Studying Spin Glasses via Combinatorial Optimization

Transcription

1 Studying Spin Glasses via Combinatorial Optimization Frauke Liers Emmy Noether Junior Research Group, Department of Computer Science, University of Cologne Dec 11 th, 2007

2 What we are interested in... develop, improve and implement algorithms for optimization problems occuring in physics: ground states of Ising spin glasses in different dimensions Potts glasses Potts glasses for q etc. study their physics together with physics colleagues We always compute exact ground states! methods we use: polynomial algorithms (matching, maximum flow algorithms, etc.) branch-and-bound or branch-and-cut algorithms with exponential worst-case running time

3 What we are interested in... develop, improve and implement algorithms for optimization problems occuring in physics: ground states of Ising spin glasses in different dimensions Potts glasses Potts glasses for q etc. study their physics together with physics colleagues We always compute exact ground states! methods we use: polynomial algorithms (matching, maximum flow algorithms, etc.) branch-and-bound or branch-and-cut algorithms with exponential worst-case running time

4 What we are interested in... develop, improve and implement algorithms for optimization problems occuring in physics: ground states of Ising spin glasses in different dimensions Potts glasses Potts glasses for q etc. study their physics together with physics colleagues We always compute exact ground states! methods we use: polynomial algorithms (matching, maximum flow algorithms, etc.) branch-and-bound or branch-and-cut algorithms with exponential worst-case running time

5 What we are interested in... develop, improve and implement algorithms for optimization problems occuring in physics: ground states of Ising spin glasses in different dimensions Potts glasses Potts glasses for q etc. study their physics together with physics colleagues We always compute exact ground states! methods we use: polynomial algorithms (matching, maximum flow algorithms, etc.) branch-and-bound or branch-and-cut algorithms with exponential worst-case running time

6 What we are interested in... develop, improve and implement algorithms for optimization problems occuring in physics: ground states of Ising spin glasses in different dimensions Potts glasses Potts glasses for q etc. study their physics together with physics colleagues We always compute exact ground states! methods we use: polynomial algorithms (matching, maximum flow algorithms, etc.) branch-and-bound or branch-and-cut algorithms with exponential worst-case running time

7 Spin Glasses e.g. Rb 2 Cu 1 x Co x F 4 experiments (Cannella & Mydosh 1972) reveal: at low temperatures: phase transition spin glass state Edwards Anderson Model (1975) short-range model interactions randomly chosen J ij {+1, 1} or Gaussian distributed H(S) = <i,j> J ijs i S j, with spin variables S i J ij ground state: min{h(s) S is spin configuration}

8 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

9 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

10 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

11 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

12 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

13 This is a Hard Problem means... NP-hard, i.e. we cannot expect to find an algorithm that solves it in time growing polynomial in the size of the input e.g., 2d Ising spin glasses with an external field or 3d lattices whereas 2d, no field, free boundaries: easy

14 This is a Hard Problem means... NP-hard, i.e. we cannot expect to find an algorithm that solves it in time growing polynomial in the size of the input e.g., 2d Ising spin glasses with an external field or 3d lattices whereas 2d, no field, free boundaries: easy

15 This is a Hard Problem means... NP-hard, i.e. we cannot expect to find an algorithm that solves it in time growing polynomial in the size of the input e.g., 2d Ising spin glasses with an external field or 3d lattices whereas 2d, no field, free boundaries: easy

16 The Exact Algorithm for Hard Instances Spinglass coupling J ij configuration Graph G=(V, E) node of G edge of G edge weight c ij + node partition V, V

17 The Exact Algorithm for Hard Instances J ij S i = 1 S i = +1

18 The Exact Algorithm for Hard Instances J ij c ij S i = 1 S i = +1 G = (V,E) H = e E J ijs i S j

19 Computing Exact Ground States J ij c ij S i = 1 S i = +1 H(S) + (i,j) E J ij = J ij (1 S i S j ) }{{} (i,j) E { 2, if S i S j = 0, otherwise = 2 S i S j J ij

20 Computing Exact Ground States J ij c ij H(S) + const = 2 S i S j J ij cut = {(i, j) E (i, j) = } its weight: (i,j) cut c ij

21 Computing Exact Ground States J ij cij H(S)+const = 2 S i S j J ij ground state minh(s) cut = {(i, j) E (i, j) = } weight (i,j) cut c ij with c ij = J ij : maximum cut in G NP-hard in general

22 Example

23 Example

24 Example

25 Example

26 Example

27 Example

28 Example

29 Example

30 Example

31 Example

32 Example

33 Example

34 Branch-and-Cut is a clever enumeration method is a general framework for solving hard combinatorial optimization problems however: specification to a certain problem is science of its own for maxcut: started by M. Jünger, G. Reinelt, G. Rinaldi improved by M. Diehl, FL ground-state server via command-line client or web interface, get result by (will be extended)

35 Branch-and-Cut is a clever enumeration method is a general framework for solving hard combinatorial optimization problems however: specification to a certain problem is science of its own for maxcut: started by M. Jünger, G. Reinelt, G. Rinaldi improved by M. Diehl, FL ground-state server via command-line client or web interface, get result by (will be extended)

36 Branch-and-Cut is a clever enumeration method is a general framework for solving hard combinatorial optimization problems however: specification to a certain problem is science of its own for maxcut: started by M. Jünger, G. Reinelt, G. Rinaldi improved by M. Diehl, FL ground-state server via command-line client or web interface, get result by (will be extended)

37 Branch-and-Cut is a clever enumeration method is a general framework for solving hard combinatorial optimization problems however: specification to a certain problem is science of its own for maxcut: started by M. Jünger, G. Reinelt, G. Rinaldi improved by M. Diehl, FL ground-state server via command-line client or web interface, get result by (will be extended)

38 Branch-and-Cut is a clever enumeration method is a general framework for solving hard combinatorial optimization problems however: specification to a certain problem is science of its own for maxcut: started by M. Jünger, G. Reinelt, G. Rinaldi improved by M. Diehl, FL ground-state server via command-line client or web interface, get result by (will be extended)

39 Branch-and-Cut Algorithm (lb): lower bound for optimum (ub): upper bound (lb) = (ub) optimality iteration upper bound lower bound optimum

40 Calculation Of (ub) For Maxcut (i, j) E 0 x ij 1 (i, j) cut x ij = 1 (i, j) cut x ij = 0 consider e.g. for P C (G) : convex hull of all cut vectors x 1 x 2 x 3

41 Calculation Of (ub) For Maxcut consider e.g. for (i, j) E 0 x ij 1 (i, j) cut x ij = 1 (i, j) cut x ij = 0 P C (G) : convex hull of all cut vectors x 1 x 2 x 3 possible cut vectors: 0 0 0

42 Calculation Of (ub) For Maxcut consider e.g. for (i, j) E 0 x ij 1 (i, j) cut x ij = 1 (i, j) cut x ij = 0 P C (G) : convex hull of all cut vectors x 1 x 2 x 3 possible cut vectors: 0 1 1

43 Calculation Of (ub) For Maxcut consider e.g. for (i, j) E 0 x ij 1 (i, j) cut x ij = 1 (i, j) cut x ij = 0 P C (G) : convex hull of all cut vectors x 1 x 2 x 3 possible cut vectors: 1 0 1

44 Calculation Of (ub) For Maxcut consider e.g. for (i, j) E 0 x ij 1 (i, j) cut x ij = 1 (i, j) cut x ij = 0 P C (G) : convex hull of all cut vectors x 1 x 2 x 3 possible cut vectors: 1 1 0

45 conv{ 0 0 0, 0 1 1, 1 0 1, } = cut polytope can be described by linear inequalities!

46 however: in higher dimensions too many would be needed, not all known solution: find part of the necessary inequalities that can easily be determined optimize over a solution space P that contains cut polytope yields (ub)

47 however: in higher dimensions too many would be needed, not all known solution: find part of the necessary inequalities that can easily be determined optimize over a solution space P that contains cut polytope yields (ub)

48 however: in higher dimensions too many would be needed, not all known solution: find part of the necessary inequalities that can easily be determined optimize over a solution space P that contains cut polytope yields (ub)

49 however: in higher dimensions too many would be needed, not all known solution: find part of the necessary inequalities that can easily be determined optimize over a solution space P that contains cut polytope yields (ub)

50 Branch And Cut Algorithm FL, M. Jünger, G. Reinelt, G. Rinaldi, in New Optimization Algorithms in Physics, A.K. Hartmann and H. Rieger (Eds.), Wiley-VCH (2004). 1 start with some solution space P P C (G) 2 solve linear program (ub) = cx =max e E c e x e, x P 3 (lb): value of any cut 4 if (ub)=(lb) or x is a cut: STOP 5 else: find better description P, goto 2) 6 if no better description can be found: BRANCH select x e with x e {0;1} x e = 0 x e = 1

51 Branch And Cut Algorithm FL, M. Jünger, G. Reinelt, G. Rinaldi, in New Optimization Algorithms in Physics, A.K. Hartmann and H. Rieger (Eds.), Wiley-VCH (2004). 1 start with some solution space P P C (G) 2 solve linear program (ub) = cx =max e E c e x e, x P 3 (lb): value of any cut 4 if (ub)=(lb) or x is a cut: STOP 5 else: find better description P, goto 2) 6 if no better description can be found: BRANCH select x e with x e {0;1} x e = 0 x e = 1

52 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

53 2d Spin Glasses in a Field with Olivier C. Martin (Paris) FL, O.C. Martin, Physical Review B, 76, 6 (2007). spin glasses exhibit subtle phase transitions in 2d: T c = 0, in 3d: T c > 0 their physics in 3d is not yet agreed upon their physics in 2d without a field agrees well with the scaling/droplet (DS) picture of Bray/Moore and Fisher/Huse (mid 80 ) for 2d with a field: previous studies found discrepancies to DS Is DS correct for 2d spin glasses in a field?

54 2d Spin Glasses in a Field with Olivier C. Martin (Paris) FL, O.C. Martin, Physical Review B, 76, 6 (2007). spin glasses exhibit subtle phase transitions in 2d: T c = 0, in 3d: T c > 0 their physics in 3d is not yet agreed upon their physics in 2d without a field agrees well with the scaling/droplet (DS) picture of Bray/Moore and Fisher/Huse (mid 80 ) for 2d with a field: previous studies found discrepancies to DS Is DS correct for 2d spin glasses in a field?

55 2d Spin Glasses in a Field with Olivier C. Martin (Paris) FL, O.C. Martin, Physical Review B, 76, 6 (2007). spin glasses exhibit subtle phase transitions in 2d: T c = 0, in 3d: T c > 0 their physics in 3d is not yet agreed upon their physics in 2d without a field agrees well with the scaling/droplet (DS) picture of Bray/Moore and Fisher/Huse (mid 80 ) for 2d with a field: previous studies found discrepancies to DS Is DS correct for 2d spin glasses in a field?

56 2d Spin Glasses in a Field with Olivier C. Martin (Paris) FL, O.C. Martin, Physical Review B, 76, 6 (2007). spin glasses exhibit subtle phase transitions in 2d: T c = 0, in 3d: T c > 0 their physics in 3d is not yet agreed upon their physics in 2d without a field agrees well with the scaling/droplet (DS) picture of Bray/Moore and Fisher/Huse (mid 80 ) for 2d with a field: previous studies found discrepancies to DS Is DS correct for 2d spin glasses in a field?

57 2d Spin Glasses in a Field with Olivier C. Martin (Paris) FL, O.C. Martin, Physical Review B, 76, 6 (2007). spin glasses exhibit subtle phase transitions in 2d: T c = 0, in 3d: T c > 0 their physics in 3d is not yet agreed upon their physics in 2d without a field agrees well with the scaling/droplet (DS) picture of Bray/Moore and Fisher/Huse (mid 80 ) for 2d with a field: previous studies found discrepancies to DS Is DS correct for 2d spin glasses in a field?

58 Our Approach exact ground-state algorithm study larger lattice sizes than before determine precise points where the ground states change as function of B study the properties of flipped clusters J ij L L lattice, periodic boundaries, Ising spins Gaussian/exponential J ij H(S) ij J ijs i S j B i S i

59 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

60 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

61 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

62 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

63 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

64 The droplet and scaling hypothesis GS low-lying excitations arise by droplet flips GS flipped l zero-field droplets l and compact. Interfacial energy is O(l θ ), total (random) magnetization goes as l d/2 B = 0: y T = θ, y T defined by ξ T 1 y T B = 0: previous studies in 2d agree with DS and find y T = θ B 0: droplet prediction in dimension d is y B = y T + d/2, y B defined by ξ B 1 y B (T = T c ) y B in d = 2. B 0: magnetization m(b) B 1/δ, and δ = y B in d = 2

65 Previous work Kinzel and Binder 1983: δ 1.39 (Monte Carlo at low T) ground-state calculations: Kawashima/Suzuki 1992: δ 1.48 Barahona 1994: δ 1.54 Rieger et al. 1996: δ 1.48 Carter et al. 2003: power scaling probably only arises for huge sizes Are there large corrections to scaling or does the droplet reasoning break down?

66 Previous work Kinzel and Binder 1983: δ 1.39 (Monte Carlo at low T) ground-state calculations: Kawashima/Suzuki 1992: δ 1.48 Barahona 1994: δ 1.54 Rieger et al. 1996: δ 1.48 Carter et al. 2003: power scaling probably only arises for huge sizes Are there large corrections to scaling or does the droplet reasoning break down?

67 Previous work Kinzel and Binder 1983: δ 1.39 (Monte Carlo at low T) ground-state calculations: Kawashima/Suzuki 1992: δ 1.48 Barahona 1994: δ 1.54 Rieger et al. 1996: δ 1.48 Carter et al. 2003: power scaling probably only arises for huge sizes Are there large corrections to scaling or does the droplet reasoning break down?

68 Details of our project Gaussian (and exponential) J ij 2500 for L = 80, 5000 for L = 70, for L 60 m sample 1 sample 2 sample B 1 compute gs at B = 0 2 determine B so that gs at B remains optimum in [B, B + B] (linear programming) 3 reoptimize at B + B + ǫ with ǫ > 0 L = 70, 80: exact gs, B = 0, 0.02, 0.04, 0.06,...

69 Details of our project Gaussian (and exponential) J ij 2500 for L = 80, 5000 for L = 70, for L 60 m sample 1 sample 2 sample B 1 compute gs at B = 0 2 determine B so that gs at B remains optimum in [B, B + B] (linear programming) 3 reoptimize at B + B + ǫ with ǫ > 0 L = 70, 80: exact gs, B = 0, 0.02, 0.04, 0.06,...

70 Exponent δ m(b) B 1/δ for δ DS = 1.28, we should see an envelope curve appear in m(b)/b 1/δ DS as a fct. of B m(b,l) / B 1/ δ [m(b)-χ 1 B]/B 1/δ B however: no flat region for L, as found earlier power-law fit yields δ = 1.45 (L = 50) 0.1 B reason for discrepancy: m has analytic and non-analytic contributions: m = χ 1 B + χ S B 1/δ +..., where χ 1 B cannot be neglected taking χ 1 B into account (inset): droplet scaling fits data very well

71 Exponent δ m(b) B 1/δ for δ DS = 1.28, we should see an envelope curve appear in m(b)/b 1/δ DS as a fct. of B m(b,l) / B 1/ δ [m(b)-χ 1 B]/B 1/δ B however: no flat region for L, as found earlier power-law fit yields δ = 1.45 (L = 50) 0.1 B reason for discrepancy: m has analytic and non-analytic contributions: m = χ 1 B + χ S B 1/δ +..., where χ 1 B cannot be neglected taking χ 1 B into account (inset): droplet scaling fits data very well

72 Exponent δ m(b) B 1/δ for δ DS = 1.28, we should see an envelope curve appear in m(b)/b 1/δ DS as a fct. of B m(b,l) / B 1/ δ [m(b)-χ 1 B]/B 1/δ B however: no flat region for L, as found earlier power-law fit yields δ = 1.45 (L = 50) 0.1 B reason for discrepancy: m has analytic and non-analytic contributions: m = χ 1 B + χ S B 1/δ +..., where χ 1 B cannot be neglected taking χ 1 B into account (inset): droplet scaling fits data very well

73 Flipping clusters are like zero-field clusters study for each realization of the disorder the largest cluster flipped for B [0, [ clusters have holes M / V 1/ S(L) L L their volume V L 2 compactness M/ V (M: cluster magnetization) insensitive to L random magnetization cluster surface L d S with d S 1.32 ( zero-field droplets: d S = 1.27) DS arguments validated

74 Flipping clusters are like zero-field clusters study for each realization of the disorder the largest cluster flipped for B [0, [ clusters have holes M / V 1/ S(L) L L their volume V L 2 compactness M/ V (M: cluster magnetization) insensitive to L random magnetization cluster surface L d S with d S 1.32 ( zero-field droplets: d S = 1.27) DS arguments validated

75 Flipping clusters are like zero-field clusters study for each realization of the disorder the largest cluster flipped for B [0, [ clusters have holes M / V 1/ S(L) L L their volume V L 2 compactness M/ V (M: cluster magnetization) insensitive to L random magnetization cluster surface L d S with d S 1.32 ( zero-field droplets: d S = 1.27) DS arguments validated

76 y B and finite size scaling of m measure y B in ξ B B 1/y B: for a sample, largest cluster flips at field B J. B = B J J biggest cluster involves L 2 spins ξ B (B J ) L B L y B pure power with y B = 1.28 works well [m(b,l)-χ 1 B]/[m(B *,L)-χ 1 B * ] 1 L 1/ δ B * /L B/B * excellent data collapse as m(b,l) χ 1 B m(b,l) χ 1 B = W(B/B ) W(0) = O(1), W(x) x 1/δ at large x. B L 1.28 as fct. of 1/L works well with O(1/L) finite size effects. B (L) = ul y B(1 + v/l) 1.28 y B 1.30

77 y B and finite size scaling of m measure y B in ξ B B 1/y B: for a sample, largest cluster flips at field B J. B = B J J biggest cluster involves L 2 spins ξ B (B J ) L B L y B pure power with y B = 1.28 works well [m(b,l)-χ 1 B]/[m(B *,L)-χ 1 B * ] 1 L 1/ δ B * /L B/B * excellent data collapse as m(b,l) χ 1 B m(b,l) χ 1 B = W(B/B ) W(0) = O(1), W(x) x 1/δ at large x. B L 1.28 as fct. of 1/L works well with O(1/L) finite size effects. B (L) = ul y B(1 + v/l) 1.28 y B 1.30

78 y B and finite size scaling of m measure y B in ξ B B 1/y B: for a sample, largest cluster flips at field B J. B = B J J biggest cluster involves L 2 spins ξ B (B J ) L B L y B pure power with y B = 1.28 works well [m(b,l)-χ 1 B]/[m(B *,L)-χ 1 B * ] 1 L 1/ δ B * /L B/B * excellent data collapse as m(b,l) χ 1 B m(b,l) χ 1 B = W(B/B ) W(0) = O(1), W(x) x 1/δ at large x. B L 1.28 as fct. of 1/L works well with O(1/L) finite size effects. B (L) = ul y B(1 + v/l) 1.28 y B 1.30

79 y B and finite size scaling of m measure y B in ξ B B 1/y B: for a sample, largest cluster flips at field B J. B = B J J biggest cluster involves L 2 spins ξ B (B J ) L B L y B pure power with y B = 1.28 works well [m(b,l)-χ 1 B]/[m(B *,L)-χ 1 B * ] 1 L 1/ δ B * /L B/B * excellent data collapse as m(b,l) χ 1 B m(b,l) χ 1 B = W(B/B ) W(0) = O(1), W(x) x 1/δ at large x. B L 1.28 as fct. of 1/L works well with O(1/L) finite size effects. B (L) = ul y B(1 + v/l) 1.28 y B 1.30

80 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

81 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

82 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

83 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

84 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

85 Conclusions for 2d Ising Spin Glasses in a Field We validated the predictions of the droplet/scaling picture: we find 1.28 δ 1.32 by more careful analysis earlier discrepances to δ = because analytic contributions to magnetization curve were not treated direct measurement of the magnetic length yields 1.28 y B 1.30 relevant spin clusters are compact, random magnetization same is true with exponentially distributed J ij

86 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

87 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

88 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

89 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

90 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

91 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

92 Potts Glasses with Bissan Ghaddar, Miguel Anjos (U. Waterloo, Canada) B. Ghaddar, M. Anjos, FL (submitted) A spin can be in k different states q 1,...q k Hamiltonian: H = i,j J ij δ qi q j we solve the problem also via branch-and-cut however: the bounds through linear optimization are very weak in practice and can be considerably improved by positive semidefinite optimization still: gs determination for Potts glasses is considerably more difficult in practice than for Ising spin glasses

93 semidefinite programming (SDP) problem: minimize a linear function of a symmetric matrix X subject to linear constraints on X, with X being positive semidefinite.

94 Branch-and-Cut Algorithm for Potts Glasses at each node of the branch-and-cut tree: 1 use pos. semidef. optimization to obtain a LB 2 add valid inequalities to get a tighter LB 3 find a feasible solution to get an UB 4 choose an edge (ij) to branch on if optimality cannot yet be proven

95 Branch-and-Cut Algorithm for Potts Glasses at each node of the branch-and-cut tree: 1 use pos. semidef. optimization to obtain a LB 2 add valid inequalities to get a tighter LB 3 find a feasible solution to get an UB 4 choose an edge (ij) to branch on if optimality cannot yet be proven

96 Branch-and-Cut Algorithm for Potts Glasses at each node of the branch-and-cut tree: 1 use pos. semidef. optimization to obtain a LB 2 add valid inequalities to get a tighter LB 3 find a feasible solution to get an UB 4 choose an edge (ij) to branch on if optimality cannot yet be proven

97 Branch-and-Cut Algorithm for Potts Glasses at each node of the branch-and-cut tree: 1 use pos. semidef. optimization to obtain a LB 2 add valid inequalities to get a tighter LB 3 find a feasible solution to get an UB 4 choose an edge (ij) to branch on if optimality cannot yet be proven

98 Results Best Solution Root Node # of Nodes - Time V Value LB UB Time to achieve 0% :00:18 2-0:00: :05:12 1-0:05: :52:08 1-0:52: :21:43 1-2:21: :35: :41: :36: :09: :48: :31: :33: :16: :01:14 1-0:01: :08:02 1-0:08: :36:18 1-0:36: :00:21 1-0:00: :26:52 1-0:26: :52:31 5-3:38: :58:15 3-1:38: :22: :12:11 Table: results for spinglass2g and spinglass3g instances where k = 3. The time is given in hr:min:sec.

99 Results k = 5 k = 7 V Objective Value Time Objective Value Time spinglass2g :23: :21: :42: :39: :09: :13: :39: :18: :14: :32:29 spinglass3g :00: :00: :08: :05: :17: :13: :00: :00: :26: :21: :04: :02: :14: :18: :49: :09: :30: :57:05 Table: results for k = 5 and 7. The time is given in hr:min:sec. doable sizes: 100 spin sites. Although the doable sizes are small, we are not aware of a faster exact algorithm. next step: replace the slow SDP-Solver by some faster routine

100 Results k = 5 k = 7 V Objective Value Time Objective Value Time spinglass2g :23: :21: :42: :39: :09: :13: :39: :18: :14: :32:29 spinglass3g :00: :00: :08: :05: :17: :13: :00: :00: :26: :21: :04: :02: :14: :18: :49: :09: :30: :57:05 Table: results for k = 5 and 7. The time is given in hr:min:sec. doable sizes: 100 spin sites. Although the doable sizes are small, we are not aware of a faster exact algorithm. next step: replace the slow SDP-Solver by some faster routine

101 Results k = 5 k = 7 V Objective Value Time Objective Value Time spinglass2g :23: :21: :42: :39: :09: :13: :39: :18: :14: :32:29 spinglass3g :00: :00: :08: :05: :17: :13: :00: :00: :26: :21: :04: :02: :14: :18: :49: :09: :30: :57:05 Table: results for k = 5 and 7. The time is given in hr:min:sec. doable sizes: 100 spin sites. Although the doable sizes are small, we are not aware of a faster exact algorithm. next step: replace the slow SDP-Solver by some faster routine

102 Results k = 5 k = 7 V Objective Value Time Objective Value Time spinglass2g :23: :21: :42: :39: :09: :13: :39: :18: :14: :32:29 spinglass3g :00: :00: :08: :05: :17: :13: :00: :00: :26: :21: :04: :02: :14: :18: :49: :09: :30: :57:05 Table: results for k = 5 and 7. The time is given in hr:min:sec. doable sizes: 100 spin sites. Although the doable sizes are small, we are not aware of a faster exact algorithm. next step: replace the slow SDP-Solver by some faster routine

103 Outline 1 Hard Ising Spin Glass Instances 2 2d Ising Spin Glasses in a Field 3 Potts Glasses 4 Potts Glasses with q

104 Potts Glasses with q with Diana Fanghänel (Cologne) D. Fanghänel, FL (in preparation) Juhasz, Rieger, Iglòi (2001) have shown: for many states the dominant contribution to the partition function is max A E(G) qf (A), f (A) = number of connected components in A(G) + i,j A(G) J ij f (A) = 16

105 Potts Glasses with q with Diana Fanghänel (Cologne) D. Fanghänel, FL (in preparation) Juhasz, Rieger, Iglòi (2001) have shown: for many states the dominant contribution to the partition function is max A E(G) qf (A), f (A) = number of connected components in A(G) + i,j A(G) J ij ass. J ij = 0.1 : f (A) =

106 Potts Glasses with q with Diana Fanghänel (Cologne) D. Fanghänel, FL (in preparation) Juhasz, Rieger, Iglòi (2001) have shown: for many states the dominant contribution to the partition function is max A E(G) qf (A), f (A) = number of connected components in A(G) + i,j A(G) J ij ass. J ij = 0.1 : f (A) =

107 Solution Approaches Angláis d Auriac et al. presented an exact algorithm it uses many maximum-flow calculations (polynomial, but takes long) our work: reduce the number of maximum-flow calculations by graph-theoretic considerations

108 Solution Approaches Angláis d Auriac et al. presented an exact algorithm it uses many maximum-flow calculations (polynomial, but takes long) our work: reduce the number of maximum-flow calculations by graph-theoretic considerations

109 Solution Approaches Angláis d Auriac et al. presented an exact algorithm it uses many maximum-flow calculations (polynomial, but takes long) our work: reduce the number of maximum-flow calculations by graph-theoretic considerations

110 Solution Approaches Angláis d Auriac et al. presented an exact algorithm it uses many maximum-flow calculations (polynomial, but takes long) our work: reduce the number of maximum-flow calculations by graph-theoretic considerations

111 Preliminary Results use coupling strengths w 1, w 2 at criticality: w 1 + w 2 = 1 number of maximum-flow calculations reduces by 1 3 L = 128: ca 1.5 minutes cpu time L = 256: < 4 h cpu time will be improved further

112 Preliminary Results use coupling strengths w 1, w 2 at criticality: w 1 + w 2 = 1 number of maximum-flow calculations reduces by 1 3 L = 128: ca 1.5 minutes cpu time L = 256: < 4 h cpu time will be improved further

113 Preliminary Results use coupling strengths w 1, w 2 at criticality: w 1 + w 2 = 1 number of maximum-flow calculations reduces by 1 3 L = 128: ca 1.5 minutes cpu time L = 256: < 4 h cpu time will be improved further

114 Preliminary Results use coupling strengths w 1, w 2 at criticality: w 1 + w 2 = 1 number of maximum-flow calculations reduces by 1 3 L = 128: ca 1.5 minutes cpu time L = 256: < 4 h cpu time will be improved further

115 Preliminary Results use coupling strengths w 1, w 2 at criticality: w 1 + w 2 = 1 number of maximum-flow calculations reduces by 1 3 L = 128: ca 1.5 minutes cpu time L = 256: < 4 h cpu time will be improved further

116 The Current Limits from difficult to easy : system currently treatable sizes Potts 100 spin sites 3d Ising (w/o field) d Ising (periodic bc) > Potts(q ) > 256 2

117 The Current Limits from difficult to easy : system currently treatable sizes Potts 100 spin sites 3d Ising (w/o field) d Ising (periodic bc) > Potts(q ) > 256 2

118 The Current Limits from difficult to easy : system currently treatable sizes Potts 100 spin sites 3d Ising (w/o field) d Ising (periodic bc) > Potts(q ) > 256 2

119 The Current Limits from difficult to easy : system currently treatable sizes Potts 100 spin sites 3d Ising (w/o field) d Ising (periodic bc) > Potts(q ) > 256 2

120 The Current Limits from difficult to easy : system currently treatable sizes Potts 100 spin sites 3d Ising (w/o field) d Ising (periodic bc) > Potts(q ) > 256 2

121 Thank you for your attention!