Introduction. / Chapter 1: |
Problem Statement and Basic Definitions / 1.1: |
Illustrative Examples / 1.2: |
Guidelines for Model Construction / 1.3: |
Exercises |
Notes and References |
Convex Analysis. / Part 1: |
Convex Sets. / Chapter 2: |
Convex Hulls / 2.1: |
Closure and Interior of a Set / 2.2: |
Weierstrass's Theorem / 2.3: |
Separation and Support of Sets / 2.4: |
Convex Cones and Polarity / 2.5: |
Polyhedral Sets, Extreme Points, and Extreme Directions / 2.6: |
Linear Programming and the Simplex Method / 2.7: |
Convex Functions and Generalizations. / Chapter 3: |
Definitions and Basic Properties / 3.1: |
Subgradients of Convex Functions / 3.2: |
Differentiable Convex Functions / 3.3: |
Minima and Maxima of Convex Functions / 3.4: |
Generalizations of Convex Functions / 3.5: |
Optimality Conditions and Duality. / Part 2: |
The Fritz John and Karush-Kuhn-Tucker Optimality Conditions. / Chapter 4: |
Unconstrained Problems / 4.1: |
Problems Having Inequality Constraints / 4.2: |
Problems Having Inequality and Equality Constraints / 4.3: |
Second-Order Necessary and Sufficient Optimality Conditions for Constrained Problems / 4.4: |
Constraint Qualifications. / Chapter 5: |
Cone of Tangents / 5.1: |
Other Constraint Qualifications / 5.2: |
Lagrangian Duality and Saddle Point Optimality Conditions. / 5.3: |
Lagrangian Dual Problem / 6.1: |
Duality Theorems and Saddle Point Optimality Conditions / 6.2: |
Properties of the Dual Function / 6.3: |
Formulating and Solving the Dual Problem / 6.4: |
Getting the Primal Solution / 6.5: |
Linear and Quadratic Programs / 6.6: |
Algorithms and Their Convergence / Part 3: |
The Concept of an Algorithm. / Chapter 7: |
Algorithms and Algorithmic Maps / 7.1: |
Closed Maps and Convergence / 7.2: |
Composition of Mappings / 7.3: |
Comparison Among Algorithms / 7.4: |
Unconstrained Optimization. / Chapter 8: |
Line Search Without Using Derivatives / 8.1: |
Line Search Using Derivatives / 8.2: |
Some Practical Line Search Methods / 8.3: |
Closedness of the Line Search Algorithmic Map / 8.4: |
Multidimensional Search Without Using Derivatives / 8.5: |
Multidimensional Search Using Derivatives / 8.6: |
Modification of Newton's Method: Levenberg-Marquardt and Trust Region Methods / 8.7: |
Methods Using Conjugate Directions: Quasi-Newton and Conjugate Gradient Methods / 8.8: |
Subgradient Optimization Methods / 8.9: |
Penalty and Barrier Functions. / Chapter 9: |
Concept of Penalty Functions / 9.1: |
Exterior Penalty Function Methods / 9.2: |
Exact Absolute Value and Augmented Lagrangian Penalty Methods / 9.3: |
Barrier Function Methods / 9.4: |
Polynomial-Time Interior Point Algorithms for Linear Programming Based on a Barrier Function / 9.5: |
Methods of Feasible Directions. / Chapter 10: |
Method of Zoutendijk / 10.1: |
Convergence Analysis of the Method of Zoutendijk / 10.2: |
Successive Linear Programming Approach / 10.3: |
Successive Quadratic Programming or Projected Lagrangian Approach / 10.4: |
Gradient Projection Method of Rosen / 10.5: |
Reduced Gradient Method of Wolfe and Generalized Reduced Gradient Method / 10.6: |
Convex-Simplex Method of Zangwill / 10.7: |
Effective First- and Second-Order Variants of the Reduced Gradient Method / 10.8: |
Linear Complementary Problem, and Quadratic, Separable, Fractional, and Geometric Programming. / Chapter 11: |
Linear Complementary Problem / 11.1: |
Convex and Nonconvex Quadratic Programming: Global Optimization Approaches / 11.2: |
Separable Programming / 11.3: |
Linear Fractional Programming / 11.4: |
Geometric Programming / 11.5: |
Mathematical Review. / Appendix A: |
Summary of Convexity, Optimality Conditions, and Duality. / Appendix B: |
Bibliography. |
Index |
Introduction |
Convex Analysis |
Convex Sets |
Convex Functions and Generalizations |
Optimality Conditions and Duality |
The Fritz John and Karush-Kuhn-Tucker Optimality Conditions |
Constraint Qualification |
Lagrangian Duality and Saddle Point Optimality Conditions |
The Concept of an Algorithm |
Unconstrained Optimization |
Penalty and Barrier Functions |
Methods of Feasible Directions |
Linear Complementary Problem, and Quadratic, Separable, Fractional, and Geometric Programming |
Mathematical Review |
Summary of Convexity, Optimality Conditions, and Duality |
Bibliography |
Introduction. / Chapter 1: |
Problem Statement and Basic Definitions / 1.1: |
Illustrative Examples / 1.2: |