane, 128 I Identifying optimal point, 140 Ill conditioned matrix, 306 Improving nonoptimal solution, 141 Inactive constraint, 94 Incremental response approach, 740 Independent events, 633 Independent random variables, 639 Indirect methods, 335, 380, 428 Indirect updated method, 361 Inequality constraints, 6, 93 Infeasibility form, 152 Infinite number of solutions, 148 Inflection point, 65 Initial value problem, 548 Input state variables, 546 Integer feasible solution, 610 Integer lattice points, 591 Integer linear programming, 589 Integer nonlinear programming, 606 Integer polynomial programming, 606 Integer programming, 3, 28, 588 Interior method, 222 Interior penalty function method, 432, 454 convergence proof, 438 extrapolation technique, 447 iterative process, 433 penalty parameter, 435 starting feasible point, 434 Interpolation methods, 253, 271 Interpretation of Lagrange multipliers, 90 Intersection of convex sets, 131 Intersection of fuzzy sets, 725 Interval halving method, 260 Interval of uncertainty, 256, 263 Introduction to optimization, 1 Inverse update formulas, 353 Inverted utility function method, 764 Iterative process of optimization, 252

6 Rao bindex.tex V3-05/21/ :46am Page Index J Jacobian, 82 Joint density function, 639 Joint distribution function, 639 Jointly distributed random variables, 639 Joint normal density function, 646 K Karmarkar s method, 222 algorithm, 226 conversion of problem, 224 statement of problem, 223 Kohonen network, 729 Kuhn-Tucker conditions, 98, 401 testing, 465 L Lagrange method, 85 necessary conditions, 86 sufficiency conditions, 87 Lagrange multipliers, 85, 675, 682 Lagrangian function, 86, 230 Learning process, 728 Lexicographic method, 765 Limit design of frames, 120 Linear convergence, 305 Linear extended penalty function, 451 Linearization of constraints, 387 Linearization of objective, 387 Linear programming, 3, 26, 119, 177 additional topics, 177 applications, 120 definitions, 127 theorems, 127 two phases, 150 Linear programming problem, 26 as a dynamic programming problem, 569 geometry, 124 infinite solutions, 126, 148 matrix form, 122 scalar form, 122 standard form, 122 unbounded solution, 127, 146 Linear simultaneous equations, 133 Line segment, 128, 202 Local minimum, 63 M Machining economics problem, 525 Marginal density function, 639 Markov processes, 3 Marquardt method, 304, 348 Mathematical programming techniques, 1, 3 MATLAB, 791 creating m-files, 793 defining matrices, 792 features, 791 introduction, 791 optimization toolbox, 793 special characters, 791 using programs, 794 MATLAB solutions, 37 binary programming problem, 624 constrained optimization problem, 474 goal attainment method, 767 interior point method, 235 linear programming problem, 156 one-dimensional problem, 294 quadratic programming problem, 237 unconstrained optimization problem, 365 Matrix methods of structural analysis, 248 Maxwell distribution, 663 Mean, 635 Membership function, 723 Merit function, 9 Method of constrained variation, 77 Method of Lagrange multipliers, 85 Methods of feasible directions, 393 Methods of operations research, 2 Metropolis criterion, 703 MIMD architecture, 761 Minimum cost pipeline, 585 Minimum drag, 672 Mixed constraints, 453 exterior penalty function method, 455 interior penalty function method, 454 Mixed equality and inequality constraints, 453 Mixed integer problem, 588 Model coordination method, 755 Modern optimization techniques, 3, 4, 693 Monotonicity, 547

7 Rao bindex.tex V3-05/21/ :46am Page 809 Index 809 Motivation of simplex method, 138 Multibay cantilever truss, 578 Multilayer feedforward network, 729 Multilevel optimization, 755 Multimodal function, 254 Multiobjective optimization, 761 Multiobjective programming, 3, 9, 33 Multiobjective programming problem, 9, 33 Multiple objective functions, 9 Multistage decision problem, 544, 548 Multistage decision process, 545 Multivariable optimization, 68 with equality constraints, 75 with inequality constraints, 93 necessary conditions, 69, 81, 94, 98 with no constraints, 68 sufficiency conditions, 70, 83 Multivariate distribution, 639 Mutation, 700 N Natural boundary conditions, 671 Necessary conditions for optimal control, 679 Negative definite matrix, 71 Network methods, 3 Neural network based methods, 693, 727 Neural networks, 3, 728 Neuron, 728 Newton method, 253, 286, 304, 345 Newton Raphson method, 287 Node, 610 Nonbasic variables, 136 Nonconvex sets, 129 Nondegenerate solution, 131 Nongradient methods, 304 Nonlinear programming, 3, 248, 301, 380 Nonlinear programming problem, 19 Nonpivotal variables, 135 Nontraditional optimization techniques, 3 Normal distribution, 643 Normality condition, 495 Normalization condition, 635 Normalization of constraints, 436 Normalized beta function integrand, 620 Norm of a matrix, 306 Norm of a vector, 305 Number of experiments, 272 Numerical integration, 458 O Objective function, 6, 9 Objective function surfaces, 9 Offspring, 700 One degree of difficulty problem, 515, 526, 532 One dimensional minimization methods, 248 Operations research, 1, 3 Optimal basic solution, 131 Optimal control problem, 16 Optimal control theory, 668, 678 Optimality criteria, 683 multiple displacement constraints, 684 single displacement constraint, 683 Optimality criteria methods, 668, 683 Optimal layout of a truss, 577 Optimal solution, 131 Optimization, 1 Optimization of fuzzy systems, 722, 725 Optimization problems classification, 14 statement, 6 Optimization techniques, 3, 35 Optimization toolbox, 36 Optimum machining conditions, 525 Orthogonal directions, 319 Orthogonality conditions, 495 Output state variables, 546 Overachievement, 766 P Parallel processing, 760 Parallel simulated annealing, 761 Parameter optimization problem, 15 Parametric constraint, 456 Parametric programming, 207 Parent, 700 Pareto optimum solution, 763 Particle swarm optimization, 3, 693, 708 computational implementation, 709 inertia term, 710 inertia weight, 711

8 Rao bindex.tex V3-05/21/ :46am Page Index Particle swarm optimization (continued) particle, 708 position, 708 velocity, 708 Pattern directions, 318 Pattern recognition, 3 Pattern search methods, 304 Penalty function, 696 Penalty function method: basic approach, 430 convergence criteria, 435 convergence proof, 438, 447 exterior method, 443 extrapolation, 447 initial value of parameter, 435 interior method, 432 iterative process, 433, 445 mixed equality and inequality constraints, 453 normalization of constraints, 436 parametric constraints, 456 penalty parameter, 431 starting feasible point, 434 Penalty parameter, 431 Performance index, 16, 679 Perturbing the design vector, 465 Phase I of simplex method, 151 Phase II of simplex method, 152 Pivot operation, 134 Pivot reduction, 135 Point in n-dimensional space, 128 Polynomial programming problem, 589 Population, 697 Positive definite matrix, 71 Positive semidefinite matrix, 71 Post optimality analysis, 207 Posynomial, 22, 492 Powell s badly scaled function, 365 Powell s method, 304, 319 algorithm, 323 convergence criterion, 326 flow chart, 325 Powell s quartic function, 364 Power screw, 57 Practical aspects of optimization, 737 Practical considerations, 293 Preassigned parameters, 6 Precision points, 535 Predual function, 500 Pressure vessel, 59 Primal and dual programs, 505, 510 Primal dual relations, 193, 501 Primal function, 500 Primal problem, 192, 509 Principle of optimality, 549 Probabilistic programming, 632 Probability, definition, 632, 633 Probability density function, 633 Probability distribution function, 634 Probability distributions, 643 Probability mass function, 634 Probability theory, 632 Problem of dimensionality, 572 Projected Lagrangian method, 425 Projection matrix, 405 Proportional damping, 749 Pseudo dual simplex method, 606 Q Quadratically convergent method, 319 Quadratic convergence, 287, 305, 325 Quadratic extended penalty function, 452 Quadratic form, 71 Quadratic interpolation method, 253, 273 Quadratic programming, 3, 24, 229 Quasi-Newton condition, 304 Quasi-Newton method, 253, 288, 350 Queueing theory, 3 R Railroad track, 48 Random jumping method, 311 Random search methods, 304, 309, 383 Random variables, 633 Random walk with direction exploitation, 313 Random walk method, 312 Rank 1 updates, 351 Rank 2 updates, 352 Rate of change of a function, 338 Rate of convergence, 305 Real valued programming problem, 28 Reanalysis, 740

9 Rao bindex.tex V3-05/21/ :46am Page 811 Index 811 Reciprocal approximation, 685 Recurrence relationship, 551 Reduced basis technique, 737 Reduction ratio, 265 Reduction of size, 737 Refitting, 277 Reflection, 328 Reflection coefficient, 330 Reflection process, 329 Regression analysis, 3 Regular simplex, 328 Relative frequency of occurrence, 636 Relative minimum, 63 Reliability theory, 3 Renewal theory, 3 Reproduction, 697 Reservoir pump installation, 505 Reservoir system, 160 Return function, 546 Revised simplex method, 177 step-by-step procedure, 182 theoretical development, 178 Rigid frame, 120 Rocket in outer space, 16 Rosenbrock s parabolic valley function, 363 Rosen s gradient projection method, 404 algorithm, 409 determination of step length, 407 projection matrix, 405 Roulette-wheel selection scheme, 698 S Saddle point, 73 Scaffolding system, 26, 57 Scaling of constraints, 787 Scaling of design variables, 305, 787 Search with accelerated step size, 255 Search with fixed step size, 254 Secant method, 253, 290 Second level problem, 757 Second order methods, 305 Semidefinite matrix, 71 Sensitivity analysis, 207 Sensitivity equations, 751 using Kuhn-Tucker conditions, 752 using the concept of feasible direction, 754 Sensitivity of optimum solution, 751 Sensitivity to problem parameters, 751 Separability, 547 Separability of functions, 31 Separable function, 31 Separable programming, 3, 31 Sequential decision problem, 544 Sequential linear discrete programming, 614 Sequential linear integer programming, 614 Sequential linear programming, 387 geometric interpretation, 388 Sequential quadratic programming, 422 derivation, 422 solution procedure, 425 Sequential unconstrained minimization, 431 Serial multistage decision process, 545 Shadow prices, 92 Shell and tube heat exchanger, 51 Side constraints, 7 Sigmoid function, 728 Signum function, 509 Simplex, 328, 384 Simplex algorithm, 139 flow chart, 143 Simplex method, 138, 304, 328 flow chart, 153 two phases, 150 Simplex multipliers, 180 Simply supported beam, 58 Simulated annealing, 3, 693, 702 Simulation methods, 3 Simultaneous equations, 133 Simultaneous search method, 257 Single stage decision problem, 546 Single variable optimization, 63 Slack variable, 123 Slider crank mechanism, 49 Solid body of revolution, 672 Solution by direct substitution, 76 Solution of linear equations, 133 Spring-cart system, 71 Stamping of circular discs, 49

10 Rao bindex.tex V3-05/21/ :46am Page Index Standard deviation, 635, 636 Standard form of LP problem, 122 Standard normal distribution, 643 Standard normal tables, 643, 645 Starting feasible point, 434 State inversion, 566 Statement of an optimization problem, 6 State transformation, 546 State variables, 16 State vector, 679 Statically determinate truss, 542 Static optimization problem, 15 Stationary point, 65 Stationary values of functionals, 669 Statistical decision theory, 3 Statistically independent events, 633 Statistical methods, 3 Steepest ascent direction, 335 Steepest descent method, 304, 339 convergence criteria, 341 Step-cone pulley, 19 Step length determination, 398, 407 Stochastic process techniques, 3 Stochastic programming, 3, 29, 632 geometric programming, 659 linear programming, 647 nonlinear programming, 652 Structural error, 535 Structural optimization packages, 787 Suboptimization, 551 SUMT, 431 Superlinear convergence, 305, 361 Surplus variable, 124 Survival of the fittest, 696 Symmetric primal-dual relations, 192 System reliability, 583 T Taylor s series expansion, 68, 642 Tentative solution, 670 Termination criteria, 401 Test functions (unconstrained nonlinear programming), 363 Beale s function, 365 Brown s badly scaled function, 365 Fletcher and Powell s helical valley, 364 Freudenstein and Roth function, 364 Powell s badly scaled function, 365 Powell s quartic function, 364 Rosenbrock s parabolic valley, 363 Wood s function, 365 Testing for concavity, 779 Testing for convexity, 779 Testing Kuhn-Tucker conditions, 465 Test problems (constrained nonlinear programming), 467 heat exchanger, 473 speed reducer (gear train), 472 three-bar truss, 467 welded beam, bar space truss, 468 Trajectory optimization problem, 16 Transformation techniques, 428 Transformation of variables, 381, 428 Transportation array, 221 Transportation problem, 177, 220 Transportation technique, 221 Transversality conditions, 682 Trapezoidal rule, 458 Travelling salesperson, 52 Trial, 254 Truss, 47, 55, 60, 248, 481, 487, 578, 623, 686, 691, 692, 741, 745, 758, 772 Tubular column design, 10, 654 Two-bar truss, 47, 55, 487, 533, 758 Two degree of difficulty problem, 526 Two phases of simplex method, 150 Two stage compressor, 67 Types of multistage decision problems, 548 U Unbounded solution, 127, 142, 146 Unconstrained minimization (GMP), 493 Unconstrained optimization problem, 6, 301 Unconstrained optimization techniques, 301

11 Rao bindex.tex V3-05/21/ :46am Page 813 Index 813 Underachievement, 766 Uniform distribution, 643 Unimodal function, 253 Union of fuzzy sets, 725 Univariate distribution, 639 Univariate method, 304, 315 Unrestricted search, 253, 254 Usable direction, 97 Usable feasible direction, 393 Utility function method, 763 V Valuation set, 723 Variable metric method, 354, 361 Variance, 636 Variation, 669 Variational operator, 669 Vector minimization problem, 762 Vector of simplex multipliers, 180 Venn diagram, 723 Vertex, 130 W Water resource system, 241 Water tank design, 550 Weighting function method, 764 Weights, 501 Well conditioned matrix, 306 Wood s function, 365 Z Zero degree of difficulty problem, 511, 525, 531 Zero-one LP problem, 608 Zero-one polynomial programming, 608 Zero-one problem, 588 Zero-one programming problems, 588, 604 Zeroth order methods, 304 Zoutendijk s method, 381, 394 determination of step length, 398 direction finding problem, 395 termination criteria, 401

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