Transformation Methods MOM (Method of Multipliers)

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Transformation MethodsMOM (Method of Multipliers)

• Study of Engineering Optimization
• Guanyao Huang
• RUbiNet - Robust and Ubiquitous Networking
  Research Group
• 2010 July 9th

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Introduction

• Motivation
• Ill-condition of subproblems in penalty
  approaches.
• Method MOM
• R is no longer iteratively updated, and
  are updated.
• Benefits
• Contour shape remains the same.

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Motivation
Inverse penalty
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• Convergence is associated with ever-increasing
  distortion of the penalty contours, which
  increases significantly the possibility of
  failure of the unconstrained search method. The
  unconstrained search might not be completed
  successfully.

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Solution

• Method of centers
• Which is equivalent to

These parameter-free methods, although attractive
on the surface, are exactly equivalent to SUMT
with a particular choice of updating rule for R.
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MOM augment the lagrangian to form an
unconstrained function whose minimum is a
Kuhn-Tucker point of the original problem
Another solution MOM
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Detailed procedure
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Property Second order derivative

• If the constraints are linear

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Link between Lagrange multiplier and KT (1)

• When iteration terminate?
• or
• then
• with
• The limit points of is KT point

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Link between Lagrange multiplier and KT (2)

• Lagrange multiplier estimates

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MOM characteristics

• We still have the problem of choosing R! (6.3.6)

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Revisit example 6.1
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Variable Bounds

• Experience simply set the out-of-bounds
  variables to their violated bounds
  simultaneously.
• An example of MOM Example 6.4.
• Some bounds are not linear.
• DFP DavidonFletcherPowell