Primal-dual interior-point methods /
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| Author / Creator: | Wright, Stephen J., 1960- |
|---|---|
| Imprint: | Philadelphia : Society for Industrial and Applied Mathematics, c1997. |
| Description: | xx, 289 p. : ill. ; 26 cm. |
| Language: | English |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/3330798 |
Table of Contents:
- Preface
- Notation
- 1. Introduction
- Linear Programming
- Primal-Dual Methods
- The Central Path
- A Primal-Dual Framework
- Path-Following Methods
- Potential-Reduction Methods
- Infeasible Starting Points
- Superlinear Convergence
- Extensions
- Mehrotra's Predictor-Corrector Algorithm
- Linear Algebra Issues
- Karmarkar's Algorithm
- 2. Background
- Linear Programming and Interior-Point Methods
- Standard Form
- Optimality Conditions, Duality, and Solution Sets
- The B {{SYMBOL 200 \f "Symbol"}} N Partition and Strict Complementarity
- A Strictly Interior Point
- Rank of the Matrix A
- Bases and Vertices
- Farkas's Lemma and a Proof of the Goldman-Tucker Result
- The Central Path
- Background
- Primal Method
- Primal-Dual Methods
- Development of the Fundamental Ideas
- Notes and References
- 3. Complexity Theory. Polynomial Versus Exponential, Worst Case vs Average Case
- Storing the Problem Data
- Dimension and Size
- The Turing Machine and Rational Arithmetic
- Primal-Dual Methods and Rational Arithmetic
- Linear Programming and Rational Numbers
- Moving to a Solution from an Interior Point
- Complexity of Simplex, Ellipsoid, and Interior-Point Methods
- Polynomial and Strongly Polynomial Algorithms
- Beyond the Turing Machine Model
- More on the Real-Number Model and Algebraic Complexity
- A General Complexity Theorem for Path-Following Methods
- Notes and References
- 4. Potential-Reduction Methods
- A Primal-Dual Potential-Reduction Algorithm
- Reducing Forces Convergence
- A Quadratic Estimate of \Phi _{{\rho }} along a Feasible Direction
- Bounding the Coefficients in The Quadratic Approximation
- An Estimate of the Reduction in \Phi _{{\rho }} and Polynomial Complexity
- What About Centrality?
- Choosing {{SYMBOL 114 \f "Symbol"}} and {{SYMBOL 97 \f "Symbol"}}
- Notes and References
- 5. Path-Following Algorithms
- The Short-Step Path-Following Algorithm
- Technical Results
- The Predictor-Corrector Method
- A Long-Step Path-Following Algorithm
- Limit Points of the Iteration Sequence
- Proof of Lemma 5.3
- Notes and References
- 6. Infeasible-Interior-Point Algorithms
- The Algorithm
- Convergence of Algorithm IPF
- Technical Results I. Bounds on \nu _k \delimiter "026B30D (x^k, s^k) \delimiter "026B30D
- Technical Results II. Bounds on (D^k)^{{-1}} \Delta x^k and D^k \Delta s^k
- Technical Results III. A Uniform Lower Bound on {{SYMBOL 97 \f "Symbol"}}k
- Proofs of Theorems 6.1 and 6.2
- Limit Points of the Iteration Sequence
- 7. Superlinear Convergence and Finite Termination
- Affine-Scaling Steps
- An Estimate of ({{SYMBOL 68 \f "Symbol"}}x, {{SYMBOL 68 \f "Symbol"}} s). The Feasible Case
- An Estimate of ({{SYMBOL 68 \f "Symbol"}} x, {{SYMBOL 68 \f "Symbol"}} s). The Infeasible Case
- Algorithm PC Is Superlinear
- Nearly Quadratic Methods
- Convergence of Algorithm LPF+
- Convergence of the Iteration Sequence; {{SYMBOL 206 \f "Symbol"}}(A, b, c) and Finite Termination
- A Finite Termination Strategy
- Recovering an Optimal Basis
- More on {{SYMBOL 206 \f "Symbol"}} (A, b, c)
- Notes and References
- 8. Extensions. The Monotone LCP
- Mixed and Horizontal LCP
- Strict Complementarity and LCP
- Convex QP
- Convex Programming
- Monotone Nonlinear Complementarity and Variational Inequalities
- Semidefinite Programming
- Proof of Theorem 8.4
- Notes and References
- 9. Detecting Infeasibility
- Self-Duality
- The Simplified HSD Form
- The HSDl Form
- Identifying a Solution-Free Region
- Implementations of the HSD Formulations
- Notes and References
- 10. Practical Aspects of Primal-Dual Algorithms
- Motivation for Mehrotra's Algorithm
- The Algorithm
- Superquadratic Convergence
- Second-Order Trajectory-Following Methods
- Higher-Order Methods
- Further Enhancements
- Notes and References
- 11. Implementations. Three Forms of the Step Equation
- The Cholesky Factorization
- Sparse Cholesky Factorization
- Minimum-Degree Orderings
- Other Orderings
- Small Pivots in the Cholesky Factorization
- Dense Columns in A
- The Augmented System Formulation
- Factoring Symmetric Indef
