Optimization Using Personal Computers: With Applications to Electrical Networks

Contents
	1. Introduction
		1.1. Scalar Functions of a Vector,
			1.1.1. Surfaces Over Two Dimensions,
			1.1.2. Quadratic Approximations to Peaks,
		1.2. Types of Optimization Problems,
			1.2.1. General Problem Statement,
			1.2.2. Objective Functions,
			1.2.3. Some Fields of Application,
		1.3. Iterative Processes,
			1.3.1. Iteration and Convergence,
			1.3.2. Numbers and Stability,
			1.3.3. Illconditioned Linear Systems,
			1.3.4. Termination and Comparison of Algorithms,
		1.4. Choices and Form,
			1.4.1. Languages and Features,
			1.4.2. Personal Computers,
			1.4.3. Point of View,
		Problems,
	2. Matrix Algebra and Algorithms
		2.1. Definitions and Operations,
			2.1.1. Vector and Matrix Notation,
			2.1.2. Utility Program MATRIX,
			2.1.3. Simple Vector and Matrix Operations,
			2.1.4. Inverse of a Square Matrix,
			2.1.5. Vector and Matrix Norms and Condition Number,
		2.2. Relationships in Vector Space,
			2.2.1. The Matrix Role in Vector Space,
			2.2.2. Orthogonal Relationships,
			2.2.3. The Matrix Eigenproblem,
			2.2.4. Special Matrix Transformations,
		Problems,
	3. Functions of Many Variables
		3.1. Systems of Linear Equations,
			3.1.1. Square Linear Systems of Full Rank,
			3.1.2. Overdetermined Linear Systems of Full Rank,
			3.1.3. Rank-Deficient Linear Systems,
		3.2. Nonlinear Functions,
			3.2.1. Quadratic and Line Functions,
			3.2.2. General Nonlinear Functions,
		3.3. Constraints,
			3.3.1. Implicit Function Theorem,
			3.3.2. Equality Constraints by Lagrange Multipliers,
			3.3.3. Constraint Qualifications—The Kuhn-Tucker Conditions,
		Problems,
	4. Newton Methods
		4.1. Obtaining and Using the Hessian Matrix,
			4.1.1. Finite Differences for Second Derivatives,
			4.1.2. Forcing Positive-Definite Factorization,
			4.1.3. Computing Quadratic Forms and Solutions,
		4.2. Trust Neighborhoods,
			4.2.1. Trust Radius,
			4.2.2. Levenberg-Marquardt Methods,
		4.3. Program NEWTON,
			4.3.1. The Algorithm and Its Implementation,
			4.3.2. Some Examples Using Program NEWTON,
			4.3.3. Simple Lower and Upper Bounds on Variables,
		4.4. Gauss-Newton Methods,
			4.4.1. Nonlinear Least-p th Objective and Gradient Functions,
			4.4.2. Positive-Definite Hessian Approximation,
			4.4.3. Weighted Least Squares and the Least-p th Method,
			4.4.4. Numerical Integration As a Sampling Strategy,
			4.4.5. Controlling the Levenberg-Marquardt Parameter,
		4.5. Program LEASTP,
			4.5.1. The Algorithm and Its Implementation,
			4.5.2. Some Examples Using Program LEASTP,
			4.5.3. Approaches to Minimax Optimization,
		Problems,
	5. Quasi-Newton Methods and Constraints
		5.1. Updating Approximations to the Hessian,
			5.1.1. General Secant Methods,
			5.1.2. Families of Quasi-Newton Matrix Updates,
			5.1.3. Invariance of Newton-like Methods to Linear Scaling,
		5.2. Line Searches,
			5.2.1. The Cutback Line Search,
			5.2.2. Quadratic Interpolation Without Derivatives,
			5.2.3. Cubic Interpolation Using Derivatives,
		5.3. Program QNEWT,
			5.3.1. The Algorithm and Its Implementation,
			5.3.2. Some Examples Using Program QNEWT,
			5.3.3. Optimization Without Explicit Derivatives,
		5.4. Constrained Optimization,
			5.4.1. Linear Constraints by Projection,
			5.4.2. Program BOXMIN for Lower and Upper Bounds,
			5.4.3. Nonlinear Constraints by Penalty Functions,
			5.4.4. Program MULTPEN for Nonlinear Constraints,
			5.4.5. Other Methods for Nonlinear Constraints,
		Problems,
	6. Network Optimization
		6.1. Network Analysis in the Sinusoidal Steady State,
			6.1.1. From Differential Equations to the Frequency Domain,
			6.1.2. Related Technical Disciplines,
			6.1.3. NetworkAnalysis,
		6.2. Constrained Optimization of Networks,
			6.2.1. Program TWEAKNET Objectives and Structure,
			6.2.2. Ladder Network Analysis,
			6.2.3. First Partial Derivatives,
			6.2.4. Summary of Program TWEAKNET with Examples,
		6.3. Exact Partial Derivatives for Linear Systems,
			6.3.1. Tellegen’s Theorem,
			6.3.2. Derivatives for Lossless Reciprocal Networks,
			6.3.3. Derivatives for Any Network Using Adjoint Networks,
			6.3.4. Derivatives Obtained by the Nodal Admittance Matrix,
		6.4. Robust Response Functions,
			6.4.1. Bilinear Functions and Forms,
			6.4.2. The Bilinear Property of Linear Networks,
			6.4.3. Sensitivity of Network Response Functions,
		Problems,
	Appendices
		Appendix A. Test Matrices
		Appendix B. Test Problems
		Appendix C. Program Listings
	References
	Index