1. Nonlinear Programming Peter Zörnig ISBN: 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston Abbildungsübersicht / List of Figures Tabellenübersicht / List of Tables

2. 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. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 3 Table 1.1. Data of the meteorological observatory.

4. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 4 Fig. 1.2. Dispersion diagram.

5. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 5 Fig. 1.3. Measurements of a box.

6. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 6 Fig. 1.4. Measures of the modified box.

7. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 7 Fig. 1.5. Locations of gas stations and demands.

8. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 8 Table 1.2. Heights of terrain and road.

9. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 9 Fig. 1.6. Terrain and optimal road height.

10. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 10 Fig. 1.7. Nonconvex feasible region.

11. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 11 Fig. 1.8. Nonconnected feasible region.

12. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 12 Fig. 1.9. Optimal point in the interior of M.

13. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 13 Fig. 1.10. Local and global minimum points.

14. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 14 Fig. 2.1. Feasible and nonfeasible directions.

15. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 15 Fig. 2.2. Cone Z(x ∗ ) of Example 2.2 (a).

16. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 16 Fig. 2.3. Cone Z(x ∗ ) of Example 2.2 (b).

17. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 17 Fig. 2.4. Cone of feasible directions for linear constraints.

18. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 18 Fig. 2.5. Cone of feasible directions with L 0 (x ∗ ) ≠ Z(x ∗ ) ≠ L(x ∗ ).

19. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 19 Fig. 2.6. Geometric illustration of Example 2.11.

20. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 20 Fig. 3.1. Line segment [x 1, x 2 ].

21. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 21 Fig. 3.2. Convex and nonconvex sets.

22. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 22 Fig. 3.3. Convex polyhedral sets.

23. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 23 Fig. 3.4. Extreme points of convex sets.

24. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 24 Fig. 3.5. Convex hull of a finite set.

25. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 25 Fig. 3.6. Convex hulls of nonfinite sets.

26. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 26 Fig. 3.7. Simplex of dimension 2 and 3.

27. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 27 Fig. 3.8. Illustration of Theorem 3.12.

28. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 28 Fig. 3.9. Convex and concave functions.

29. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 29 Fig. 3.10. Illustration of Theorem 3.25.

30. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 30 Fig. 3.11. The epigraph of a function.

31. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 31 Fig. 3.12. Supporting hyperplanes of convex sets.

32. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 32 Fig. 3.13. Support function of f at x ∗.

33. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 33 Fig. 3.14. Support functions of the function f of Example 3.35.

34. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 34 Fig. 3.15. The function f (x1, x2) = |x1| + |x2|.

35. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 35 Fig. 3.16. Illustration of Theorem 3.48.

36. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 36 Fig. 3.17. Minimum point of a concave function.

37. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 37 Fig. 4.1. Nonregular point x ∗ = 0.

38. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 38 Fig. 4.2. Illustration of the Farkas lemma.

39. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 39 Fig. 4.3. Geometrical illustration of KKT conditions.

40. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 40 Fig. 4.4. Illustration of the relation (4.22).

41. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 41 Fig. 4.5. Illustration of Example 4.8.

42. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 42 Fig. 4.6. Illustration of Example 4.14.

43. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 43 Fig. 4.7. Saddle point.

44. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 44 Fig. 4.8. Geometric illustration of (P).

45. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 45 Fig. 4.9. Geometric illustration of (D).

46. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 46 Fig. 4.10. Function h 1 (x 1 ) for various values of u.

47. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 47 Fig. 4.11. Dual objective function.

48. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 48 Fig. 4.12. Geometrical solution of Example 4.47 (case a).

49. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 49 Fig. 4.13. Geometrical solution of Example 4.47 (case b).

50. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 50 Fig. 4.14. Illustration of Example 4.50.

51. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 51 Fig. 6.1. Unimodal function.

52. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 52 Fig. 6.2. Locating of x.

53. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 53 Fig. 6.3. Construction of the new interval [a k+1, b k+1 ].

54. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 54 Table 6.1. Golden Section.

55. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 55 Fig. 6.4. Iteration of Newton’s method.

56. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 56 Fig. 6.5. Construction of a useless value.

57. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 57 Fig. 6.6. Cycling of Newton’s method.

58. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 58 Fig. 6.7. Interpolation polynomials of Example 6.17.

59. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 59 Fig. 6.8. Interpolation polynomials of Example 6.20.

60. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 60 Fig. 7.1. Orthogonal coordinate transformations.

61. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 61 Fig. 7.2. Level curves of the function 2x 2 1 + 4x 1 x 2 + 5x 2 2.

62. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 62 Fig. 7.3. Level surface z = 6 for the function 7x 2 1 − 2x 1 x 2 + 4x 1 x 3 + 7x 2 2 − 4x 2 x 3 + 10x 2 3.

63. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 63 Table 7.1. Gradient method.

64. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 64 Fig. 7.4. Minimization process of the gradient method.

65. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 65 Table 7.2. Conjugate gradient method (quadratic problem).

66. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 66 Table 7.3. Conjugate gradient method (nonquadratic problem).

67. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 67 Fig. 7.5. Solution of Example 7.30 by the conjugate gradient method.

68. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 68 Table 7.4. DFP method.

69. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 69 Table 7.5. Cyclic minimization.

70. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 70 Fig. 7.6. Solution of Example 7.38.

71. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 71 Table 7.6. Inexact line search (case a).

72. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 72 Fig. 7.7. Convergence to a nonoptimal point.

73. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 73 Table 7.7. Inexact line search (case b).

74. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 74 Fig. 7.8. Oscillation of the algorithm.

75. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 75 Fig. 7.9. Reversal strategy.

76. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 76 Fig. 7.10. Solution of the quadratic subproblem.

77. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 77 Fig. 7.11. Trust region method.

78. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 78 Fig. 8.1. Construction of the search direction.

79. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 79 Fig. 8.2. Resolution of Example 8.2 by Rosen’s method.

80. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 80 Fig. 8.3. Optimal directions.

81. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 81 Fig. 8.4. Zigzag movement.

82. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 82 Table 8.1. Pivoting of Example 8.17.

83. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 83 Fig. 9.1. Active set method.

84. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 84 Table 9.1. Initial tableau of Lemke’s method.

85. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 85 Table 9.2. Pivot steps of Example 9.4.

86. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 86 Table 9.3. Pivot steps of Example 9.5.

87. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 87 Fig. 9.2. Unbounded optimal solution.

88. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 88 Fig. 9.3. Locations of gas stations.

89. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 89 Fig. 10.1. Penalty terms.

90. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 90 Fig. 10.2. Function q(x, r) of Example 10.2.

91. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 91 Fig. 10.3. Function q(x, r) of Example 10.2.

92. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 92 Fig. 10.4. Penalty method.

93. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 93 Fig. 10.5. Exact penalty function (10.12).

94. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 94 Fig. 10.6. Illustration of robustness.

95. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 95 Fig. 10.7. Barrier terms.

96. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 96 Fig. 10.8. Barrier function s(x, c) of Example 10.16.

97. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 97 Table 10.1. Solution of Example 10.16.

98. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 98 Table 10.2. Solution of Example 10.17.

99. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 99 Fig. 10.9. Barrier method.

100. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 100 Fig. 11.1. Partial cost function.

101. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 101 Fig. 11.2. Nondifferentiable function of a minimax problem.

102. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 102 Fig. 11.3. Nondifferentiable bidimensional objective function.

103. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 103 Fig. 11.4. Graphical solution of problem (11.9).

104. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 104 Fig. 11.5. Objective function of problem (11.8).

105. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 105 Fig. 11.6. Minimization of function (11.10) with the gradient method.

106. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 106 Fig. 11.7. Minimization of function (11.11) with the gradient method.

107. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 107 Fig. 11.8. Piecewise linear function f and differential approximation g for ε = 0.1.

108. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 108 Table 11.1. Minimization by differentiable approximation.

109. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 109 Fig. 11.9. Local and global minima of problem (11.28).

110. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 110 Fig. 11.10. D.c. function.

111. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 111 Fig. 11.11. Branch-and-bound method.

112. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 112 Fig. 11.11. Branch-and-bound method (continuation).

113. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 113 Fig. 11.12. Subdivisions of the feasible region.

114. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 114 Fig. 11.13. Optimization problem (11.36).

115. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 115 Table 11.2. Population P(0).

116. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 116 Table 11.3. Population P(1).

117. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 117 Fig. 2.7. Feasible set and gradients.

118. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 118 Fig. 2.8. Feasible region and gradients.

119. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 119 Fig. 2.9. Feasible directions at x ∗ in diverse cases.

120. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 120 Fig. 3.18. Algebraic sum.

121. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 121 Fig. 3.19. Feasible direction.

122. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 122 Fig. 3.20. Discontinuous concave function.

123. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 123 Fig. 3.21. Convex function.

124. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 124 Fig. 3.22. Geometrical illustration of the subdifferential.

125. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 125 Fig. 3.23. Dispersion diagram.

126. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 126 Fig. 3.24. Feasible region.

127. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 127 Fig. 4.15. Regular point x ∗ = 0 with linearly dependent gradients.

128. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 128 Fig. 4.16. Geometric resolution of Exercise 4.9.

129. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 129 Fig. 4.17. Geometric solution of Exercise 4.15.

130. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 130 Fig. 4.18. Geometric solution of Exercise 4.33.

131. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 131 Fig. 4.19. Geometric solution of Exercise 4.34.

132. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 132 Table 5.1. Error sequence.

133. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 133 Table 5.2. Linear and quadratic convergence.

134. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 134 Table 6.2. Golden Section.

135. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 135 Table 6.3. Bisection algorithm.

136. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 136 Fig. 6.9. Premature convergence of the bisection method.

137. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 137 Fig. 6.10. Hermite interpolation polynomial.

138. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 138 Fig. 7.12. Level curves of Exercise 7.7.

139. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 139 Fig. 7.13. Displaced ellipses.

140. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 140 Fig. 7.14. Level curves of quadratic functions.

141. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 141 Table 7.8. Gradient method.

142. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 142 Table 7.9. Newton’s method.

143. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 143 Fig. 7.15. Minimization process of Newton’s method.

144. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 144 Table 7.10. Conjugate gradient method.

145. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 145 Table 7.11. DFP method.

146. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 146 Table 7.12. Conjugate gradient method.

147. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 147 Table 7.13. DFP method.

148. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 148 Table 7.14. Variant of the DFP method.

149. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 149 Table 7.15. Cyclic minimization.

150. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 150 Fig. 7.16. Level curves of the first quadratic model of Example 7.50.

151. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 151 Fig. 7.17. Level curves of the function of Exercise 7.54 (b).

152. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 152 Fig. 8.5. Geometric solution of Exercise 8.7.

153. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 153 Fig. 8.6. Feasible region.

154. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 154 Fig. 8.7. Zoutendijk’s method.

155. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 155 Fig. 8.8. Graphical illustration of both methods.

156. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 156 Table 8.2. Pivoting of Example 8.18.

157. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 157 Table 9.4. Lemke’s method.

158. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 158 Table 9.5. Lemke’s method.

159. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 159 Table 10.3. Barrier method.

160. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 160 Fig. 10.10. Solution process.

161. Nonlinear Programming, Peter Zörnig ISBN 978-3-11-031527-1 © 2014 by Walter de Gruyter GmbH, Berlin/Boston 161 Table 10.4. Barrier method.