Robust Integer Programming

Transcription

1 Robust Integer Programming Technion Israel Institute of Technology

2 Outline. Overview our theory of Graver bases for integer programming (with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel)

3 Outline. Overview our theory of Graver bases for integer programming (with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel) 2. Robust integer programming (Arxiv, 24)

4 Outline. Overview our theory of Graver bases for integer programming (with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel) 2. Robust integer programming (Arxiv, 24). Application - multiway tables

5 Outline. Overview our theory of Graver bases for integer programming (with Berstein, De Loera, Hemmecke, Lee, Romanchuk, Rothblum, Weismantel) 2. Robust integer programming (Arxiv, 24). Application - multiway tables 4. Huge multiway tables - P versus NP and conp (Arxiv, 24)

6 (Non)-Linear Integer Programming The problem is: min { f(x) : Ax = b, l x u, x in Z n } with data: A: integer m x n matrix b: right-hand side in Z m l,u: lower/upper bounds in Z n f: function from Z n to R

7 (Non)-Linear Integer Programming The problem is: min { f(x) : Ax = b, l x u, x in Z n } with data: A: integer m x n matrix b: right-hand side in Z m l,u: lower/upper bounds in Z n f: function from Z n to R Our theory enables polynomial time solution of broad natural universal (non)-linear integer programs in variable dimension

8 Graver Bases and Nonlinear Integer Programming

9 Graver Bases The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax =. x is conformal-minimal if no other y in same orthant has all y i x i

10 Graver Bases The Graver basis of an integer matrix A is the finite set G(A) of conformal-minimal nonzero integer vectors x satisfying Ax =. Example: Consider A=( 2 ). Then G(A) consists of circuits: ±(2 - ), ±( -), ±( -2) non-circuits: ±( - )

11 Some Theorems on (Non)-Linear Integer Programming Theorem : linear optimization in polytime with G(A): min { wx : Ax = b, l x u, x in Z n } Reference: N-fold integer programming (De Loera, Hemmecke, Onn, Weismantel) Discrete Optimization (Volume in memory of George Dantzig), 28

12 Some Theorems on (Non)-Linear Integer Programming Theorem 2: separable convex minimization in polytime with G(A): min {Σ f i (x i ) : Ax = b, l x u, x in Z n } Reference: A polynomial oracle-time algorithm for convex integer minimization (Hemmecke, Onn, Weismantel) Mathematical Programming, 2

13 Some Theorems on (Non)-Linear Integer Programming Theorem : polynomial minimization in polytime with G(A): min {p(x) : Ax = b, l x u, x in Z n } for a certain class of possibly non-convex multivariate polynomials. Reference: The quadratic Graver cone, quadratic integer minimization & extensions (Lee, Onn, Romanchuk, Weismantel), Mathematical Programming, 22

14 The Iterative Algorithm

15 Proof of Theorem 2 To solve min {f(x) : Ax = b, l x u, x in Z n } R n Do: with the Graver basis G(A) f R

16 Proof of Theorem 2 To solve min {f(x) : Ax = b, l x u, x in Z n } R n f Do: with the Graver basis G(A). Find initial point by auxiliary program R

17 Proof of Theorem 2 To solve min {f(x) : Ax = b, l x u, x in Z n } R n R f with the Graver basis G(A) Do:. Find initial point by auxiliary program 2. Iteratively improve by Graver-best steps, that is, by best cz with c in Z and z in G(A).

18 Proof of Theorem 2 To solve min {f(x) : Ax = b, l x u, x in Z n } R n R f with the Graver basis G(A) Do:. Find initial point by auxiliary program 2. Iteratively improve by Graver-best steps, that is, by best cz with c in Z and z in G(A). Using supermodality of f and integer Caratheodory theorem (Cook-Fonlupt-Schrijver, Sebo) we can show polytime convergence to some optimal solution

19 Robust Integer Programming

20 Robust Integer Programming Player (decision maker) chooses point x feasible in X := {x Z n : Ax = b, l x u }

21 Robust Integer Programming Player (decision maker) chooses point x feasible in X := {x Z n : Ax = b, l x u } Player 2 (nature) reveals cost c in a set of one of two forms: Box: C := {c Z n : d c e } List: C := {c,..., c k }

22 Robust Integer Programming Player (decision maker) chooses point x feasible in X := {x Z n : Ax = b, l x u } Player 2 (nature) reveals cost c in a set of one of two forms: Box: C := {c Z n : d c e } List: C := {c,..., c k } The problem is: min x X max c C cx

23 Robust Integer Programming Player (decision maker) chooses point x feasible in X := {x Z n : Ax = b, l x u } Player 2 (nature) reveals cost c in a set of one of two forms: Box: C := {c Z n : d c e } List: C := {c,..., c k } The problem is: min x X max c C cx The dual problem is: max c C min x X cx

24 Robust Integer Programming Theorem: The following hold for Robust IP with G(A) given: Box C List C min x X max c C cx polynomial time NP-hard max c C min x X cx NP-hard polynomial time Reference: Robust integer programming (Onn) Arxiv, 24

25 Multiway Tables

26 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X m k+ tables with given margins

27 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X m k+ tables with given margins. If all m i fixed then solvable by fixed dimension theory

28 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n

29 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n g If one side n is variable then we have the following: 9 Lemma: For fixed m,..., m k there is Graver complexity g:=g(m,..., m k ) such that the relevant Graver basis consists of all O(n g ) lifts of the relevant Graver basis of m X... X m k X g tables.

30 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n g If one side n is variable then we have the following: 9 Lemma: For fixed m,..., m k there is Graver complexity g:=g(m,..., m k ) such that the relevant Graver basis consists of all O(n g ) lifts of the relevant Graver basis of m X... X m k X g tables. g(,)=9, g(,4) = 27, g(,5) =? Ω(2 m ) = g(,m) = Ο(6 m )

31 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n g Corollary: (Non)-linear optimization over m X... X m k X n tables with given margins is doable in polynomial time O(n g(m,,m k ) L).

32 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n Corollary: (Non)-linear optimization over m X... X m k X n tables with given margins is doable in polynomial time O(n g(m,,m k ) L). Corollary: Robust optimization with Box C over m X... X m k X n tables with given margins is doable in polynomial time O(n g(m,,m k ) L).

33 Multiway Tables The multiway table problem of Motzkin (952) concerns minimization over m X... X m k X n tables with given margins. n Corollary: (Non)-linear optimization over m X... X m k X n tables with given margins is doable in polynomial time O(n g(m,,m k ) L). However, if two sides are variable then have the Universality Theorem: Every integer program is one over X m X n tables with given line-sums.

34 Multiway Tables Theorem: Linear optimization over m X... X m k X n tables with given margins is doable in fixed-parameter cubic time O(n L). Reference: N-fold integer programming in cubic time (Hemmecke, Onn, Romanchuk) Mathematical Programming, 2

35 Huge Multiway Tables

36 Huge Multiway Tables Consider the problem of deciding existence of an l X m X n huge table with t given types of row-sums and column-sums having n k layers of each type k with the n k encoded in binary. n

37 Huge Multiway Tables Consider the problem of deciding existence of an l X m X n huge table with t given types of row-sums and column-sums having n k layers of each type k with the n k encoded in binary. n Theorem: For fixed t the problem is in P. For variable t it is in NP and conp. Reference: Huge Multiway Table Problems (Onn) Arxiv, 24

38 Some Bibliography (available at - The complexity of -way tables (SIAM J. Comp.) - Convex combinatorial optimization (Disc. Comp. Geom.) - Markov bases of -way tables (J. Symb. Comp.) - All linear and integer programs are slim -way programs (SIAM J. Opt.) - Graver complexity of integer programming (Annals Combin.) - N-fold integer programming (Disc. Opt. in memory of Dantzig) - Convex integer maximization via Graver bases (J. Pure App. Algebra) - Polynomial oracle-time convex integer minimization (Math. Prog.) - The quadratic Graver cone, quadratic integer minimization & extensions (Math Prog.) - N-fold integer programming in cubic time (Math. Prog.) - Robust integer programming (Arxiv) - Huge multiway table problems (Arxiv)

39 Comprehensive development is in my monograph available electronically from my homepage (with kind permission of EMS) (excluding Robust IP and Huge Multiway Tables