Solving Curved Linear Programs via the Shadow Simplex Method

Transcription

1 Solving Curved Linear rograms via the Shadow Simplex Method Daniel Dadush 1 Nicolai Hähnle 2 1 Centrum Wiskunde & Informatica (CWI) 2 Bonn Universität CWI Scientific Meeting 01/15

2 Outline 1 Introduction Linear rogramming and its Applications The Simplex Method Results 2 The Shadow Simplex Method ivot Rule 3-Step Shadow Simplex ath 3 Conclusions D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 2 / 24

3 Outline 1 Introduction Linear rogramming and its Applications The Simplex Method Results 2 The Shadow Simplex Method ivot Rule 3-Step Shadow Simplex ath 3 Conclusions D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 3 / 24

4 Linear rogramming Linear rogramming (L): linear constraints & linear objective with continuous variables. max c T x subject to Ax b, x R n (A has m rows, n columns) D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 4 / 24

5 Linear rogramming Linear rogramming (L): linear constraints & linear objective with continuous variables. max c T x subject to Ax b, x R n (A has m rows, n columns) c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 4 / 24

6 Linear rogramming Linear rogramming (L): linear constraints & linear objective with continuous variables. max c T x subject to Ax b, x R n (A has m rows, n columns) c Amazingly versatile modeling language. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 4 / 24

7 Linear rogramming Linear rogramming (L): linear constraints & linear objective with continuous variables. max c T x subject to Ax b, x R n (A has m rows, n columns) c Amazingly versatile modeling language. Generally provides a relaxed view of a desired optimization problem, but can be solved in polynomial time! D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 4 / 24

8 Mixed Integer rogramming Mixed Integer rogramming (MI): models both continuous and discrete choices. max c T x + d T y subject to Ax + By b, x R n 1, y Z n 2 D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 5 / 24

9 Mixed Integer rogramming Mixed Integer rogramming (MI): models both continuous and discrete choices. max c T x + d T y subject to Ax + By b, x R n 1, y Z n 2 D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 5 / 24

10 Mixed Integer rogramming Mixed Integer rogramming (MI): models both continuous and discrete choices. max c T x + d T y subject to Ax + By b, x R n 1, y Z n 2 One of the most successful modeling language for many real world applications. While instances can be extremely hard to solve (MI is N-hard), many practical instances are not. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 5 / 24

11 Mixed Integer rogramming Mixed Integer rogramming (MI): models both continuous and discrete choices. max c T x + d T y subject to Ax + By b, x R n 1, y Z n 2 Many sophisticated software packages exist for these models (CLEX, Gurobi, etc.). MI solving is now considered a mature and practical technology. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 5 / 24

12 Sample Applications Routing delivery / pickup trucks for customers. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 6 / 24

13 Sample Applications Optimizing supply chain logistics. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 6 / 24

14 Standard Framework for Solving MIs Relax integrality of the variables. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

15 Standard Framework for Solving MIs Relax integrality of the variables. max c T x + d T y subject to Ax + By b, x R n 1, y Z n 2 D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

16 Standard Framework for Solving MIs Relax integrality of the variables. max c T x + d T y subject to Ax + By b, x R n 1, y R n 2 D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

17 Standard Framework for Solving MIs Relax integrality of the variables. max c T x + d T y subject to Ax + By b, x R n 1, y R n 2 Solve the L. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

18 Standard Framework for Solving MIs Relax integrality of the variables. max Solve the L. c T x + d T y subject to Ax + By b, x R n 1, y R n 2 Add extra constraints to tighten the L or guess the values of some of the integer variables. Repeat. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

19 Standard Framework for Solving MIs Relax integrality of the variables. max Solve the L. c T x + d T y subject to Ax + By b, x R n 1, y R n 2 Add extra constraints to tighten the L or guess the values of some of the integer variables. Repeat. Need to solve a lot of Ls quickly. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 7 / 24

20 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

21 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 1 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

22 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 2 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

23 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 3 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

24 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 4 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

25 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 5 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

26 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 6 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

27 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 7 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

28 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n Simplex Method: move from vertex to vertex along the graph of until the optimal solution is found. v 8 c D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 8 / 24

29 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. Question Why is simplex so popular? D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 9 / 24

30 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. Question Why is simplex so popular? Easy to reoptimize when adding an extra constraint or variable. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 9 / 24

31 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. Question Why is simplex so popular? Easy to reoptimize when adding an extra constraint or variable. Vertex solutions are often nice (e.g. sparse, easy to interpret). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 9 / 24

32 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. Question Why is simplex so popular? Easy to reoptimize when adding an extra constraint or variable. Vertex solutions are often nice (e.g. sparse, easy to interpret). Simplex pivots implementable using simple linear algebra. Worst case O(mn) time per iteration. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 9 / 24

33 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. roblem No known pivot rule is proven to converge in polynomial time!!! D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 10 / 24

34 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. roblem No known pivot rule is proven to converge in polynomial time!!! Simplex lower bounds: Klee-Minty (1972): designed deformed cubes, providing worst case examples for many pivot rules. Friedmann et al. (2011): systematically designed bad examples using Markov decision processes. In these examples, the pivot rule is tricked into taking an (sub)exponentially long path, even though short paths exists. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 10 / 24

35 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. roblem No known pivot rule is proven to converge in polynomial time!!! Simplex upper bounds: Kalai (1992), Matousek-Sharir-Welzl ( ): Random facet rule requires 2 O( n log m) pivots on expectation. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 11 / 24

36 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. roblem No known pivot rule is proven to converge in polynomial time!!! olynomial bounds for random or smoothed linear programs: Borgwardt (82), Smale (83), Adler (83), Todd (83), Haimovich (83), Meggido (86), Adler-Shamir-Karp (86,87), Spielman-Teng (01,04), Spielman-Deshpande (05), Spielman-Kelner (06), Vershynin (06) D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 11 / 24

37 Linear rogramming via the Simplex Method max c T x subject to Ax b, x R n A has n columns, m rows. roblem No known pivot rule is proven to converge in polynomial time!!! olynomial bounds for random or smoothed linear programs: Borgwardt (82), Smale (83), Adler (83), Todd (83), Haimovich (83), Meggido (86), Adler-Shamir-Karp (86,87), Spielman-Teng (01,04), Spielman-Deshpande (05), Spielman-Kelner (06), Vershynin (06) These works rely on the shadow simplex method. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 11 / 24

38 Linear rogramming and the Hirsch Conjecture = {x R n : Ax b}, A R m n lives in R n (ambient dimension is n) and has m constraints. Besides the computational efficiency of the simplex method, an even more basic question is not understood: Question How can we bound the length of paths on the graph of? I.e. how to bound the diameter of? D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 12 / 24

39 Linear rogramming and the Hirsch Conjecture = {x R n : Ax b}, A R m n lives in R n (ambient dimension is n) and has m constraints. Conjecture (olynomial Hirsch Conjecture) The diameter of is bounded by a polynomial in the dimension n and number of constraints m. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 13 / 24

40 Linear rogramming and the Hirsch Conjecture = {x R n : Ax b}, A R m n lives in R n (ambient dimension is n) and has m constraints. Conjecture (olynomial Hirsch Conjecture) The diameter of is bounded by a polynomial in the dimension n and number of constraints m. Diameter upper bounds: Barnette, Larman (1974): 1 3 2n 2 (m n ). Kalai, Kleitman (1992), Todd (2014): (m n) log n. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 13 / 24

41 Well-Conditioned olytopes = {x R n : Ax b}, A Z m n Definition (Bounded Subdeterminants) An integer matrix A has subdeterminants bounded by if every square submatrix B of A satisfies det(b). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 14 / 24

42 Well-Conditioned olytopes = {x R n : Ax b}, A Z m n Definition (Bounded Subdeterminants) An integer matrix A has subdeterminants bounded by if every square submatrix B of A satisfies det(b). Diameter Bound: Bonifas, Di Summa, Eisenbrand, Hähnle, Niemeier (2012): non-constructive O(n log n ) bound. Brunsch,Röglin (2013): shadow simplex method finds paths of length O(mn 3 4 ). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 14 / 24

43 Well-Conditioned olytopes = {x R n : Ax b}, A Z m n Definition (Bounded Subdeterminants) An integer matrix A has subdeterminants bounded by if every square submatrix B of A satisfies det(b). Diameter Bound: Bonifas, Di Summa, Eisenbrand, Hähnle, Niemeier (2012): non-constructive O(n log n ) bound. Brunsch,Röglin (2013): shadow simplex method finds paths of length O(mn 3 4 ). Optimization: Dyer, Frieze (1994): = 1 (totally unimodular) random walk simplex uses O(m 16 n 6 log(mn) 3 ) pivots. Eisenbrand, Vempala (2014): random walk simplex uses O(mpoly(n, )) pivots. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 14 / 24

44 Faster Shadow Simplex = {x R n : Ax b}, A Z m n Subdeterminants of A bounded by. Theorem (D., Hähnle 2014+) Diameter is bounded by O(n 3 2 ln(n )). Can solve L using O(n 5 2 ln(n )) pivots on expectation. Based on a new analysis and variant of the shadow simplex method. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 15 / 24

45 τ-wide olyhedra v 1 v 3 v 2 v 4 D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

46 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 Normal cone at v is all objectives maximized at v. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

47 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 N 1 N 3 N 4 Normal cone at v is all objectives maximized at v. Normal fan is the set of normal cones. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

48 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 τ N 1 N 3 N 4 Normal cone at v is all objectives maximized at v. Normal fan is the set of normal cones. is τ-wide if each normal cone contains a unit ball of radius τ centered on the unit sphere. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

49 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 τ N 1 N 3 N 4 Normal cone at v is all objectives maximized at v. Normal fan is the set of normal cones. is τ-wide if each normal cone contains a unit ball of radius τ centered on the unit sphere. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

50 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 τ N 1 N 3 N 4 has subdeterminant bound is τ-wide for τ = 1/(n ) 2. Normal cone at v is all objectives maximized at v. Normal fan is the set of normal cones. is τ-wide if each normal cone contains a unit ball of radius τ centered on the unit sphere. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 16 / 24

51 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 τ N 1 N 3 N 4 In 2 dimensions all interior angles are at most π 2τ, i.e. sharp. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 17 / 24

52 τ-wide olyhedra N 2 N 1 v 2 v 3 N 3 v 1 v 4 N 4 N 2 τ N 1 N 3 N 4 In 2 dimensions all interior angles are at most π 2τ, i.e. sharp. Theorem (D.-Hähnle 2014) If is τ-wide then the diameter of is bounded by O(n/τ ln(1/τ)). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 17 / 24

53 Outline 1 Introduction Linear rogramming and its Applications The Simplex Method Results 2 The Shadow Simplex Method ivot Rule 3-Step Shadow Simplex ath 3 Conclusions D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 18 / 24

54 ivot Rule c v 1 c d v 3 v 2 v 4 d Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

55 ivot Rule c v 1 c d v 3 v 2 v 4 d Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

56 ivot Rule c d 1 v 1 c d v 2 v 3 v 4 d d 1 Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

57 ivot Rule c d d 1 v 2 v 3 v 1 v 4 d d 1 c Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

58 ivot Rule c d d 1 d 2 v 1 v d 1 2 d 2 v 3 v 4 d c Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

59 ivot Rule c v 1 c d d 2 v 2 v 3 v 4 d d 1 d 2 Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

60 ivot Rule c v 1 c d v 2 d 2 d 2 v 3 v 4 d Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

61 ivot Rule c v 1 c d v 3 v 2 v 4 d Move from v 1 to v 3 by following [c, d] through the normal fan. ivot steps correspond to crossing facets of the normal fan. Question How can we bound the number of intersections with the normal fan? D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 19 / 24

62 ivot Rule c v 1 c d v 2 v 3 v 4 d Question How can we bound the number of intersections with the normal fan? olynomial bounds for random and smoothed linear programs: Borgwardt (82), Smale (83), Adler (83), Todd (83), Haimovich (83), Meggido (86), Adler-Shamir-Karp (86,87), Spielman-Teng (01,04), Spielman-Deshpande (05), Spielman-Kelner (06), Vershynin (06) D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 20 / 24

63 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

64 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. c is an objective maximized at the starting vertex. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

65 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. c is an objective maximized at the starting vertex. d is the L objective function. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

66 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. c is an objective maximized at the starting vertex. d is the L objective function. Theorem (D.-Hähnle 2014) Assume is an n-dimensional τ-wide polytope. hase (b). Expected number of crossings of [c + X, d + X ] with normal fan is O( c d /τ). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

67 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. c is an objective maximized at the starting vertex. d is the L objective function. Theorem (D.-Hähnle 2014) Assume is an n-dimensional τ-wide polytope. hase (b). Expected number of crossings of [c + X, d + X ] with normal fan is O( c d /τ). hase (a+c). Expected number of crossings of [c + αx, c + X ] (same for d), for α (0, 1], with normal fan is O(n/τ ln(1/α)). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

68 3-Step Shadow Simplex ath We use a 3-step shadow simplex path: c (a) c + X (b) d + X (c) d where X is distributed proportional to e x. c is an objective maximized at the starting vertex. d is the L objective function. Theorem (D.-Hähnle 2014) Assume is an n-dimensional τ-wide polytope. hase (b). Expected number of crossings of [c + X, d + X ] with normal fan is O( c d /τ). hase (a+c). Expected number of crossings of [c + αx, c + X ] (same for d), for α (0, 1], with normal fan is O(n/τ ln(1/α)). Main ingredient for all our results. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 21 / 24

69 Outline 1 Introduction Linear rogramming and its Applications The Simplex Method Results 2 The Shadow Simplex Method ivot Rule 3-Step Shadow Simplex ath 3 Conclusions D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 22 / 24

70 Navigation over the Voronoi Graph y t x Z Z + t Figure: Randomized Straight Line algorithm Closest Vector roblem (CV): Find closest lattice vector y to t. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 23 / 24

71 Navigation over the Voronoi Graph y t x Z Z + t Figure: Randomized Straight Line algorithm Closest Vector roblem (CV): Find closest lattice vector y to t. Can reduce CV to efficient navigation over the Voronoi graph (Sommer,Feder,Shalvi 09; Micciancio,Voulgaris 10-13). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 23 / 24

72 Navigation over the Voronoi Graph y t x Z Z + t Figure: Randomized Straight Line algorithm Closest Vector roblem (CV): Find closest lattice vector y to t. Can move between nearby lattice points using a polynomial number of steps (Bonifas, D. 14). D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 23 / 24

73 Summary New and simpler analysis and variant of the Shadow Simplex method. Improved diameter bounds and simplex algorithm for curved polyhedra. D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 24 / 24

74 Summary New and simpler analysis and variant of the Shadow Simplex method. Improved diameter bounds and simplex algorithm for curved polyhedra. Thank you! D. Dadush, N. Hähnle Shadow Simplex Scientific Meeting 24 / 24