This reference book gives the reader a complete but comprehensive presentation of the foundations of convex analysis and presents applications to significant situations in engineering. The presentation of the theory is self-contained and the proof of all the essential results is given. The examples consider meaningful situations such as the modeling of curvilinear structures, the motion of a mass of people or the solidification of a material. Non convex situations are considered by means of relaxation methods and the connections between probability and convexity are explored and exploited in order to generate numerical algorithms.

This reference book gives the reader a complete but comprehensive presentation of the foundations of convex analysis and presents applications to significant situations in engineering. The presentation of the theory is self-contained and the proof of all the essential results is given. The examples consider meaningful situations such as the modeling of curvilinear structures, the motion of a mass of people or the solidification of a material. Non convex situations are considered by means of relaxation methods and the connections between probability and convexity are explored and exploited in order to generate numerical algorithms.

Introduction ixPART 1 MOTIVATION: EXAMPLES AND APPLICATIONS 1Chapter 1 Curvilinear Continuous Media 31.1 One-dimensional curvilinear media 41.2 Supple membranes 22Chapter 2 Unilateral System Dynamics 332.1 Dynamics of ideally flexible strings 342.2 Contact dynamics 40Chapter 3 A Simplified Model of Fusion/Solidification 533.1 A simplified model of phase transition 53Chapter 4 Minimization of a Non-Convex Function 614.1 Probabilities, convexity and global optimization 61Chapter 5 Simple Models of Plasticity 695.1 Ideal elastoplasticity 72PART 2 THEORETICAL ELEMENTS 77Chapter 6 Elements of Set Theory 796.1 Elementary notions and operations on sets 806.2 The axiomof choice 836.3 Zorn's lemma 89Chapter 7 Real Hilbert Spaces 977.1 Scalar product and norm 997.2 Bases anddimensions 1077.3 Open sets and closed sets 1147.4 Sequences 1237.5 Linear functionals 1377.6 Complete space 1467.7 Orthogonal projection onto a vector subspace 1607.8 Riesz's representationtheory 1677.9 Weak topology 1737.10 Separable spaces: Hilbert bases and series 184Chapter 8 Convex Sets 2018.1 Hyperplanes 2018.2 Convexsets 2088.3 Convexhulls 2128.4 Orthogonal projection on a convex set 2178.5 Separationtheorems 2288.6 Convexcone 241Chapter 9 Functionals on a Hilbert Space 2539.1 Basic notions 2549.2 Convexfunctionals 2619.3 Semi-continuous functionals 2719.4 Affine functionals 2989.5 Convexification and LSC regularization 3039.6 Conjugate functionals 3209.7 Subdifferentiability 331Chapter 10 Optimization 36110.1 The optimization problem 36110.2 Basic notions 36210.3 Fundamental results 374Chapter 11 Variational Problems 42111.1 Fundamental notions 42111.2 Zeros of operators 45511.3 Variational inequations 46311.4 Evolutionequations 469Bibliography 487Index 495