A First Course in Optimization

Optimization without Calculus Chapter Summary The Arithmetic Mean-Geometric Mean Inequality An Application of the AGM Inequality: the Number e Extending the AGM Inequality Optimization Using the AGM Inequality The Holder and Minkowski Inequalities Cauchy's Inequality Optimizing using Cauchy's Inequality An Inner Product for Square Matrices Discrete Allocation Problems Geometric Programming Chapter Summary An Example of a GP Problem Posynomials and the GP Problem The Dual GP Problem Solving the GP Problem Solving the DGP Problem Constrained Geometric Programming Basic Analysis Chapter Summary Minima and Infima Limits Completeness Continuity Limsup and Liminf Another View Semi-Continuity Convex Sets Chapter Summary The Geometry of Real Euclidean Space A Bit of Topology Convex Sets in RJ More on Projections Linear and Affine Operators on RJ The Fundamental Theorems Block-Matrix Notation Theorems of the Alternative Another Proof of Farkas' Lemma Gordan's Theorem Revisited Vector Spaces and Matrices Chapter Summary Vector Spaces Basic Linear Algebra LU and QR Factorization The LU Factorization Linear Programming Chapter Summary Primal and Dual Converting a Problem to PS Form Duality Theorems The Basic Strong Duality Theorem Another Proof Proof of Gale's Strong Duality Theorem Some Examples The Simplex Method Yet Another Proof The Sherman-Morrison-Woodbury Identity An Example of the Simplex Method Another Example Some Possible Difficulties Topics for Projects Matrix Games and Optimization Chapter Summary Two-Person Zero-Sum Games Deterministic Solutions Randomized Solutions Symmetric Games Positive Games Example: The "Bluffing" Game Learning the Game Non-Constant-Sum Games Differentiation Chapter Summary Directional Derivative Partial Derivatives Some Examples Gateaux Derivative Frechet Derivative The Chain Rule Convex Functions Chapter Summary Functions of a Single Real Variable Functions of Several Real Variables Sub-Differentials and Sub-Gradients Sub-Gradients and Directional Derivatives Functions and Operators Convex Sets and Convex Functions Convex Programming Chapter Summary The Primal Problem From Constrained to Unconstrained Saddle Points The Karush-Kuhn-Tucker Theorem On Existence of Lagrange Multipliers The Problem of Equality Constraints Two Examples The Dual Problem Nonnegative Least-Squares Solutions An Example in Image Reconstruction Solving the Dual Problem Minimum One-Norm Solutions Iterative Optimization Chapter Summary The Need for Iterative Methods Optimizing Functions of a Single Real Variable The Newton-Raphson Approach Approximate Newton-Raphson Methods Derivative-Free Methods Rates of Convergence Descent Methods Optimizing Functions of Several Real Variables Auxiliary-Function Methods Projected Gradient-Descent Methods Feasible-Point Methods Quadratic Programming Simulated Annealing Solving Systems of Linear Equations Chapter Summary Arbitrary Systems of Linear Equations Regularization Nonnegative Systems of Linear Equations Regularized SMART and EMML Block-Iterative Methods Conjugate-Direction Methods Chapter Summary Iterative Minimization Quadratic Optimization Conjugate Bases for RJ The Conjugate Gradient Method Krylov Subspaces Extensions of the CGM Operators Chapter Summary Operators Contraction Operators Orthogonal-Projection Operators Two Useful Identities Averaged Operators Gradient Operators Affine-Linear Operators Paracontractive Operators Matrix Norms Looking Ahead Chapter Summary Sequential Unconstrained Minimization Examples of SUM Auxiliary-Function Methods The SUMMA Class of AF Methods Bibliography Index Exercises appear at the end of each chapter.