Decomposition based techniques in mathematical programming

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Decomposition based techniques in mathematical
programming
Chandra A. Poojari
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Outline

• Structure of a two-stage Stochastic programming
  model.
• Description of the Benders decomposition.
• Application of Benders on an illustrative
  example.
• Extensions to the Benders algorithm.
• Other decomposition based techniques.

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Two-stage SP with linear recourse
random event Probability Second-stage
cost Technical matrix Recourse
matrix Right-hand side First-stage
decisions Second-stage decisions.
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Subject to
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Recourse model using scenarios
Properties
1. Piece-wise Convex with non-linear objective
function
2. Requires multi-dimensional summation.
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Solving a dual block angular system
Scenario sub-problems
Master problem
x
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Benders decomposition
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Feasibility of the solution
Cone (Ds)
is feasible if
Cone (Ds)
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Feasibility of the solution
Assume
such that
Cone
Therefore,
such that
Cone
and
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How do we get
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Generation of the feasibility cut
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Generation of the feasibility cut
is the feasibility cut.
Therefore
Let r be the index of the feasibility cuts
rr1
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The refined master
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Optimality of the solution
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Generation of the optimality cut
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Generation of the optimality cut
Therefore
is the optimality cut
Let t be the index of the optimality cuts
tt1
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Illustrative example
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Illustrative example
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Illustrative example
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Illustrative example
Sub problem for scenario 1
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Shadow price
Technical matrix
Rhs
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Illustrative example
Sub problem for scenario 1
Feasibility cut
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Illustrative example
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Illustrative example
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Illustrative example
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Illustrative example
The solution to the master problem is
feasible. Is it optimal for the entire model ?
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Illustrative Example
Upper Bound
Lower Bound
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Illustrative example
Optimality cut for scenario 1
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Illustrative example
Optimality cut for scenario 2
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Illustrative Example
The aggregated optimality cut
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Illustrative example
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Illustrative example
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Illustrative example
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Illustrative example
The solution to the master problem is
feasible. Is it optimal for the entire model ?
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Illustrative Example
Upper Bound
Lower Bound
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Illustrative example
Optimality cut for scenario 1
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Illustrative example
Optimality cut for scenario 2
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Illustrative Example
The aggregated optimality cut
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Illustrative example
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Illustrative example
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Illustrative example
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Illustrative example
Sub problem for scenario 2
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Shadow price
Technical matrix
Rhs
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Illustrative example
Sub problem for scenario 2
Feasibility cut
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Illustrative example
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Illustrative example
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Illustrative example
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Illustrative example
Sub problem for scenario 2
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Shadow price
Technical matrix
Rhs
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Illustrative example
Sub problem for scenario 2
Feasibility cut
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Illustrative example
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Illustrative example
The solution to the master problem is
feasible. Is it optimal for the entire model ?
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Illustrative Example
Upper Bound
Lower Bound
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Illustrative Example
Upper Bound
Lower Bound
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Regularised Benders decomposition
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Regularised Benders decomposition
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Feasibility/ Optimality cuts
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Augmented lagrangian
Scenario sub-problems
The non-anticipativity constraints
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Thank you