Basic Mathematical Programming Theory

Prof. Giorgio Giorgi teaches Mathematics at the Faculty of Economics of the University of Pavia. His research interests essentially focus on mathematical economics, generalized convexity, and optimization.Bienvenido Jiménez and Vicente Novo are professors of Applied Mathematics at the National University of Distance Education, Madrid, Spain. Their research focus on smooth and nonsmooth optimization, mathematical programming and multiobjective, vector and set optimization.

• Preface.- Chapter 1. Basic Notions and Definitions.- 1.1. Introduction.- 1.2. Basic Notions of Analysis and Linear Algebra.- 1.3. Basic Notions and Properties of Optimization Problems.- Chapter 2. Elements of Convex Analysis. Theorems of the Alternative for Linear Systems. Tangent Cones.- 2.1. Elements of Convex Analysis.- 2.2. Theorems of the Alternative for Linear Systems.- 2.3. Tangent Cones.- Chapter 3. Convex Functions and Generalized Convex Functions.- 3.1. Convex Functions.- 3.2. Generalized Convex Functions.- 3.3. Optimality Properties of Convex and Generalized Convex Functions. Theorems of the Alternative for Nonlinear Systems.- Chapter 4. Unconstrained Optimization Problems. Set-Constrained Optimization Problems. Classical Constrained Optimization Problems.- 4.1. Unconstrained Optimization Problems.- 4.2. Set-Constrained Optimization Problems.- 4.3. Optimization Problems with Equality Constraints (“Classical Constrained Optimization Problems”).- Chapter 5. Constrained Optimization Problems with Inequality Constraints.- 5.1. First-Order Conditions.- 5.2. Constraint Qualifications.- 5.3. Second-Order Conditions.- 5.4. Other Formulations of the Problem. Some Examples.- Chapter 6. Constrained Optimization Problems with Mixed Constraints.- 6.1. First-Order Conditions.- 6.2. Constraint Qualifications.- 6.3. Second-Order Conditions.- 6.4. Problems with a Set Constraint. Asymptotic Optimality Conditions.- Chapter 7.Sensitivity Analysis.- 7.1. General Results.- 7.2. Sensitivity Results for Right-Hand Side Perturbations.- Chapter 8. Convex Optimization: Saddle Points Characterization and Introduction to Duality.- 8.1. Convex Optimization: Saddle Points Characterization.- 8.2. Introduction to Duality.- Chapter 9. Linear Programming and Quadratic Programming.- 9.1. Linear Programming.- 9.2. Duality for Linear Programming.- 9.3. Quadratic Programming.- Chapter 10. Introduction to Nonsmooth Optimization Problems.- 10.1. The Convex Case.- 10.2. The Lipschitz Case.- 10.3.The Axiomatic Approach of K.-H. Elster and J. Thierfelder to Nonsmooth Optimization.- Chapter 11. Introduction to Multiobjective Optimization.- 11.1. Optimality Notions.- 11.2. The Weighted Sum Method and Optimality Conditions.- References.- Index.

“This book serves as a comprehensive and indispensable tool for Ph.D. students and young researchers in the Field of nonlinear programming. … it offers a substantial coverage of theoretical results concerning optimality conditions in nonlinear programming. Certain chapters are particularly valuable as teaching materials for advanced master's courses.” (Giovanni Fasano, Mathematical Reviews, Decemeber, 2024)