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Advanced Topics in Optimization PowerPoint Presentation

Advanced Topics in Optimization

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Presentation Transcript

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

Optimization Methods: M8L3

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