Description:

This second edition provides an enhanced exposition of the long-overlooked Hadamard semidifferential calculus, first introduced in the 1920s by mathematicians Jacques Hadamard and Maurice René Fréchet. Hadamard semidifferential calculus is possibly the largest family of nondifferentiable functions that retains all the features of classical differential calculus, including the chain rule, making it a natural framework for initiating a large audience of undergraduates and non-mathematicians into the world of nondifferentiable optimization.

Introduction to Optimization and Hadamard Semidifferential Calculus, Second Edition:

• builds upon its prior edition’s foundations in Hadamard semidifferential calculus, showcasing new material linked to convex analysis and nonsmooth optimization;
• presents a modern treatment of optimization and Hadamard semidifferential calculus while remaining at a level that is accessible to undergraduate students; and
• challenges students with exercises related to problems in such fields as engineering, mechanics, medicine, physics, and economics and supplies answers in Appendix B.

Contents:

List of Figures

Preface to the Second Edition • A Great and Beautiful Subject • Intended Audience and Objectives of the Book • A New Revised, Reorganized, and Enriched Edition • Numbering and Referencing Systems • Acknowledgments

Preface to the First Edition • A Great and Beautiful Subject • Intended Audience and Objectives of the Book • Numbering and Referencing Systems • Acknowledgments

Chapter 1. Introduction • Minima and Maxima • Calculus of Variations and Its Offspring • Contents of the Book • Some Background Material in Classical Analysis and Linear Algebra • Greatest Lower Bound and Least Upper Bound • Euclidean Space • Cartesian Product, Balls, and Continuity • Open and Closed Sets 7 • Cauchy and Convergent Sequences • Closed Sets and Closure • Open Covering and Compact Sets 8 • Complement and Boundary of a Set • Mapping, Function, Continuity, and Linearity • Real-Valued and Vector-Valued Functions • Continuity and Continuous Extension • Linear Functions, Transposed, and Associated Matrices • Determinant • Eigenvalues • Exercises

Chapter 2. Existence, Convexities, and Convexification • Introduction • Weierstrass Existence Theorem • Extreme of Function with Extented Values • Lower and Upper Semicontinuities • Existence of Minimizers in U • U Compact • U Closed but Not Necessarily Bounded • Growth Property at Infinity • Some Properties of the Set of Minimizers • Ekeland’s Variational Principle • Convexity, Quasiconvexity, Strict Convexity, and Uniqueness • Definitions: Convex Sets and Functions • Linear and Affine Subspaces and Convex Hull • Linear and Affine Subspaces • Relative Interior of a Convex Set • Convex Hull and Theorem of Caratheodory • Convex Functions • Some Properties and Characterizations • Operations on Convex Functions • Set of Minimizers of a Convex Function • Domain of a Convex Function and Its Relative Interior • Quasiconvex Functions • Strictly Convex Functions and Uniqueness • Convexification and Fenchel—Legendre Transform • Affine Functions • Convex lsc Functions as Upper Envelopes of Affine Functions • Fenchel—Legendre Transform • lsc Convexification and Fenchel—Legendre Bitransform • Infima of the Objective Function and of Its lsc Convexified Function • Primal and Dual Problems and Fenchel Duality Theorem • Exercises

Chapter 3. Semidifferentiability, Differentiability, Continuity, and Convexities • Introduction • Real-Valued Functions of a Real Variable • Continuity and Differentiability • Mean Value Theorem • Property of the Derivative of a Function Differentiable Everywhere • Taylor’s Theorem • Real-Valued Functions of Several Real Variables • Geometrical Approach via the Differential • Semidifferentials, Differentials, Gradient, and Partial Derivatives • Definitions • Examples and Counterexamples • Gradient, Jacobian Mapping, and Jacobian Matrix • Frechet Differential • Hadamard Differential and Semidifferential • Operations on Semidifferentiable Functions • Algebraic Operations, Lower and Upper Envelopes • Chain Rule for the Composition of Functions • Lipschitzian Functions • Definitions and Their Hadamard Semidifferential • Dini and Hadamard Upper and Lower Semidifferentials • Clarke Upper and Lower Semidifferentials • Properties of Upper and Lower Subdifferentials • Continuity, Hadamard Semidifferential, and Frechet Differential • Mean Value Theorem for Functions of Several Variables • Functions of Classes C^(°) and C⁽¹⁾ • Functions of  Class C^(p) and Hessian Matrix • Convex and Semiconvex Functions • Directionally Differentiable Convex Functions • Semidifferentiability and Continuity of Convex Functions • Convexity and Semidifferentiability • Convexity and Continuity • Lower Hadamard Semidifferential at a Boundary Point of the Domain • Semiconvex Functions and Hadamard Semidifferentiability • Preliminaries • Infimum with Respect to One Parameter • Existence and Expression of the Right-hand Side Derivative • Infimum of a Parametrized Quadratic Function • Infimum with Respect to Several Parameters • Sublinear and Superlinear Functions • General Theorem (Y Not Necessarily Compact) • Theorem of Danskin and Its Variations: Y Compact • Concave Case • Semidifferentiable Case • Summary of Semidifferentiability and Differentiability • Exercises

Chapter 4. Optimality Conditions • Introduction • Unconstrained Differentiable Optimization • Some Basic Results and Examples • Least and Greatest Eigenvalues of a Symmetric Matrix • Hadamard Semidifferential of the Least Eigenvalue • Optimality Conditions for U Convex • Cones • Convex Gateaux Differentiable Objective Function • Convex Objective Function without Semidifferentiability • Semidifferentiable Objective Function • Admissible Directions and Tangent Cones to U • Set of Admissible Directions or Half-Tangents • Properties of the Tangent Cones T_u (x) and S_u (x) • Clarke’s and Other Tangent Cones • Orthogonality, Transposition, and Dual Cones • Orthogonality and Transposition • Dual Cones • Dual Cones for Linear Constraints • A Single Constraint • Several Constraints of the Same Types • Mixed Equality and Inequality Constraints • Necessary Optimality Conditions for U Arbitrary • Necessary Optimality Condition • Hadamard Semidifferentiable Objective Function • Arbitrary Objective Function • Dual Necessary Optimality Condition • Affine Equality and Inequality Constraints • Orientation and Notation • Characterization of Tu (x) and of Tu (x)* • Frechet Differentiable Objective Function • Quadratic Objective Function • Theorem of Frank—Wolfe • Convex Objective Function • Nonconvex Objective Function: Equality Constraints • Interpolation: Kriging and Radial Basis Functions • Lagrange and Fenchel Problems: Dual Problems • Some Elements of Two-Person Zero-Sum Games • Linear Programming • Primal and Dual Problems • Different Forms of the LP Problem • Primal and Dual Lagrangian Problems: The Four Cases • Quadratic Programming • Primal and Dual Problems • Different Forms of the QP Problem • Primal and Dual Lagrangian Problems: The Four Cases • Glimpse at Optimality via Subdifferentials • Exercises

Chapter 5. Differentiable and Semidifferentiable Constrained Optimization • Introduction • Equality Constraints: Lagrange • Tangent Cone of Admissible Directions • Jacobian Matrix and Rank or Lyusternik Theorem • Differentiable Objective Function • Lagrange Multipliers Theorem • Examples • Second Order Optimality Conditions • Semidifferentiable Objective Function • Hahn-Banach Theorem • Theorem of Lagrange Revisited • Inequality Constraints: Karush, John, and Kuhn-Tucker • Orientation • Frechet Differentiable Objective and Constraints • Fundamental T .emma and Main Theorem • Sufficient Conditions for Constraint Qualification • Examples • Theorem of E John • Semidifferentiable Objective and Constraint Functions • Auxiliary Lemma • Fundamental Lemma • Main Theorem • Sufficient Conditions for Constraint Qualification • Elements of Bibliography Index of Notation Index • Mixed Equality and Inequality Constraints: MangasarianFromovitz • Fréchet Differentiable Objective and Constraint Functions • Preliminaries and Notation • Fundamental Lemma • Main Theorem • Sufficient Conditions for Constraint Qualification • Examples • Semidifferentiable • Objective and Inequality Constraint Functions • Fundamental Lemma • Main Theorem • Sufficient Conditions for Constraint Qualification • Exercises

Appendix A. Inverse Function Theorem • Inverse Function Theorem • A Version of the Rank Theorem or Lyusternik’s Theorem

Appendix B. Answers to Exercises • Exercises of Chapter 1 • Exercises of Chapter 2 • Exercises of Chapter 3 • Exercises of Chapter 4 • Exercises of Chapter 5

Appendix C. Hadamard Semidifferential: Functions Defined on Arbitrary Sets • Introduction • Functions Defined on a Vector Space • Functions Defined on Arbitrary Sets

Elements of Bibliography

Index of Notation

Index

About the Author:

Michel C. Delfour is a professor of mathematics and statistics at the University of Montreal and is the author or coauthor of 13 books and about 200 papers. Delfour’s areas of research include shape and topological optimal design, analysis and control of delay and distributed parameter systems, control and stabilization of large flexible space structures, numerical methods in differential equations and optimization, and transfinite interpolation. He is a Fellow of SIAM, the Canadian Mathematical Society, the Academy of Science at the Royal Society of Canada, and, formerly, the Guggenheim Foundation.

Target Audience:

Students of mathematics, physics, engineering, economics, and other disciplines that demand a basic knowledge of mathematical analysis and linear algebra will find this a fitting primary or companion resource for their studies.