This text provides a modern and mathematically rigorous treatment of semidifferential calculus in the context of optimization. Semidifferentials are a natural tool for solving certain problems in non-differentiable optimization. Classical notions in convex analysis are introduced (convexification, duality, linear and quadratic programming, two-person zero-sum games, Lagrange primal and dual problems, semiconvex and semiconcave functions) and the theory is developed to a sophisticated enough level to tackle finite-dimensional versions of problems in the calculus of variations. This text is designed for a one-term course at the undergraduate level in a wide variety of numerate disciplines and includes sufficient background material in calculus and linear algebra to be self-contained. Additional material beyond the scope of a basic undergraduate course is included to further develop the theory. The theoretical content of the text is enriched by numerous examples and exercises, for which solutions are included.
• A self-contained one-term course in semidifferential optimisation • Highly applicable in fields such as physics, engineering, medicine and economics • Mathematically rigorous, with definitions, theorems and proofs
Contents
1. Introduction; 2. Existence, convexities, and convexification; 3. Semi-differentiability, differentiability, continuity, and convexities; 4. Optimality conditions; 5. Constrained differentiable optimization; Appendix A. Inverse and implicit function theorems; Appendix B. Answers to exercises.
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