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

06/23/09 16:22Information Security2 Outline Kelly Betting Channel Capacity Reformulated Kelly Purpose What's Next

06/23/09 16:22Information Security3 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

06/23/09 16:22Information Security4 J.L. Kelly, Jr “A New Interpretation of Information Rate” (1956) Growth rate is V n /V 0 =2 nG 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 V k (w) = k I(S;R) + log w Kelly Betting Public KnowledgeTransmitted Results (S) Intercepted Results ( R) noise

06/23/09 16:22Information Security5 Channel Capacity From Information Theory: – The maximum mutual information over all input probabilities is the Channel Capacity. Formed as an optimization problem

06/23/09 16:22Information Security6 Channel Capacity: Dual If we take the Lagrange Dual of the CCP we get the following Geometric Program (in convex form)

06/23/09 16:22Information Security7 Reformulated Kelly Remember our Kelly formula Then reformulate as an optimization problem And finally the Kelly Dual as a GP (in convex form)

06/23/09 16:22Information Security8 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

06/23/09 16:22Information Security9 What's Next Variations on source data distributions – Input costs – Encoding / Noise To create a more general version of Kelly