The approximate minimization of functionals

Title Page......Page 2
Preface......Page 3
Contents......Page 5
1.1. Introduction\0......Page 8
1.3. Basic Functional Analysis\0......Page 10
1.4. General Functional Analysis for Minimization Problems\0......Page 14
1.5. The Role of Convexity\0......Page 20
1.6. Minimizing Sequences\0......Page 24
2.2. Constrained Minimization\0......Page 35
2.3. Unconstrained Minimization\0......Page 38
2.4. Remarks on Operator Equations\0......Page 41
3.2. Regularization\0......Page 44
3.3. A Numerical Method for Optimal Control Problems\0......Page 48
3.4. Chebyshev Solution of Differential Equations\0......Page 59
3.5. Calculus of Variations\0......Page 62
3.6. Two-Point Boundary Value Problems\0......Page 66
3.7. The Ritz Method\0......Page 72
4.1. Introduction\0......Page 77
4.2. Criticizing Sequences and Convergence in General\0......Page 78
4.3. Glgbal Minimum along the Line\0......Page 83
4.4. First Local Minimum along the Line: Positive Weight Thereon\0......Page 84
4.5. A Simple Interval along the Line\0......Page 87
4.6. A Range Function along the Line\0......Page 91
4.7. Search Methods along the Line\0......Page 96
4.8. Specialization to Steepest Descent\0......Page 100
4.9. Step-Size Algorithms for Constrained Problems\0......Page 102
4.10. Direction Algorithms for Constrained Problems\0......Page 112
4.11. Other Methods for Constrained Problems\0......Page 117
5.2. Conjugate Directions for Quadratic Functionals\0......Page 121
5.3. Conjugate Gradients for Quadratic Functionals\0......Page 124
5.4. Conjugate Gradients as an Optimal Process\0......Page 126
5.5. The Projected-Gradient Viewpoint\0......Page 129
5.6. Conjugate Gradients for General Functionals\0......Page 132
5.7. Local-Convergence Rates\0......Page 134
5.8. Computational Modifications\0......Page 143
6.2. Convergence of x,+ i - x, to zero\0......Page 149
6.3. The Limit Set of {x,}\0......Page 152
6.4. Improved Convergence Results\0......Page 157
6.6. Minimization along the Line\0......Page 161
7.2. Variable-Metric Directions\0......Page 166
7.3. Exact Methods for Quadratics\0......Page 172
7.4. Some Particular Methods\0......Page 175
7.5. Constrained Problems\0......Page 185
8.1. Introduction\0......Page 187
8.2. Newton and Newton-like Methods\0......Page 188
8.3. Generalized Linear Iterations\0......Page 192
8.4. Least-Squares Problems\0......Page 197
9.2. Modifying Davidon\'s First Method\0......Page 201
9.3. Modifying Newton\'s Method\0......Page 204
9.4. Modifying the Gauss-Newton Method\0......Page 207
9.5. Modifying the Gauss-Newton-Gradient Compromise\0......Page 208
9.6. Methods Ignoring Derivatives\0......Page 210
EPILOGUE\0......Page 218
REFERENCES\0......Page 220
INDEX\0......Page 232