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Preface | |
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Preface to the Second Edition | |
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Introduction | |
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Mathematical Formulation | |
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Example: A Transportation Problem | |
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Continuous versus Discrete Optimization | |
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Constrained and Unconstrained Optimization | |
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Global and Local Optimization | |
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Stochastic and Deterministic Optimization | |
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Convexity | |
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Optimization Algorithms | |
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Notes and References | |
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Fundamentals of Unconstrained Optimization | |
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What Is a Solution? | |
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Recognizing a Local Minimum | |
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Nonsmooth Problems | |
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Overview of Algorithms | |
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Two Strategies: Line Search and Trust Region | |
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Search Directions for Line Search Methods | |
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Models for Trust-Region Methods | |
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Scaling | |
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Exercises | |
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Line Search Methods | |
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Step Length | |
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The Wolfe Conditions | |
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The Goldstein Conditions | |
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Sufficient Decrease and Backtracking | |
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Convergence of Line Search Methods | |
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Rate of Convergence | |
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Convergence Rate of Steepest Descent | |
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Newton's Method | |
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Quasi-Newton Methods | |
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Newton's Method with Hessian Modification | |
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Eigenvalue Modification | |
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Adding a Multiple of the Identity | |
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Modified Cholesky Factorization | |
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Modified Symmetric Indefinite Factorization | |
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Step-Length Selection Algorithms | |
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Interpolation | |
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Initial Step Length | |
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A Line Search Algorithm for the Wolfe Conditions | |
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Notes and References | |
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Exercises | |
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Trust-Region Methods | |
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Outline of the Trust-Region Approach | |
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Algorithms Based on the Cauchy Point | |
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The Cauchy Point | |
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Improving on the Cauchy Point | |
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The Dogleg Method | |
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Two-Dimensional Subspace Minimization | |
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Global Convergence | |
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Reduction Obtained by the Cauchy Point | |
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Convergence to Stationary Points | |
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Iterative Solution of the Subproblem | |
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The Hard Case | |
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Proof of Theorem 4.1 | |
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Convergence of Algorithms Based on Nearly Exact Solutions | |
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Local Convergence of Trust-Region Newton Methods | |
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Other Enhancements | |
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Scaling | |
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Trust Regions in Other Norms | |
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Notes and References | |
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Exercises | |
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Conjugate Gradient Methods | |
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The Linear Conjugate Gradient Method | |
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Conjugate Direction Methods | |
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Basic Properties of the Conjugate Gradient Method | |
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A Practical Form of the Conjugate Gradient Method | |
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Rate of Convergence | |
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Preconditioning | |
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Practical Preconditioners | |
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Nonlinear Conjugate Gradient Methods | |
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The Fletcher-Reeves Method | |
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The Polak-Ribiere Method and Variants | |
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Quadratic Termination and Restarts | |
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Behavior of the Fletcher-Reeves Method | |
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Global Convergence | |
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Numerical Performance | |
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Notes and References | |
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Exercises | |
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Quasi-Newton Methods | |
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The BFGS Method | |
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Properties of the BFGS Method | |
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Implementation | |
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The SR1 Method | |
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Properties of SR1 Updating | |
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The Broyden Class | |
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Convergence Analysis | |
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Global Convergence of the BFGS Method | |
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Superlinear Convergence of the BFGS Method | |
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Convergence Analysis of the SR1 Method | |
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Notes and References | |
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Exercises | |
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Large-Scale Unconstrained Optimization | |
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Inexact Newton Methods | |
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Local Convergence of Inexact Newton Methods | |
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Line Search Newton-CG Method | |
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Trust-Region Newton-CG Method | |
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Preconditioning the Trust-Region Newton-CG Method | |
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Trust-Region Newton-Lanczos Method | |
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Limited-Memory Quasi-Newton Methods | |
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Limited-Memory BFGS | |
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Relationship with Conjugate Gradient Methods | |
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General Limited-Memory Updating | |
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Compact Representation of BFGS Updating | |
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Unrolling the Update | |
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Sparse Quasi-Newton Updates | |
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Algorithms for Partially Separable Functions | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Calculating Derivatives | |
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Finite-Difference Derivative Approximations | |
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Approximating the Gradient | |
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Approximating a Sparse Jacobian | |
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Approximating the Hessian | |
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Approximating a Sparse Hessian | |
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Automatic Differentiation | |
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An Example | |
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The Forward Mode | |
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The Reverse Mode | |
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Vector Functions and Partial Separability | |
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Calculating Jacobians of Vector Functions | |
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Calculating Hessians: Forward Mode | |
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Calculating Hessians: Reverse Mode | |
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Current Limitations | |
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Notes and References | |
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Exercises | |
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Derivative-Free Optimization | |
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Finite Differences and Noise | |
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Model-Based Methods | |
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Interpolation and Polynomial Bases | |
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Updating the Interpolation Set | |
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A Method Based on Minimum-Change Updating | |
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Coordinate and Pattern-Search Methods | |
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Coordinate Search Method | |
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Pattern-Search Methods | |
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A Conjugate-Direction Method | |
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Nelder-Mead Method | |
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Implicit Filtering | |
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Notes and References | |
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Exercises | |
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Least-Squares Problems | |
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Background | |
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Linear Least-Squares Problems | |
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Algorithms for Nonlinear Least-Squares Problems | |
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The Gauss-Newton Method | |
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Convergence of the Gauss-Newton Method | |
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The Levenberg-Marquardt Method | |
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Implementation of the Levenberg-Marquardt Method | |
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Convergence of the Levenberg-Marquardt Method | |
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Methods for Large-Residual Problems | |
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Orthogonal Distance Regression | |
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Notes and References | |
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Exercises | |
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Nonlinear Equations | |
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Local Algorithms | |
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Newton's Method for Nonlinear Equations | |
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Inexact Newton Methods | |
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Broyden's Method | |
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Tensor Methods | |
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Practical Methods | |
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Merit Functions | |
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Line Search Methods | |
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Trust-Region Methods | |
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Continuation/Homotopy Methods | |
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Motivation | |
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Practical Continuation Methods | |
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Notes and References | |
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Exercises | |
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Theory of Constrained Optimization | |
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Local and Global Solutions | |
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Smoothness | |
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Examples | |
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A Single Equality Constraint | |
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A Single Inequality Constraint | |
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Two Inequality Constraints | |
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Tangent Cone and Constraint Qualifications | |
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First-Order Optimality Conditions | |
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First-Order Optimality Conditions: Proof | |
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Relating the Tangent Cone and the First-Order Feasible Direction Set | |
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A Fundamental Necessary Condition | |
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Farkas' Lemma | |
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Proof of Theorem 12.1 | |
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Second-Order Conditions | |
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Second-Order Conditions and Projected Hessians | |
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Other Constraint Qualifications | |
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A Geometric Viewpoint | |
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Lagrange Multipliers and Sensitivity | |
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Duality | |
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Notes and References | |
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Exercises | |
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Linear Programming: The Simplex Method | |
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Linear Programming | |
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Optimality and Duality | |
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Optimality Conditions | |
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The Dual Problem | |
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Geometry of the Feasible Set | |
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Bases and Basic Feasible Points | |
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Vertices of the Feasible Polytope | |
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The Simplex Method | |
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Outline | |
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A Single Step of the Method | |
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Linear Algebra in the Simplex Method | |
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Other Important Details | |
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Pricing and Selection of the Entering Index | |
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Starting the Simplex Method | |
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Degenerate Steps and Cycling | |
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The Dual Simplex Method | |
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Presolving | |
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Where Does the Simplex Method Fit? | |
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Notes and References | |
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Exercises | |
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Linear Programming: Interior-Point Methods | |
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Primal-Dual Methods | |
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Outline | |
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The Central Path | |
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Central Path Neighborhoods and Path-Following Methods | |
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Practical Primal-Dual Algorithms | |
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Corrector and Centering Steps | |
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Step Lengths | |
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Starting Point | |
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A Practical Algorithm | |
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Solving the Linear Systems | |
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Other Primal-Dual Algorithms and Extensions | |
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Other Path-Following Methods | |
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Potential-Reduction Methods | |
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Extensions | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Fundamentals of Algorithms for Nonlinear Constrained Optimization | |
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Categorizing Optimization Algorithms | |
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The Combinatorial Difficulty of Inequality-Constrained Problems | |
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Elimination of Variables | |
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Simple Elimination using Linear Constraints | |
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General Reduction Strategies for Linear Constraints | |
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Effect of Inequality Constraints | |
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Merit Functions and Filters | |
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Merit Functions | |
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Filters | |
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The Maratos Effect | |
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Second-Order Correction and Nonmonotone Techniques | |
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Nonmonotone (Watchdog) Strategy | |
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Notes and References | |
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Exercises | |
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Quadratic Programming | |
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Equality-Constrained Quadratic Programs | |
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Properties of Equality-Constrained QPs | |
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Direct Solution of the KKT System | |
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Factoring the Full KKT System | |
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Schur-Complement Method | |
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Null-Space Method | |
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Iterative Solution of the KKT System | |
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CG Applied to the Reduced System | |
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The Projected CG Method | |
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Inequality-Constrained Problems | |
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Optimality Conditions for Inequality-Constrained Problems | |
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Degeneracy | |
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Active-Set Methods for Convex QPs | |
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Specification of the Active-Set Method for Convex QP | |
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Further Remarks on the Active-Set Method | |
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Finite Termination of Active-Set Algorithm on Strictly Convex QPs | |
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Updating Factorizations | |
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Interior-Point Methods | |
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Solving the Primal-Dual System | |
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Step Length Selection | |
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A Practical Primal-Dual Method | |
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The Gradient Projection Method | |
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Cauchy Point Computation | |
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Subspace Minimization | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Penalty and Augmented Lagrangian Methods | |
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The Quadratic Penalty Method | |
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Motivation | |
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Algorithmic Framework | |
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Convergence of the Quadratic Penalty Method | |
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Ill Conditioning and Reformulations | |
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Nonsmooth Penalty Functions | |
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A Practical l[subscript 1] Penalty Method | |
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A General Class of Nonsmooth Penalty Methods | |
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Augmented Lagrangian Method: Equality Constraints | |
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Motivation and Algorithmic Framework | |
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Properties of the Augmented Lagrangian | |
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Practical Augmented Lagrangian Methods | |
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Bound-Constrained Formulation | |
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Linearly Constrained Formulation | |
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Unconstrained Formulation | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Sequential Quadratic Programming | |
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Local SQP Method | |
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SQP Framework | |
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Inequality Constraints | |
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Preview of Practical SQP Methods | |
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IQP and EQP | |
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Enforcing Convergence | |
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Algorithmic Development | |
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Handling Inconsistent Linearizations | |
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Full Quasi-Newton Approximations | |
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Reduced-Hessian Quasi-Newton Approximations | |
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Merit Functions | |
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Second-Order Correction | |
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A Practical Line Search SQP Method | |
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Trust-Region SQP Methods | |
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A Relaxation Method for Equality-Constrained Optimization | |
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Sl[subscript 1] QP (Sequential l[subscript 1] Quadratic Programming) | |
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Sequential Linear-Quadratic Programming (SLQP) | |
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A Technique for Updating the Penalty Parameter | |
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Nonlinear Gradient Projection | |
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Convergence Analysis | |
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Rate of Convergence | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Interior-Point Methods for Nonlinear Programming | |
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Two Interpretations | |
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A Basic Interior-Point Algorithm | |
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Algorithmic Development | |
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Primal vs. Primal-Dual System | |
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Solving the Primal-Dual System | |
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Updating the Barrier Parameter | |
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Handling Nonconvexity and Singularity | |
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Step Acceptance: Merit Functions and Filters | |
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Quasi-Newton Approximations | |
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Feasible Interior-Point Methods | |
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A Line Search Interior-Point Method | |
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A Trust-Region Interior-Point Method | |
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An Algorithm for Solving the Barrier Problem | |
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Step Computation | |
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Lagrange Multipliers Estimates and Step Acceptance | |
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Description of a Trust-Region Interior-Point Method | |
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The Primal Log-Barrier Method | |
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Global Convergence Properties | |
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Failure of the Line Search Approach | |
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Modified Line Search Methods | |
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Global Convergence of the Trust-Region Approach | |
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Superlinear Convergence | |
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Perspectives and Software | |
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Notes and References | |
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Exercises | |
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Background Material | |
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Elements of Linear Algebra | |
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Vectors and Matrices | |
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Norms | |
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Subspaces | |
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Eigenvalues, Eigenvectors, and the Singular-Value Decomposition | |
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Determinant and Trace | |
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Matrix Factorizations: Cholesky, LU, QR | |
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Symmetric Indefinite Factorization | |
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Sherman-Morrison-Woodbury Formula | |
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Interlacing Eigenvalue Theorem | |
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Error Analysis and Floating-Point Arithmetic | |
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Conditioning and Stability | |
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Elements of Analysis, Geometry, Topology | |
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Sequences | |
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Rates of Convergence | |
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Topology of the Euclidean Space R[superscript n] | |
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Convex Sets in R[superscript n] | |
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Continuity and Limits | |
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Derivatives | |
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Directional Derivatives | |
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Mean Value Theorem | |
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Implicit Function Theorem | |
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Order Notation | |
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Root-Finding for Scalar Equations | |
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A Regularization Procedure | |
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References | |
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Index | |