Minimization of CVar using Kelley's cutting plane algorithm

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Minimization of C-Var using Kelley's cutting
plane algorithm

• Suchandan Guha

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Presentation Layout

• Introduction
• General Cutting Plane methods
• Kelleys cutting plane method
• Application on test problem
• Implementation and results
• Conclusion and Future work

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Introduction

• Minimization of this function gives
• CVaRoptimum value
• VaRvalue of zeta where optimum is obtained
• Problems
• Function is non linear and non differentiable

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Introduction

• However, the function is convex
• Cutting plane methods help in converting a convex
  optimization problem to LP
• Kelleys cutting plane algorithm

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Cutting Plane Method

• Both f(x) and g(x) are convex and
  non-differentiable
• Due to convexity at any point x0 in the domain of
  f(x), there exists at least one supporting
  hyperplane given by

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Cutting Plane Method
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Cutting Plane Method
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Cutting Plane Algorithm

• Initialize
• Get an initial upper bound and a lower bound for
  the optimal solution
• Get an initial relaxed master problem MP0.
• Iterate (k)
• Get a query point and a corresponding lower
  bound
• If is infeasible to NDP (g(xk) gt0), then the
  oracle of g generates a feasibility cut of type

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Cutting Plane Algorithm

• If is feasible to NDP (g(xk) 0), then the
  oracle of f generates an optimality cut of type
  and an upper bound
• Update the bounds
• if is feasible to NDP, then
• Update index sets
• If a feasibility cut is added then and
• If an optimality cut is added then and
• Either STOP or k k 1

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Kelleys Cutting Plane Algorithm

• Query point for next iteration solution of
  current LP
• Lower bound Optimum value obtained from current
  LP
• Upper bound f(current point)
• GAP Upper bound Lower bound
• STOP if GAP lt certain value

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Kelleys Cutting Plane Algorithm
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Cutting Plane Method
Min found here
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First Point
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C-VaR minimization using Kelleys CPM
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Test Problem

• Blended rug co
• Produces 2 types of carpets
• Leases a machine for a fixed amount of time
  decided beforehand
• Runs 2 processes on the machine
• Proc1 / day-gt consumes 500 units of wool yarn and
  800 units of synthetic yarn and produces 400
  yards of A carpet and 200 yards of B carpet
• Proc2 / day-gt consumes 600 units of wool yarn and
  700 units of synthetic yarn and produces 200
  yards of A carpet and 400 yards of B carpet

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Test Problem

• Decision Variables
• x1number of days for which process 1 is run
• x2number of days for which process 2 is run
• y1number of yards of carpet 1 produced
• y2number of yards of carpet 2 produced
• r1number of units of wool yarn used
• r2number of units of synthetic yarn used
• Ttotal no of days machine is leased for

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Convex optimization problem
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Prices in different scenarios
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Test Problem Sub-differentials
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Implementation Results

• C with Concert

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Conclusion Future work

• Results matching with those of excel
• Can be extended to the continuous case
• Especially useful when the number of scenarios is
  very large

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• Questions??????