Primal-Dual Algorithms for Rational Convex Programs I. Algorithmic Game Theory and Internet Computing. Vijay V. Vazirani Georgia Tech. Rational convex program. A nonlinear convex program that always has a rational solution, using polynomially many bits,

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Algorithmic Game Theory and Internet Computing

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Primal-Dual Algorithms for Rational Convex Programs I Algorithmic Game Theoryand Internet Computing Vijay V. Vazirani Georgia Tech

Rational convex program • A nonlinear convex program that always has a rational solution, using polynomially many bits, if all parameters are rational.

Rational convex program • A nonlinear convex program that always has a rational solution, using polynomially many bits, if all parameters are rational. • i.e., it “behaves” like an LP!

Linear Fisher Market • Assume: • Buyer i’s total utility, • mi: money of buyer i. • One unit of each good j. • n buyers and g goods. • Findequilibrium prices!

Why remarkable? • Equilibrium simultaneously optimizes for all agents. • How is this done via a single objective function?

Eisenberg-Gale Program Theorem: Optimal soln. gives equilibrium allocations, dual gives prices. Equilibrium utilities and prices are unique. Show: Each buyer spends all her money & buys an optimal bundle of goods. All goods fully sold.

Auction for Google’s TV ads N. Nisan et. al, 2009: • Used market equilibrium based approach. • Combinatorial algorithms for linear case provided “inspiration”.

Network N(p) p(1) m(1) p(2) m(2) t s p(3) m(3) m(4) p(4) infinite capacities

Max flow in N(p) p(1) m(1) p(2) m(2) p(3) m(3) m(4) p(4) p: equilibrium prices iff both cuts saturated

Idea of algorithm • “primal” variables: allocations • “dual” variables: prices of goods • Approach equilibrium prices from below: • start with very low prices; buyers have surplus money • iteratively keep raising prices and decreasing surplus

An important consideration • The price of a good never exceeds its equilibrium price • Invariant: s is a min-cut

Primal-dual algorithms so far • Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching.

Primal-dual algorithms so far • Raise dual variables greedily. (Lot of effort spent on designing more sophisticated dual processes.) • Only exception: Edmonds, 1965: algorithm for max weight matching. • Otherwise primal objects go tight and loose. Difficult to account for these reversals -- in the running time.

Our algorithm • Dual variables (prices) are raised greedily • Yet, primal objects go tight and loose • Because of enhanced KKT conditions