Computational Methods and Optimization

This course provides an introduction to computational methods for solving optimization problems. Students develop the vector calculus foundations needed for optimization, explore applications in operations research and engineering, and learn linear programming and constrained optimization techniques. Laboratory work gives students practice applying these methods through projects.

The course is divided into four parts.

1. Numerical methods: root-finding techniques, which also underpin gradient-descent methods, and numerical integration of complex functions.
2. Vector calculus: multivariate functions, higher-order derivatives, extrema, Taylor expansions, gradients, divergence, curl, and their physical significance.
3. Constrained and unconstrained optimization: Hessians and Lagrange multipliers for solving optimization problems.
4. Optimization paradigms: linear programming for linear objectives with linear equality and inequality constraints, and dynamic programming for breaking complex decisions into smaller steps. The travelling-salesperson problem is used as a case study.

1. Become familiar with constrained and unconstrained optimization techniques.
2. Apply vector calculus concepts, including gradients, divergence, and the Hessian matrix, to analyze and solve optimization problems.
3. Implement optimization algorithms using Python.
4. Critically analyze the performance and effectiveness of optimization methods.
5. Explore operations research and mathematical modeling through applied engineering projects.

By the end of the course, students will be able to:
1. Compute integrals of complex mathematical functions using numerical techniques.
2. Compute roots of mathematical functions using computational methods.
3. Analyze vector-valued functions and identify extrema and saddle points.
4. Compute the curl and divergence of vector fields.
5. Solve operations research problems using techniques such as linear and dynamic programming.

Numerical Analysis, Richard L. Burden and J. Douglas Faires, 9th edition, Cengage Learning, 2010.
Thomas’ Calculus, J. Hass, C. Heil and M. D. Weir, 15th edition, Pearson Education, 2024.
Calculus, H. Anton, I. Bivens and S. Davis, 10th edition, John Wiley & Sons, 2016.
Introduction to Operations Research, Frederick S. Hillier and Gerald J. Lieberman, 10th edition, McGraw-Hill Education, 2014.