Course

Unit 1

Introduction: Mathematical optimization, Convex optimization, Least-squares and linear programming, Simplex method, Two phase method, Integer linear programming, Nonlinear optimization

Unit 2

Convex sets: Affine and convex sets. Some important examples. Operations that preserve convexity. Generalized inequalities. Separating and supporting hyperplanes. Dual cones and generalized inequalities

Unit 3

Convex functions: Basic properties and examples. Operations that preserve convexity. The conjugate function. Quasi-convex functions. Log-concave and log-convex functions. Convexity with respect to generalized inequalities

Unit 4

Convex optimization problems. Optimization problems. Convex optimization. Linear optimization problems. Quadratic optimization problems. Geometric programming. Generalized inequality constraints. Vector optimization.

Unit 5

The Lagrange dual function. The Lagrange dual problem. Geometric interpretation. Saddle-point interpretation. Optimality conditions. Perturbation and sensitivity analysis. Theorems of alternatives. Generalized inequalities.

Text Books / References

Textbooks:

1. Stephen Boyd, Lieven Vandenberghe, Convex Optimization, Cambridge University Press, 2009.

References:

1. Dimitri P. Bertsekas, Convex Optimization Theory, University Press, 2016.
2. Hamdy A. Taha, “Operations Research-An Introduction”, Prentice Hall, 9th Edition, 2010.
3. Edwin K.P. Chong, Stanislaw H. Zak, “An Introduction to Optimization”, 2nd edition, Wiley, 2021.

This course introduces the fundamentals of convex sets and convex functions to study the convex optimization problems. Also, the different types of linear and nonlinear constrained optimization problems with their methods are included. Topics include various optimization problems for linear and nonlinear programming problems, affine and convex sets with their important properties, convex functions and their special types of functions, convex optimization, geometric programming, dual and Lagrange dual functions with equality and inequality constraints, along with their applications.

Course Outcomes: After successful completion of this course, students will be able to

• CO1 : Understand the basic concepts and different types of optimizations related to linear as well as nonlinear problems in engineering. Learn the concept of affine and convex sets and their importance.
• CO2 : Understand the properties, generalized inequalities, and operations that preserve convexity in convex sets as well as convex functions. Learn the concepts of separating and supporting hyperplanes, dual cones, and generalized inequalities.  
• CO3 : Understand and learn the applications for the different optimization problems such as convex optimization, optimization problems, linear optimization, quadratic optimization, geometric programming, generalized inequality constraints, and vector optimization.
• CO4 : Understand the duality and Lagrange dual function problem. Learn the geometric interpretation, saddle point interpretation, perturbation, sensitivity analysis, generalized inequalities, and their applications.

CO-PO Mapping: