| Preface | p. vii |
| Background | |
| Linear Algebra | p. 3 |
| Vectors | p. 3 |
| Matrices and Matrix Operations | p. 5 |
| Matrices and Mappings | p. 6 |
| Inverse and Generalized Inverse Matrices | p. 8 |
| Direct Methods for Solving Linear Equations | p. 9 |
| Norms | p. 12 |
| Scalar Products | p. 14 |
| Eigenvalues and Eigenvectors | p. 17 |
| Matrix Decompositions | p. 19 |
| Penalized Matrices | p. 22 |
| Optimization | p. 27 |
| Optimization Problems and Solutions | p. 27 |
| Unconstrained Quadratic Programming | p. 28 |
| Quadratic Cost Functions | p. 28 |
| Unconstrained Minimization of Quadratic Functions | p. 29 |
| Convexity | p. 31 |
| Convex Quadratic Functions | p. 32 |
| Local and Global Minimizers of Convex Function | p. 34 |
| Existence of Minimizers | p. 35 |
| Projections to Convex Sets | p. 36 |
| Equality Constrained Problems | p. 38 |
| Optimality Conditions | p. 39 |
| Existence and Uniqueness | p. 41 |
| KKT Systems | p. 42 |
| Min-max, Dual, and Saddle Point Problems | p. 44 |
| Sensitivity | p. 46 |
| Error Analysis | p. 47 |
| Inequality Constrained Problems | p. 49 |
| Polyhedral Sets | p. 49 |
| Farkas's Lemma | p. 50 |
| Necessary Optimality Conditions for Local Solutions | p. 51 |
| Existence and Uniqueness | p. 52 |
| Optimality Conditions for Convex Problems | p. 54 |
| Optimality Conditions for Bound Constrained Problems | p. 55 |
| Min-max, Dual, and Saddle Point Problems | p. 55 |
| Equality and Inequality Constrained Problems | p. 57 |
| Optimality Conditions | p. 58 |
| Existence and Uniqueness | p. 59 |
| Partially Bound and Equality Constrained Problems | p. 59 |
| Duality for Dependent Constraints | p. 61 |
| Duality for Semicoercive Problems | p. 64 |
| Linear Programming | p. 69 |
| Solvability and Localization of Solutions | p. 69 |
| Duality in Linear Programming | p. 70 |
| Algorithms | |
| Conjugate Gradients for Unconstrained Minimization | p. 73 |
| Conjugate Directions and Minimization | p. 74 |
| Generating Conjugate Directions and Krylov Spaces | p. 77 |
| Conjugate Gradient Method | p. 78 |
| Restarted CG and the Gradient Method | p. 81 |
| Rate of Convergence and Optimality | p. 82 |
| Min-max Estimate | p. 82 |
| Estimate in the Condition Number | p. 84 |
| Convergence Rate of the Gradient Method | p. 86 |
| Optimality | p. 87 |
| Preconditioned Conjugate Gradients | p. 87 |
| Preconditioning by Conjugate Projector | p. 90 |
| Conjugate Projectors | p. 90 |
| Minimization in Subspace | p. 91 |
| Conjugate Gradients in Conjugate Complement | p. 92 |
| Preconditioning Effect | p. 94 |
| Conjugate Gradients for More General Problems | p. 96 |
| Convergence in Presence of Rounding Errors | p. 97 |
| Numerical Experiments | p. 98 |
| Basic CG and Preconditioning | p. 98 |
| Numerical Demonstration of Optimality | p. 99 |
| Comments and Conclusions | p. 100 |
| Equality Constrained Minimization | p. 103 |
| Review of Alternative Methods | p. 105 |
| Penalty Method | p. 107 |
| Minimization of Augmented Lagrangian | p. 108 |
| An Optimal Feasibility Error Estimate for Homogeneous Constraints | p. 109 |
| Approximation Error and Convergence | p. 111 |
| Improved Feasibility Error Estimate | p. 112 |
| Improved Approximation Error Estimate | p. 113 |
| Preconditioning Preserving Gap in the Spectrum | p. 115 |
| Exact Augmented Lagrangian Method | p. 116 |
| Algorithm | p. 117 |
| Convergence of Lagrange Multipliers | p. 119 |
| Effect of the Steplength | p. 120 |
| Convergence of the Feasibility Error | p. 124 |
| Convergence of Primal Variables | p. 124 |
| Implementation | p. 125 |
| Asymptotically Exact Augmented Lagrangian Method | p. 126 |
| Algorithm | p. 126 |
| Auxiliary Estimates | p. 127 |
| Convergence Analysis | p. 128 |
| Adaptive Augmented Lagrangian Method | p. 130 |
| Algorithm | p. 131 |
| Convergence of Lagrange Multipliers for Large <$>\varrho<$> | p. 132 |
| R-Linear Convergence for Any Initialization of <$>\varrho<$> | p. 134 |
| Semimonotonic Augmented Lagrangians (SMALE) | p. 135 |
| SMALE Algorithm | p. 136 |
| Relations for Augmented Lagrangians | p. 137 |
| Convergence and Monotonicity | p. 139 |
| Linear Convergence for Large <$>\varrho_0<$> | p. 142 |
| Optimality of the Outer Loop | p. 143 |
| Optimality of SMALE with Conjugate Gradients | p. 145 |
| Solution of More General Problems | p. 147 |
| Implementation of Inexact Augmented Lagrangians | p. 148 |
| Stopping, Modification of Constraints, and Preconditioning | p. 148 |
| Initialization of Constants | p. 148 |
| Numerical Experiments | p. 150 |
| Uzawa, Exact Augmented Lagrangians, and SMALE | p. 150 |
| Numerical Demonstration of Optimality | p. 151 |
| Comments and References | p. 152 |
| Bound Constrained Minimization | p. 155 |
| Review of Alternative Methods | p. 157 |
| KKT Conditions and Related Inequalities | p. 158 |
| The Working Set Method with Exact Solutions | p. 160 |
| Auxiliary Problems | p. 160 |
| Algorithm | p. 161 |
| Finite Termination | p. 164 |
| Polyak's Algorithm | p. 165 |
| Basic Algorithm | p. 165 |
| Finite Termination | p. 166 |
| Characteristics of Polyak's Algorithm | p. 167 |
| Inexact Polyak's Algorithm | p. 167 |
| Looking Ahead and Estimate | p. 167 |
| Looking Ahead Polyak's Algorithm | p. 170 |
| Easy Re-release Polyak's Algorithm | p. 171 |
| Properties of Modified Polyak's Algorithms | p. 172 |
| Gradient Projection Method | p. 173 |
| Conjugate Gradient Versus Gradient Projections | p. 174 |
| Contraction in the Euclidean Norm | p. 175 |
| The Fixed Steplength Gradient Projection Method | p. 177 |
| Quadratic Functions with Identity Hessian | p. 178 |
| Dominating Function and Decrease of the Cost Function | p. 181 |
| Modified Proportioning with Gradient Projections | p. 184 |
| MPGP Schema | p. 184 |
| Rate of Convergence | p. 186 |
| Modified Proportioning with Reduced Gradient Projections | p. 189 |
| MPRGP Schema | p. 189 |
| Rate of Convergence | p. 190 |
| Rate of Convergence of Projected Gradient | p. 193 |
| Optimality | p. 197 |
| Identification Lemma and Finite Termination | p. 198 |
| Finite Termination for Dual Degenerate Solution | p. 201 |
| Implementation of MPRGP with Optional Modifications | p. 204 |
| Expansion Step with Feasible Half-Step | p. 204 |
| MPRGP Algorithm | p. 205 |
| Unfeasible MPRGP | p. 206 |
| Choice of Parameters | p. 208 |
| Dynamic Release Coefficient | p. 209 |
| Preconditioning | p. 210 |
| Preconditioning in Face | p. 210 |
| Preconditioning by Conjugate Projector | p. 212 |
| Numerical Experiments | p. 216 |
| Polyak, MPRGP, and Preconditioned MPRGP | p. 216 |
| Numerical Demonstration of Optimality | p. 217 |
| Comments and References | p. 218 |
| Bound and Equality Constrained Minimization | p. 221 |
| Review of the Methods for Bound and Equality Constrained Problems | p. 222 |
| SMALBE Algorithm for Bound and Equality Constraints | p. 223 |
| KKT Conditions and Projected Gradient | p. 223 |
| SMALBE Algorithm | p. 223 |
| Inequalities Involving the Augmented Lagrangian | p. 225 |
| Monotonicity and Feasibility | p. 227 |
| Boundedness | p. 229 |
| Convergence | p. 233 |
| Optimality of the Outer Loop | p. 235 |
| Optimality of the Inner Loop | p. 237 |
| Solution of More General Problems | p. 239 |
| Implementation | p. 240 |
| SMALBE-M | p. 241 |
| Numerical Experiments | p. 242 |
| Balanced Reduction of Feasibility and Gradient Errors | p. 242 |
| Numerical Demonstration of Optimality | p. 243 |
| Comments and References | p. 244 |
| Applications to Variational Inequalities | |
| Solution of a Coercive Variational Inequality by FETI-DP Method | p. 249 |
| Model Coercive Variational Inequality | p. 250 |
| FETI-DP Domain Decomposition and Discretization | p. 251 |
| Optimality | p. 254 |
| Numerical Experiments | p. 255 |
| Comments and References | p. 256 |
| Solution of a Semicoercive Variational Inequality by TFETI Method | p. 259 |
| Model Semicoercive Variational Inequality | p. 260 |
| TFETI Domain Decomposition and Discretization | p. 261 |
| Natural Coarse Grid | p. 264 |
| Optimality | p. 265 |
| Numerical Experiments | p. 267 |
| Comments and References | p. 269 |
| References | p. 271 |
| Index | p. 281 |
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