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# Advanced Topics in Optimization PowerPoint PPT Presentation

Advanced Topics in Optimization. Multilevel Optimization. Objectives. To discuss about Multilevel Optimization To describe a decomposition method for nonlinear optimization problems, known as model-coordination method. Introduction.

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#### Presentation Transcript

Multilevel Optimization

Optimization Methods: M8L3

### Objectives

• To discuss about Multilevel Optimization

• To describe a decomposition method for nonlinear optimization problems, known as model-coordination method

Optimization Methods: M8L3

### Introduction

• In practical situations, an optimization problem involves a large number of variables and constraints

• In multilevel optimization, such large sized problems are decomposed into smaller independent problems

• The overall optimum solution is obtained by solving each sub-problem independently

Optimization Methods: M8L3

### Model Coordination Method

• Consider an minimization optimization problem consisting of n variables,

• subjected to constraints

• where and represents the lower and upper bound of the decision variable

• Decision variable vector:

Optimization Methods: M8L3

### Model Coordination Method …contd.

• For applying model coordination method, the vector X should be divided into two sub-vectors, Y and Z

• Y contains the coordination variables between the subsystems i.e., variables that are common to the sub-problems

• Z vector contains the free or confined variables of sub-problems

• If the problem is partitioned into ‘P’ sub-problems, then vector Z can also be partitioned into ‘P’ variable sets, each set corresponding to each sub-problem

Optimization Methods: M8L3

### Model Coordination Method …contd.

• Thus the objective function can be partitioned into ‘P’ parts

where denotes the objective function of the kthsub-problem

• The coordination variable Ywill appear in all sub-objective functions and Zk will appear only in kth sub-objective function

Optimization Methods: M8L3

### Model Coordination Method …contd.

• Similarly the constraints are also decomposed as

• The lower and upper bound constraints are

• The problem is decomposed and solved using a two level approach

Optimization Methods: M8L3

### Model Coordination Method: Procedure

• First level:

• Fix the value of the coordination variables, Y at some value, say Yopt

• Solve each independent sub-problem and find the value of Zk

subject to

Optimization Methods: M8L3

### Model Coordination Method: Procedure …contd.

• Let the values of Zk obtained by solving this problem be Zkopt

• Second level:

• Now consider the problem after substituting the Zkopt values

• subject to

Optimization Methods: M8L3

### Model Coordination Method: Procedure …contd.

• Solve this problem to find a new Yopt

• Again solve the first level problems

• This process is repeated until convergence is achieved

Optimization Methods: M8L3

Thank You

Optimization Methods: M8L3