Applications of Geometric Programming in Information Security Chris

Applications of Geometric Programming in Information Security Chris Ware University of Victoria

Outline Kelly Betting Channel Capacity Reformulated Kelly Purpose What's Next June 27, 2009 Information Security 2

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 June 27, 2009 Information Security 3

Kelly Betting J. L. Kelly, Jr “A New Interpretation of Information Rate” (1956) Transmitted Results (S) Public Knowledge noise Intercepted Results ( R) Growth rate is Vn/V 0=2 n. G Maximum exponential rate of growth of gambler's capital • G → Mutual Information: I(S; R) Maximum expected logarithm of gambler's capital after k rounds • Vk(w) = k I(S; R) + log w June 27, 2009 Information Security 4

Channel Capacity From Information Theory: – The maximum mutual information over all input probabilities is the Channel Capacity. Formed as an optimization problem June 27, 2009 Information Security 5

Channel Capacity: Dual If we take the Lagrange Dual of the CCP we get the following Geometric Program (in convex form) June 27, 2009 Information Security 6

Reformulated Kelly Remember our Kelly formula Then reformulate as an optimization problem And finally the Kelly Dual as a GP (in convex form) June 27, 2009 Information Security 7

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 June 27, 2009 Information Security 8

What's Next Variations on source data distributions – Input costs – Encoding / Noise To create a more general version of Kelly June 27, 2009 Information Security 9