The Mathematics of Nonlinear Programming

1 Unconstrained Optimization via Calculus.- 1.1. Functions of One Variable.- 1.2. Functions of Several Variables.- 1.3. Positive and Negative Definite Matrices and Optimization.- 1.4. Coercive Functions and Global Minimizers.- 1.5. Eigenvalues and Positive Definite Matrices.- Exercises.- 2 Convex Sets and Convex Functions.- 2.1. Convex Sets.- 2.2. Some Illustrations of Convex Sets in Economics— Linear Production Models.- 2.3. Convex Functions.- 2.4. Convexity and the Arithmetic-Geometric Mean Inequality— An Introduction to Geometric Programming.- 2.5. Unconstrained Geometric Programming.- 2.6. Convexity and Other Inequalities.- Exercises.- 3 Iterative Methods for Unconstrained Optimization.- 3.1. Newton’s Method.- 3.2. The Method of Steepest Descent.- 3.3. Beyond Steepest Descent.- 3.4. Broyden’s Method.- 3.5. Secant Methods for Minimization.- Exercises.- 4 Least Squares Optimization.- 4.1. Least Squares Fit.- 4.2. Subspaces and Projections.- 4.3. Minimum Norm Solutions of Underdetermined Linear Systems.- 4.4. Generalized Inner Products and Norms; The Portfolio Problem.- Exercises.- 5 Convex Programming and the Karush-Kuhn-Tucker Conditions.- 5.1. Separation and Support Theorems for Convex Sets.- 5.2. Convex Programming; The Karush-Kuhn-Tucker Theorem.- 5.3. The Karush-Kuhn-Tucker Theorem and Constrained Geometric Programming.- 5.4. Dual Convex Programs.- 5.5. Trust Regions.- Exercises.- 6 Penalty Methods.- 6.1. Penalty Functions.- 6.2. The Penalty Method.- 6.3. Applications of the Penalty Function Method to Convex Programs.- Exercises.- 7 Optimization with Equality Constraints.- 7.1. Surfaces and Their Tangent Planes.- 7.2. Lagrange Multipliers and the Karush-Kuhn-Tucker Theorem for Mixed Constraints.- 7.3. Quadratic Programming.- Exercises.