1st Edition

# Elements of Classical and Geometric Optimization

By Debasish Roy, G Visweswara Rao Copyright 2024
524 Pages 31 Color & 197 B/W Illustrations
by CRC Press

524 Pages 31 Color & 197 B/W Illustrations
by CRC Press

Also available as eBook on:

This comprehensive textbook covers both classical and geometric aspects of optimization using methods, deterministic and stochastic, in a single volume and in a language accessible to non-mathematicians. It will help serve as an ideal study material for senior undergraduate and graduate students in the fields of civil, mechanical, aerospace, electrical, electronics, and communication engineering.

The book includes:

• Derivative-based Methods of Optimization.
• Direct Search Methods of Optimization.
• Basics of Riemannian Differential Geometry.
• Geometric Methods of Optimization using Riemannian Langevin Dynamics.
• Stochastic Analysis on Manifolds and Geometric Optimization Methods.

This textbook comprehensively treats both classical and geometric optimization methods, including deterministic and stochastic (Monte Carlo) schemes. It offers an extensive coverage of important topics including derivative-based methods, penalty function methods, method of gradient projection, evolutionary methods, geometric search using Riemannian Langevin dynamics and stochastic dynamics on manifolds. The textbook is accompanied by online resources including MATLAB codes which are uploaded on our website. The textbook is primarily written for senior undergraduate and graduate students in all applied science and engineering disciplines and can be used as a main or supplementary text for courses on classical and geometric optimization.

Contents

Chapter 1 Optimization methods – A preview
1.1 Introduction
1.2  The continuous case – mathematical formulation
1.3  The discrete case – The travelling salesman problem
1.4  Basics of probability theory and random number generation
1.5 The brachistochrone problem
1.6 More on functional optimization: Hamilton’s principle
1.7 Constrained optimization problems and optimality conditions
1.8. Functional optimization and optimal control
Concluding Remarks
Exercises

Notations
References

Chapter 2  Classical derivative-based methods of optimization
2.1 Introduction
2.3 Quasi-Newton methods
2.4 Penalty function methods
2.5 Linear programming (LP)
2.6. Method of generalized reduced gradients
2.7 Method of feasible directions
Concluding remarks
Exercises
Notations
References

Chapter 3 – Classical derivative-free methods of optimization
3.1 Introduction
3.2 Direct search methods
3.3   Other direct search methods
3.4  Metaheuristics - Evolutionary methods

Concluding remarks
Exercises
Notations
References

Chapter 4 Elements of Riemannian Differential Geometry and geometric methods of optimization

4.1 Introduction
4.2 Tangent vectors and tangent space on manifolds
4.3 Riemannian (geometric) version of some classical gradient methods
4.4. Statistical estimation by geometrical method of optimization
4.5. Stochastic processes, stochastic calculus and solution of SDEs
4.6. Analogy between statistical sampling and stochastic optimization
4.7. Geometric method of optimization by Riemannian Langevin dynamics
Concluding remarks
Exercises
Notations
References

Chapter 5 Stochastic analysis on a manifold and more on geometric optimization methods
5.1. Introduction
5.2 Stochastic development on a manifold
5.3. Non-convex function optimization based on stochastic development
5.4.  Parameter estimation by GALA
Concluding remarks
Notations
References

### Biography

Debasish Roy, Professor, Department of Civil Engineering, Indian Institute of Science, Bangalore. He obtained his Ph.D. from the Indian Institute of Science, followed by post-doctoral research at the University of Innsbruck, Austria. He has published over 140 research papers in journals of national and international repute. His current research areas include geometrically inspired and gauge theories for continuum mechanics of solids, non-equilibrium thermodynamics of solids and fluctuation relations valid far from equilibrium, defect engineering and metamaterials with acoustic band gaps and optimization based on stochastic search on Riemannian manifolds.

G. Visweswara Rao is currently working as an engineering consultant in Bangalore, India. He received his Ph.D. from the Indian Institute of Science, Bangalore, in 1989. He has published several research papers in the areas of structural dynamics specific to earthquake engineering, nonlinear and random vibration, and structural control. His areas of research include non-linear and stochastic structural dynamics.