nonlinear programming peter zörnig isbn: 978-3-11-031527-1 © 2014 by walter de gruyter gmbh,...

Nonlinear Programming Peter ZörnigISBN: 978-3-11-031527-1

© 2014 by Walter de Gruyter GmbH, Berlin/Boston

Abbildungsübersicht / List of Figures

Tabellenübersicht / List of Tables

Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1© 2014 by Walter de Gruyter GmbH, Berlin/Boston

Fig. 1.1. Graphical illustration of the transportation problem.

2


3

Table 1.1. Data of the meteorological observatory.


4

Fig. 1.2. Dispersion diagram.


5

Fig. 1.3. Measurements of a box.


6

Fig. 1.4. Measures of the modified box.


7

Fig. 1.5. Locations of gas stations and demands.


8

Table 1.2. Heights of terrain and road.


9

Fig. 1.6. Terrain and optimal road height.


10

Fig. 1.7. Nonconvex feasible region.


11

Fig. 1.8. Nonconnected feasible region.


12

Fig. 1.9. Optimal point in the interior of M.


13

Fig. 1.10. Local and global minimum points.


14

Fig. 2.1. Feasible and nonfeasible directions.


15

Fig. 2.2. Cone Z(x∗) of Example 2.2 (a).


16

Fig. 2.3. Cone Z(x∗) of Example 2.2 (b).


17

Fig. 2.4. Cone of feasible directions for linear constraints.


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Fig. 2.5. Cone of feasible directions with L0(x∗) ≠ Z(x∗) ≠ L(x∗).


19

Fig. 2.6. Geometric illustration of Example 2.11.


20

Fig. 3.1. Line segment [x1, x2].


21

Fig. 3.2. Convex and nonconvex sets.


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Fig. 3.3. Convex polyhedral sets.


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Fig. 3.4. Extreme points of convex sets.


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Fig. 3.5. Convex hull of a finite set.


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Fig. 3.6. Convex hulls of nonfinite sets.


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Fig. 3.7. Simplex of dimension 2 and 3.


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Fig. 3.8. Illustration of Theorem 3.12.


28

Fig. 3.9. Convex and concave functions.


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30

Fig. 3.11. The epigraph of a function.


31

Fig. 3.12. Supporting hyperplanes of convex sets.


32

Fig. 3.13. Support function of f at x∗.


33

Fig. 3.14. Support functions of the function f of Example 3.35.


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Fig. 3.15. The function f (x1, x2) = |x1| + |x2|.


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36

Fig. 3.17. Minimum point of a concave function.


37

Fig. 4.1. Nonregular point x∗ = 0.


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Fig. 4.2. Illustration of the Farkas lemma.


39

Fig. 4.3. Geometrical illustration of KKT conditions.


40

Fig. 4.4. Illustration of the relation (4.22).


41

Fig. 4.5. Illustration of Example 4.8.


42



43

Fig. 4.7. Saddle point.


44

Fig. 4.8. Geometric illustration of (P).


45

Fig. 4.9. Geometric illustration of (D).


46

Fig. 4.10. Function h1(x1) for various values of u.


47

Fig. 4.11. Dual objective function.


48

Fig. 4.12. Geometrical solution of Example 4.47 (case a).


49

Fig. 4.13. Geometrical solution of Example 4.47 (case b).


50



51

Fig. 6.1. Unimodal function.


52

Fig. 6.2. Locating of x.


53

Fig. 6.3. Construction of the new interval [ak+1, bk+1].


54

Table 6.1. Golden Section.


55

Fig. 6.4. Iteration of Newton’s method.


56

Fig. 6.5. Construction of a useless value.


57

Fig. 6.6. Cycling of Newton’s method.


58

Fig. 6.7. Interpolation polynomials of Example 6.17.


59

Fig. 6.8. Interpolation polynomials of Example 6.20.


60

Fig. 7.1. Orthogonal coordinate transformations.


61

Fig. 7.2. Level curves of the function 2x21 + 4x1x2 + 5x2

2.


62

Fig. 7.3. Level surface z = 6 for the function 7x21 − 2x1x2 + 4x1x3 + 7x2

2 − 4x2x3 + 10x2

3.


63

Table 7.1. Gradient method.


64

Fig. 7.4. Minimization process of the gradient method.


65

Table 7.2. Conjugate gradient method (quadratic problem).


66

Table 7.3. Conjugate gradient method (nonquadratic problem).


67

Fig. 7.5. Solution of Example 7.30 by the conjugate gradient method.


68

Table 7.4. DFP method.


69

Table 7.5. Cyclic minimization.


70

Fig. 7.6. Solution of Example 7.38.


71

Table 7.6. Inexact line search (case a).


72

Fig. 7.7. Convergence to a nonoptimal point.


73

Table 7.7. Inexact line search (case b).


74

Fig. 7.8. Oscillation of the algorithm.


75

Fig. 7.9. Reversal strategy.


76

Fig. 7.10. Solution of the quadratic subproblem.


77

Fig. 7.11. Trust region method.


78

Fig. 8.1. Construction of the search direction.


79

Fig. 8.2. Resolution of Example 8.2 by Rosen’s method.


80

Fig. 8.3. Optimal directions.


81

Fig. 8.4. Zigzag movement.


82

Table 8.1. Pivoting of Example 8.17.


83

Fig. 9.1. Active set method.


84

Table 9.1. Initial tableau of Lemke’s method.


85

Table 9.2. Pivot steps of Example 9.4.


86

Table 9.3. Pivot steps of Example 9.5.


87

Fig. 9.2. Unbounded optimal solution.


88

Fig. 9.3. Locations of gas stations.


89

Fig. 10.1. Penalty terms.


90

Fig. 10.2. Function q(x, r) of Example 10.2.


91

Fig. 10.3. Function q(x, r) of Example 10.2.


92

Fig. 10.4. Penalty method.


93

Fig. 10.5. Exact penalty function (10.12).


94

Fig. 10.6. Illustration of robustness.


95

Fig. 10.7. Barrier terms.


96

Fig. 10.8. Barrier function s(x, c) of Example 10.16.


97

Table 10.1. Solution of Example 10.16.


98

Table 10.2. Solution of Example 10.17.


99

Fig. 10.9. Barrier method.


100

Fig. 11.1. Partial cost function.


101

Fig. 11.2. Nondifferentiable function of a minimax problem.


102

Fig. 11.3. Nondifferentiable bidimensional objective function.


103

Fig. 11.4. Graphical solution of problem (11.9).


104

Fig. 11.5. Objective function of problem (11.8).


105

Fig. 11.6. Minimization of function (11.10) with the

gradient method.


106

Fig. 11.7. Minimization of function (11.11) with the

gradient method.


107

Fig. 11.8. Piecewise linear function f and differential approximation g for ε = 0.1.


108

Table 11.1. Minimization by differentiable approximation.


109

Fig. 11.9. Local and global minima of problem (11.28).


110

Fig. 11.10. D.c. function.


111

Fig. 11.11. Branch-and-bound method.


112

Fig. 11.11. Branch-and-bound method (continuation).


113

Fig. 11.12. Subdivisions of the feasible region.


114

Fig. 11.13. Optimization problem (11.36).


115

Table 11.2. Population P(0).


116

Table 11.3. Population P(1).


117

Fig. 2.7. Feasible set and gradients.


118

Fig. 2.8. Feasible region and gradients.


119

Fig. 2.9. Feasible directions at x∗ in diverse cases.


120

Fig. 3.18. Algebraic sum.


121

Fig. 3.19. Feasible direction.


122

Fig. 3.20. Discontinuous concave function.


123

Fig. 3.21. Convex function.


124

Fig. 3.22. Geometrical illustration of the subdifferential.


125

Fig. 3.23. Dispersion diagram.


126

Fig. 3.24. Feasible region.


127

Fig. 4.15. Regular point x∗ = 0 with linearly dependent gradients.


128

Fig. 4.16. Geometric resolution of Exercise 4.9.


129

Fig. 4.17. Geometric solution of Exercise 4.15.


130



131



132

Table 5.1. Error sequence.


133

Table 5.2. Linear and quadratic convergence.


134

Table 6.2. Golden Section.


135

Table 6.3. Bisection algorithm.


136

Fig. 6.9. Premature convergence of the bisection method.


137

Fig. 6.10. Hermite interpolation polynomial.


138

Fig. 7.12. Level curves of Exercise 7.7.


139

Fig. 7.13. Displaced ellipses.


140

Fig. 7.14. Level curves of quadratic functions.


141

Table 7.8. Gradient method.


142

Table 7.9. Newton’s method.


143

Fig. 7.15. Minimization process of Newton’s method.


144

Table 7.10. Conjugate gradient method.


145



146

Table 7.12. Conjugate gradient method.


147



148

Table 7.14. Variant of the DFP method.


149

Table 7.15. Cyclic minimization.


150

Fig. 7.16. Level curves of the first quadratic model of

Example 7.50.


151

Fig. 7.17. Level curves of the function of Exercise 7.54

(b).


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153

Fig. 8.6. Feasible region.


154

Fig. 8.7. Zoutendijk’s method.


155

Fig. 8.8. Graphical illustration of both methods.


156

Table 8.2. Pivoting of Example 8.18.


157

Table 9.4. Lemke’s method.


158

Table 9.5. Lemke’s method.


159

Table 10.3. Barrier method.


160

Fig. 10.10. Solution process.


161

Table 10.4. Barrier method.

nonlinear programming peter zörnig isbn: 978-3-11-031527-1 © 2014 by walter de gruyter gmbh,...

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