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

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

3. 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!

4. 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??

7. KKT optimality conditions

8. Possible RCPs

9. Why important??

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

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

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

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

17. 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.

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

19. Quadratic RCPs

20. 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.

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

22. 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?

23. Logarithmic RCPs

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

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

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

27. Do markets even admit equilibrium prices?

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

29. Supply-demand curves

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

31. Irving Fisher, 1891
Defined a fundamental
market model

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

35. total utility

36. For given prices, find optimal bundle of goods

37. Several buyers with different utility functions and moneys.

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

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

40. 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.

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

42. 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!

43. 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.

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

45. 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.

46. Eisenberg-Gale Program, 1959

47. Eisenberg-Gale Program, 1959
prices pj

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

49. KKT conditions

50. 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.

51. KKT conditions

52. 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

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

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

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

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

57. 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)

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

59. 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.”

60. 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

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

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

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

64. Are there other classes of RCPs?

65. Sturmfels & Uhler, 2009:

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

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

68. 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?