**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.