Advanced topics in optimization
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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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Advanced Topics in Optimization

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Advanced topics in optimization

Advanced Topics in Optimization

Multilevel Optimization

Optimization Methods: M8L3


Objectives

Objectives

  • To discuss about Multilevel Optimization

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

Optimization Methods: M8L3


Introduction

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

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

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 contd1

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 contd2

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

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

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 contd1

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


Advanced topics in optimization

Thank You

Optimization Methods: M8L3


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