Vijay V. Vazirani Georgia Tech Rational Convex Programs Vijay V. Vazirani Georgia Tech

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

Rational convex program Always has a rational solution (if feasible) using polynomially many bits, if all parameters are rational. i.e., it “behaves” like an LP!

Rational convex program Always has a rational solution (if feasible) using polynomially many bits, if all parameters are rational. i.e., it “behaves” like an LP! Do they exist??

KKT optimality conditions

Possible RCPs

Why important??

Combinatorial optimization Central problems have LP-relaxations that always have integer optimal solutions!

Combinatorial optimization Central problems have LP-relaxations that always have integer optimal solutions! ILP: Integral LP

Cornerstone problems in P Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

Solve ILP LP-solver suffices. Why design combinatorial algorithms, especially today that LP-solvers are so fast?

Combinatorial algorithms Very rich theory Gave field of algorithms some of its formative and fundamental notions, e.g. P Helped spawn off algorithmic areas, e.g., approximation algs, parallel algs. Preferable in applications, since efficient and malleable.

Importance of rationality Program A: Combinatorial, polynomial time (strongly poly.) algorithm Program B: Polynomial time (strongly poly.) algorithm, given LP-oracle.

Quadratic RCPs

Combinatorial Algorithms Helgason, Kennington & Lall, 1980 Single constraint Minoux, 1984 Minimum quadratic cost flow Frank & Karzanov, 1992 Closest point from origin to bipartite perfect matching polytope. Karzanov & McCormic, 1997 Any totally unimodular matrix.

Ben-Tal & Nemirovski, 1999 Polyhedral approximation of second-order cone Main technique: Solves any quadratic RCP in polynomial time, given an LP-oracle.

Ben-Tal & Nemirovski, 1999 Polyhedral approximation of second-order cone Main technique: Solves any quadratic RCP in polynomial time, given an LP-oracle. Strongly polynomial algorithm?

Logarithmic RCPs

Logarithmic RCPs Optimal solutions to such RCPs capture equilibria for various market models!

Interesting fact So far, all markets whose equilibria can be computed efficiently admit convex or quasiconvex programs!

Central Tenet within Mathematical Economics Markets should operate at equilibrium i.e., prices s.t. Parity between supply and demand

Do markets even admit equilibrium prices?

Do markets even admit equilibrium prices? Easy if only one good!

Supply-demand curves

Do markets even admit equilibrium prices? What if there are multiple goods and multiple buyers with diverse desires and different buying power?

Irving Fisher, 1891 Defined a fundamental market model

Concave utility function (for good j) amount of j utility

total utility

For given prices, find optimal bundle of goods

Several buyers with different utility functions and moneys.

Several buyers with different utility functions and moneys Several buyers with different utility functions and moneys. Equilibrium prices

Several buyers with different utility functions and moneys Several buyers with different utility functions and moneys. Show equilibrium prices exist.

Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

Several buyers with different utility functions and moneys Several buyers with different utility functions and moneys. Find equilibrium prices.

Arrow-Debreu Theorem, 1954 Highly non-constructive! Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Highly non-constructive!

Arrow-Debreu Theorem, 1954 Continuous, quasiconcave, Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Continuous, quasiconcave, satisfying non-satiation.

Complexity-theoretic question For “reasonable” utility fns., can market equilibrium be computed in P? If not, what is its complexity?

Linear Fisher Market DPSV, 2002: First polynomial time algorithm Assume: Buyer i’s total utility, mi : money of buyer i. One unit of each good j.

Eisenberg-Gale Program, 1959

Eisenberg-Gale Program, 1959 prices pj

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

KKT conditions

Proof of rationality Guess positive allocation variables (say k). Substitute 1/pj by a new variable. LP with (k + g) equations and non-negativity constraint for each variable.

KKT conditions

Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 By extending primal-dual paradigm to setting of convex programs & KKT conditions

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

Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, plc utilities?

Piecewise linear, concave utility Additively separable over goods amount of j

Markets with separable, plc utilities are PPAD-complete Chen, Dai, Du, Teng, 2009 Chen & Teng, 2009 V. & Yannakakis, 2009

Markets with separable, plc utilities are PPAD-complete Chen, Dai, Du, Teng, 2009 Chen & Teng, 2009 V. & Yannakakis, 2009 (Building on combinatorial insights from DPSV)

Theorem (V., 2002): Generalized linear Fisher market to Spending constraint utilities. Polynomial time algorithm for computing equilibrium.

Is there a convex program for this model? “We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

Eisenberg-Gale Markets Jain & V., 2007 EG[2] Markets Chakrabarty, Devanur & V. 2008 Price disc. Market Goel & V. Spending constraint market V., 2005 Nash Bargaining V., 2008 EG convex program = Devanur’s program

V., 2010: Assuming perfect price discrimination, can handle: Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation.

V., 2010: Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation. Compare with Arrow-Debreu utilities!! continuous, quasiconcave, satisfying non-satiation.

A new development Orlin, 2009: Strongly polynomial algorithm for Fisher’s linear case, using scaling. Open: For rest

Are there other classes of RCPs?

Sturmfels & Uhler, 2009:

Linear Arrow-Debreu market Eaves, 1976: Has a rational solution. Jain, 2004: Convex program Does not yield proof of rationality

Linear Arrow-Debreu market Eaves, 1976: Has a rational solution. Jain, 2004: Convex program Does not yield proof of rationality Open: Find RCP

Can Fisher’s linear case (or any of the other problems) Open Can Fisher’s linear case (or any of the other problems) be captured via an LP?