Convexity and Optimization in Rn

Preface.

I: Topics in Real Analysis.

1. Introduction.

2. Vectors in R".

3. Algebra of Sets.

4. Metric Topology of R".

5. Limits and Continuity.

6. Basic Propertyof Real Numbers.

7. Compactness.

8. Equivalent Norms and Cartesian Products.

9. Fundamental Existence Theorem.

10. Linear Transformations.

11. Differentiation in R".

II: Convex Sets in R".

1. Lines and Hyperplanes in R".

2. Properties of Convex Sets.

3. Separation Theorems.

4. Supporting Hyperplanes:Extreme Points.

5. Systems of Linear Inequalities:Theorems of the Alternative.

6. Affine Geometry.

7. More on Separation and Support.

III: Convex Functions.

1. Definition and Elementary Properties.

2. Subgradients.

3. Differentiable Convex Functions.

4. Alternative Theorems for Convex Functions.

5. Application to Game Theory.

IV: Optimization Problems.

1. Introduction.

2. Differentiable Unconstrained Problems.

3. Optimization of Convex Functions.

4. Linear Programming Problems.

5. First-Order Conditions for Differentiable NonlinearProgrammingProblems.

6. Second-Order Conditions.

V: Convex Programming and Duality.

1. Problem Statement.

2. Necessary Conditions and Sufficient Conditions.

3. Perturbation Theory.

4. Lagrangian Duality.

5. Geometric Interpretation.

6. Quadratic Programming.

7. Dualityin Linear Programming.

VI: Simplex Method.

1. Introduction.

2. Extreme Points of Feasible Set.

3. Preliminaries to Simplex Method.

4. Phase II of Simplex Method.

5. Termination and Cycling.

6. Phase I of Simplex Method.

7. Revised Simplex Method.

Bibliography.

Index.