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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