# Mov Lecture 25 - Mathematical proof of the Lagrange multiplier method

Sumit Feb 10, 2015 By:

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Lecture 25 - Mathematical proof of the Lagrange multiplier method

This video is part of the Module Design and Optimization of Energy systems and Lecture Series on Design and Optimization of Energy Systems by Prof. C. Balaji , Department of Mechanical Engineering, IIT Madras

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COURSE OUTLINE

Introduction to system design - Regression analysis and curve fitting - modeling of thermal equipment (very brief) - system simulation (successive substitution - Newton - Raphson method) - examples - economic analysis - optimization - Lagrange multipliers, search methods, linear programming, geometric programming- New generation optimization techniques - simulated annealing, Genetic Algorithms, Bayesian statistics.

Examples applied to heat transfer problems and energy systems such as gas and steam power plants, refrigeration systems, heat pumps and so on.

COURSE DETAIL

Sl.No.1

Module 1: Introduction

Introduction to design and specifically system design.

Morphology of design with a flow chart.

Very brief discussion on market analysis, profit, time value of money, an example of discounted cash flow technique.

Concept of workable design, practical example on workable system and optimal design.

Sl.No.2

Module 2 : System Simulation

Classification.

Successive substitution method - examples.

Newton Raphson method - one unknown - examples.

Newton Raphson method - multiple unknowns - examples.

Gauss Seidel method - examples.

Rudiments of finite difference method for partial differential equations, with an example.

Sl.No.3

Module 3: Regression and Curve Fitting

Need for regression in simulation and optimization.

Concept of best fit and exact fit.

Exact fit - Lagrange interpolation, Newton's divided difference - examples.

Least square regression - theory, examples from linear regression with one and more unknowns - examples.

Power law forms - examples.

Gauss Newton method for non-linear least squares regression - examples.

Sl.No.4

Module 4: Optimization

Introduction.

Formulation of optimization problems – examples.

Calculus techniques – Lagrange multiplier method – proof, examples.

Search methods – Concept of interval of uncertainty, reduction ratio, reduction ratios of simple search techniques like exhaustive search, dichotomous search, Fibonacci search and Golden section search – numerical examples.

Method of steepest ascent/ steepest descent, conjugate gradient method – examples.

Geometric programming – examples.

Dynamic programming – examples.

Linear programming – two variable problem –graphical solution.

New generation optimization techniques – Genetic algorithm and simulated annealing - examples.

Introduction to Bayesian framework for optimization- examples.