Unit I

Introduction to Optimization, Historical Development, Applications of Optimization, Statement of an Optimization Problem, Classification of Optimization Problems, Optimization Techniques.

Unit II

Single Variable Optimization- Optimality criteria, bracketing methods-exhaustive search method, bounding phase method- region elimination methods- interval halving, Fibonacci search, golden section search, point estimation-successive quadratic search, gradient based methods.

Unit III

Multivariable Optimization, optimality criteria, unconstrained optimization-solution by direct substitution, unidirectional search-direct search methods, evolutionary search method, simplex search method, Hook-Jeeves pattern search method.

Unit IV

Gradient based methods-steepest descent, Cauchy’s steepest descent method, Newton’s method, conjugate gradient method-constrained optimization Multivariable Optimization with no constraints, Multivariable Optimization with Equality Constraints, Solution by Direct Substitution

Unit V

Solution by the Method of Lagrange Multipliers- Multivariable Optimization with Inequality Constraints, Kuhn–Tucker Conditions, Constraint Qualification, Convex Programming Problem.

Course Outcomes:

CO1: Understand different types of Optimization Techniques in engineering problems. Learn Optimization methods such as Bracketing methods, Region elimination methods, Point estimation methods.
CO2: Learn gradient based Optimizations Techniques in single variables as well as multi-variables (non-linear).
CO3: Understand the Optimality criteria for functions in several variables and learn to apply OT methods like unidirectional search and direct search methods.
CO4: Learn constrained optimization techniques. Learn to verify Kuhn-Tucker conditions and Lagrangian Method.
CO5: Familiarize the concept of optimization in practical applications to find the best feasible solutions in practical applications

Practical/Lab to be Performed Using MATLAB/Python

1. Solution of optimization problems using Karush-Kuhn-Tucker conditions
2. Test whether the given matrix is positive definite/negative definite/semi positive Definite/ semi negative definite
3. Test whether the given function is concave/convex.
4. To determine local/Relative optima of a given unconstrained problem.
   1. Find optimal solution of single variable functions using (i)Exhaustive search methods,
      1. Bounding phase method
      2. Region elimination method interval halving,
      3. Fibonacci search
      4. Golden section search
      5. Point estimation-successive quadratic search
      6. Gradient based methods
   2. Find optimal solution of two variable problems based on the methods
      1. (i)Hook-Jeeves pattern search method
      2. (ii) Gradient based methods-steepest descent
      3. (iii)Cauchy’s steepest descent method
      4. (iv)Newton’s method
      5. Conjugate gradient method-constrained optimization

Text Book

1. Edwin K.P. Chong, Stanislaw H. Zak, “An Introduction to Optimization”, 2nd edition, Wiley, 2013.
2. S.S. Rao, “Optimization Theory and Applications”, Second Edition, New Age International (P) Limited Publishers, 1995.

References

• Kalyanmoy Deb, “Optimization for Engineering Design Algorithms and Examples”, Prentice Hall of India, New Delhi, 2004.
• M. Asghar Bhatti (2000), Practical optimization methods, Springer.