Analysis of Algorithms

1
Applications of Geometric Programming in
Information Security
Chris Ware University of Victoria
2
Outline

• Kelly Betting
• Channel Capacity
• Reformulated Kelly
• Purpose
• What's Next

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Optimization Problems

• General Optimization Problem
• We define a monomial as
• Then a Geometric Program is an optimization
  problem where all f are posynomials and all h are
  monomials

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Kelly Betting

• J.L. Kelly, Jr A New Interpretation of
  Information Rate (1956)
• Growth rate is Vn/V02nG
• Maximum exponential rate of growth of gambler's
  capital
• G ? Mutual Information I(SR)
• Maximum expected logarithm of gambler's capital
  after k rounds
• Vk(w) k I(SR) log w

Public Knowledge
Transmitted Results (S)
noise
Intercepted Results ( R)
5
Channel Capacity

• From Information Theory
• The maximum mutual information over all input
  probabilities is the Channel Capacity.
• Formed as an optimization problem

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Channel Capacity Dual

• If we take the Lagrange Dual of the CCP
• we get the following Geometric Program (in convex
  form)

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Reformulated Kelly

• Remember our Kelly formula
• Then reformulate as an optimization problem
• And finally the Kelly Dual as a GP (in convex
  form)

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Benefits of CCP as a GP

• Weak Duality any feasible solution produces an
  upper bound on channel capacity
• Strong Duality the optimal solution is the
  channel capacity
• Both primal and dual problems can be
  simultaneously and efficiently solved through the
  primal-dual interior point method

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What's Next

• Variations on source data distributions
• Input costs
• Encoding / Noise
• To create a more general version of Kelly