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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

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

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Thank You

Optimization Methods: M8L3

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