PEE218: COMPUTATIONAL TECHNIQUES AND STATISTICAL METHODS

 

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3

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3.5


Course Objectives:

1.      To explain the necessary conditions of convex and concave optimisation problems

2.      To describe various types of conventional solution techniques applied to optimisation problems

3.      To impart knowledge about the probability concepts

4.      To demonstrate the ANOVA process

Introduction: Review of Linear Programming concepts.

 

Unconstrained Problems:  First-Order Necessary Conditions, Second-Order Conditions Convex and Concave Functions, Minimization and Maximization of Convex Functions Zero-Order Conditions, Global Convergence of Descent Algorithms, Speed of Convergence.

 

Basic Descent Methods: Fibonacci and Golden Section Search, Line Search by Curve fitting, Global Convergence of Curve Fitting, Closedness of Line Search Algorithms, Inaccurate Line Search, The Method of Steepest Descent, Newton's Method, Coordinate Descent Methods, Spacer Steps.

 

Conjugate Direction Methods: Conjugate Directions, Descent Properties of the Conjugate Direction Method, The Conjugate Gradient Method, The CûG Method as an Optimal Process, The Partial Conjugate Gradient Method, Extension to Nonquadratic Problems,

 

Quasi-Newton Methods: Modified Newton Method, Construction of the Inverse, DavidonûFletcherûPowell Method, The Broyden Family, Convergence Properties, Scaling, Memoryless Quasi-Newton Methods, Combination of Steepest Descent and Newton's Method.

 

Constrained optimization: Kuhn-Tucker conditions, Quadratic programming problems, Algorithm for constrained optimization, Gradient projection method, Dual of quadratic programming problems.

 

Review of Probability: Appraisal of axiomatic approach of probability, Conditional probability, Baye’s rule, Conditional distributions, and conditional expectations.

 

Markov Chains: Basics of markov chains, Finite state space, Markov chains, Transition and stationary markov chains. Continuous time markov process: continuous time branching processes, Kolmogorov, Forward and backward equations, Pure birth, Pure death, Birth and death process.

 

Analysis of Variance: One Way Classification: ANOVA for fixed effect model, ANOVA for Random Effect Model, Two-way Classification (one observation per cell): ANOVA for fixed effect model, ANOVA for Random Effect Model.

 

Design of Experiments: Completely Randomised Design, Randomised Block Design, Latin Square Design, their statistical analysis and variance of estimates, Analysis of Covariance.

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Course Learning Outcomes (COL):

On the completion of the course, the student will be able:

1.      To apply the conventional techniques of direct search to solve un constrained optimisation problems

2.      To solve constrined optimisation problems.

3.      To apply ANOVA test to assess the degree of variance,

4.      To analyse design of experiments through variance and covariance estimates

5.      To apply the concept of proabilibity to understand the frequency of occurrence and adequacy estimate.

Recommended Books:

1.        Luenberger D.G., Linear and Nonlinear Programming, Addison Wesley (2003).

2.        Fletcher R., Practical methods of Optimization, John Wiley (1980).

3.        Taha H.A., Operation research- An Introduction, PHI (2007).

4.        Billy E. Gillett, Introduction to operation Research – A computer oriented algorithmic approach, Tata McGraw Hill (1993).

5.        Populis,A., Random Variables and Stochastic Processes, Tata McGraw Hill (2002).

6.        Montgomery, Introduction to Statistical Quality Control, John Wiley and Sons (2005).