The Degree of Central Curve in Quadratic Programming

Transcription

1 The in Quadratic Programming Mathematics Department San Francisco State University 15 October 2014

2 Quadratic Programs minimize 1 2 x t Qx + x t c subject to Ax = b x 0 where Q is n n positive definite matrix c R n A is d n matrix of rank d b R + (A)

3 Quadratic Programs minimize 1 2 x t Qx + x t c subject to Ax = b x 0 where Q is n n positive definite matrix c R n A is d n matrix of rank d b R + (A) A convex optimization problem!

4 Log-barrier and a curve A related convex problem to be solved for λ 0 minimize 1 2 x t Qx + x t c λ log(x i ) subject to Ax = b

5 Log-barrier and a curve A related convex problem to be solved for λ 0 minimize 1 2 x t Qx + x t c λ log(x i ) subject to Ax = b... and its Karush-Kuhn-Tucker (KKT) Equations: Ax = b, x 0, Qx + c λ x i e i A t y = 0

6 Log-barrier and a curve A related convex problem to be solved for λ 0 minimize 1 2 x t Qx + x t c λ log(x i ) subject to Ax = b... and its Karush-Kuhn-Tucker (KKT) Equations: Ax = b, x 0, Qx + c λ x i e i A t y = 0 {(x (λ), λ, y (λ))} for λ > 0 is a curve in R n+1+d.

7 Example minimize 1 2 (2x x x x 2 4 ) +4x 1 x 2 + 3x 3 2x 4 subject to 2x 1 3x 2 +x 3 = 9 x 1 2x 2 +x 4 = 6 x 1, x 2, x 3, x 4 0

8 The Central Path Definition Assume {x R n : Ax = b, x 0} bounded. Then the projection of {(x (λ), λ, y (λ)) for λ > 0 on R n is called the central path of the quadratic program.

9 The Central Path Definition Assume {x R n : Ax = b, x 0} bounded. Then the projection of {(x (λ), λ, y (λ)) for λ > 0 on R n is called the central path of the quadratic program. 2x 1 +4 λ x 1 2y 1 +y 2 = 0 3x 2 1 λ x 2 +3y 1 +2y 2 = 0 3x 3 +3 λ x 3 y 1 = 0 4x 4 2 λ x 4 y 2 = 0 2x 1 3x 2 +x 3 = 9 x 1 2x 2 +x 4 = 6 x 1 x 2 x 3 x 4 0

10 The Central Path

11 The Central Curve Definition The Zariski closure of the projection onto C n of the solutions to (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, and Ax b = 0 is called the central curve of the quadradic program.

12 The Central Curve Definition The Zariski closure of the projection onto C n of the solutions to (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, and Ax b = 0 is called the central curve of the quadradic program. 2x x 1 λ 2y 1 x 1 +y 2 x 1 = 0 3x 2 2 x 2 λ +3y 1 x 2 +2y 2 x 2 = 0 3x x 3 λ y 1 x 3 = 0 4x 4 4 2x 4 λ y 2 x 4 = 0 2x 1 3x 2 +x 3 = 9 x 1 2x 2 +x 4 = 6

13 The Central Curve Definition The Zariski closure of the projection onto C n of the solutions to (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, and Ax b = 0 is called the central curve of the quadradic program.

14 The Central Curve Definition The Zariski closure of the projection onto C n of the solutions to (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, and Ax b = 0 is called the central curve of the quadradic program. We denote the ideal generated by the above equations by J. And we denote the ideal of the central curve by I C.

15 The Central Curve In our running example the central curve is defined by Ax = b and 4x 2 1 x 2x 3 3x 1 x 2 2 x 3 21x 1 x 2 x x 2 1 x 2x 4 + 6x 1 x 2 2 x 4 + 2x 2 1 x 3x 4 3x 2 2 x 3x 4 6x 1 x 2 3 x 4 9x 2 x 2 3 x x 1 x 2 x x 1x 3 x 2 4 8x 2x 3 x x 1 x 2 x 3 4x 1 x 2 x 4 4x 1 x 3 x 4 4x 2 x 3 x 4 = 0

16 The Central Curve

17 The Degree of the Central Curve Goal: Compute equations for and the degree (of the projective closure) of the central curve of quadratic programs (for generic Q, c, A, b).

18 The Degree of the Central Curve Goal: Compute equations for and the degree (of the projective closure) of the central curve of quadratic programs (for generic Q, c, A, b). Motivation: Work started by Bayer and Lagarias Dedieu, Malajovich, and Shub (2005) considered the total curvature of the central path for a given linear program De Loera, Sturmfels, and Vinzant (2012) determined the equations for the central curve of linear programs and computed the degree to be ( ) ( n 1 d. This implies a bound of 2π(n d 1) n 1 d 1) on the total curvature of the central curve of a generic LP Continuous Hirsch Conjecture is false: Allemigeon, Benchimol, Gaubert, and Joswig (2014) Quadratic programming is an intermediate stage from linear to semidefinite programs

19 Clearing Denominators Proposition V(J : (x 1 x 2 x n ) ) = V(J)

20 Clearing Denominators Proposition V(J : (x 1 x 2 x n ) ) = V(J) The central curve does not have components in coordinate hyperplanes. The LP central curve is an irreducible curve. The QP central curve should be irreducible as well.

21 Degree and the Solutions in the Torus Theorem When Q, A, c, and b are generic then the degree of the central curve is equal to the number of solutions in (C ) n+d+1 to the system (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f

22 Degree and the Solutions in the Torus Theorem When Q, A, c, and b are generic then the degree of the central curve is equal to the number of solutions in (C ) n+d+1 to the system Theorem (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f The number of solutions in (C ) n+d+1 to the above system is equal to the number of solutions in (C ) n+d+1 to the system when Q is generic and diagonal.

23 Computational Data for the Diagonal Case d/n

24 Computing Mixed Volume: LP case The equations: c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f

25 Computing Mixed Volume: LP case The equations: c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f Using the (d + 1) linear equations make a substitution x = v 0 + t 1 v t n d 1 v n d 1 and get a system of n equations in n unknowns: t 1,..., t n d 1, λ, y 1,..., y d where the support of each equation is λ, 1, t 1,..., t n d 1, y 1, y 1 t 1,..., y 1 t n d 1,..., y d, y d t 1,..., y d t n d 1

26 Computing Mixed Volume: LP case So the mixed volume for the LP system is the volume of n d 1 d.

27 Computing Mixed Volume: LP case So the mixed volume for the LP system is the volume of n d 1 d. Theorem The degree of the LP central curve for generic data is ( ) n 1 = d n d 1 k=0 ( ) n 2 k d 1

28 Computing Mixed Volume: LP case So the mixed volume for the LP system is the volume of n d 1 d. Theorem The degree of the LP central curve for generic data is ( ) n 1 = d n d 1 k=0 ( ) n 2 k d 1 For the right hand side of the formula use, for instance, staircase triangulation.

29 Computing Mixed Volume: QP case The equations: (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f

30 Computing Mixed Volume: QP case The equations: (Qx) i x i + c i x i λ (y t A i )x i = 0, i = 1,..., n, Ax = b ex = f Using the (d + 1) linear equations make a substitution x = v 0 + t 1 v t n d 1 v n d 1 and get a system of n equations in n unknowns: t 1,..., t n d 1, λ, y 1,..., y d where the support of each equation is λ, 1, t 1,..., t n d 1, t 2 1, t 1 t 2,..., t 2 n d 1 y 1, y 1 t 1,..., y 1 t n d 1,..., y d, y d t 1,..., y d t n d 1

31 Computing Mixed Volume: QP case Theorem The degree of the QP central curve for generic data is n d 1 k=0 ( n 2 k d 1 ) 2 k

32 Computing Mixed Volume: QP case Theorem The degree of the QP central curve for generic data is n d 1 k=0 ( n 2 k d 1 ) 2 k Again use staircase triangulation but with the right volumes of the simplices.

33 Equations If a degree reverse lex x 1 < x 2 < < x n is used, the reduced Gröbner bases look like this (d = 2, n = 5): 26x 2 2 x 3x 4 16x 2 x 2 3 x 4 44x 2 x 3 x x 2 2 x 3x x 2 x 2 3 x x 2 1 x 3x 4 42x 1 x 2 3 x 4 44x 1 x 3 x x 2 1 x 3x 5 46x 1 x 2 3 x x 2 1 x 4x 5 24x 2 1 x 2x 4 42x 1 x 2 2 x x 1 x 2 x x 2 1 x 2x 5 46x 1 x 2 2 x x 2 1 x 4x 5 33x 2 1 x 2x 3 22x 1 x 2 2 x 3 22x 1 x 2 x x 2 1 x 2x x 1 x 2 2 x x 2 1 x 3x 5

34 Equations If a degree reverse lex x 1 < x 2 < < x n is used, the reduced Gröbner bases look like this (d = 2, n = 5): 26x 2 2 x 3x 4 16x 2 x 2 3 x 4 44x 2 x 3 x x 2 2 x 3x x 2 x 2 3 x x 2 1 x 3x 4 42x 1 x 2 3 x 4 44x 1 x 3 x x 2 1 x 3x 5 46x 1 x 2 3 x x 2 1 x 4x 5 24x 2 1 x 2x 4 42x 1 x 2 2 x x 1 x 2 x x 2 1 x 2x 5 46x 1 x 2 2 x x 2 1 x 4x 5 33x 2 1 x 2x 3 22x 1 x 2 2 x 3 22x 1 x 2 x x 2 1 x 2x x 1 x 2 2 x x 2 1 x 3x 5 Initial ideal = x 2 2 x 3x 4, x 2 1 x 3x 4, x 2 1 x 2x 4, x 2 1 x 2x 3 = x 3, x 4 x 2, x 4 x 2 1, x 4 x 2, x 3 x 2 1, x 3 x 2 1, x 2 2.

35 Elimination in the Quadratic Case In order to eliminate λ and y 1,..., y d from d i x 2 i + c i x i λ (y t A i )x i = 0, i = 1,..., n eliminate y 1,..., y d and z 1,..., z n from d i (w i +z i ) = y t A i c i, i = 1,..., n and d 1 w 1 z 1 = d 2 w 2 z 2 = = d n w n z n

36 Elimination in the Quadratic Case In order to eliminate λ and y 1,..., y d from d i x 2 i + c i x i λ (y t A i )x i = 0, i = 1,..., n eliminate y 1,..., y d and z 1,..., z n from d i (w i +z i ) = y t A i c i, i = 1,..., n and d 1 w 1 z 1 = d 2 w 2 z 2 = = d n w n z n Use each circuit C of A to eliminate y from the first set of equations, then use z k = (d 1 w 1 z 1 )/(d k w k ) to write the resulting equation as f C (w 1,..., w n, z 1 ). each f C is linear in z 1 now using two carefully chosen C and C eliminate z 1 to obtain g C,C (w 1,..., w n ) in the elimination ideal.

37 Elimination in the Quadratic Case Theorem The ideal J = g C,C (w 1,..., w n ) is contained in the elimination ideal I and M = in < (g C,C ) is contained in in < (I ). The ideal M is equal to M = w 2 i w j1 w j2... w jd : 1 i n d 1, i < j 1 < j 2 < < j d < n. Morever, M has the irreducible decomposition M = n d 1 k=0 T {k+2,...,n}, T =n d k 1 w 2 j : 0 < j < k+1 + w t : t T

38 Equations of the Central Curve in the Diagonal Case Corollary The degree of M is equal to n d 1 k=0 ( n 2 k d 1 ) 2 k

39 Equations of the Central Curve in the Diagonal Case Corollary The degree of M is equal to n d 1 k=0 ( n 2 k d 1 ) 2 k Theorem For generic Q, A, b, c and Q a diagonal matrix the central curve for quadratic programming is defined by J = g C,C (w 1,..., w n )