Easy and not so easy multifacility location problems... (In 20 minutes.)

Easy and not so easy multifacility location problems... (In 20 minutes.) MINLP 2014 Pittsburgh, June 2014 Justo Puerto Institute of Mathematics (IMUS) Universidad de Sevilla

Outline 1 Introduction (In 3 minutes) 2 Discrete ordered median problem (in 7 minutes) 3 Continuous multifacility problems (10 minutes) 4 Dimensionality reduction of the SDP-rel for multifacility location problems (in no time) Mutifacility Location Problems MINLP 2014 2

A few minutes of motivation Mutifacility Location Problems MINLP 2014 3

A few minutes of motivation Logistic problem or in another jargon Clustering or Classification Problem Mutifacility Location Problems MINLP 2014 3

Discrete ordered median problem: Modeling framework Mutifacility Location Problems MINLP 2014 4

Discrete ordered median problem: Modeling framework r 1 r 2 r 3 r 4 r 5 λ 1r 1 + λ 2r 2 + λ 3r 3 + λ 4r 4 + λ 5r 5 Mutifacility Location Problems MINLP 2014 4

Ordered Median Functions Definition Let x R n and λ R n. The ordered median function, f λ is defined as: f λ (x) = λ, sort(x) where sort(x) = (x (1),..., x (n) ) with x (i) {x 1,..., x n} and such that x (1) x (2) x (n). Mutifacility Location Problems MINLP 2014 5

Special Cases Mean ( 1 n,, 1 n ) 1 n Minimum (1, 0,, 0) min i 1 i n Maximum (0, 0,, 1) max i 1 i n {}}{ k-centrum (0,, 0, 1,, 1) {}}{ anti-k-centrum ( 1,, 1, 0,, 0) {}}{{}}{ (k 1, k 2)-Trimmed mean ( 0,, 0, 1,, 1, 0,, 0) k 1 k k k 2 i=n k+1 k n k 2 i=k 1 +1 Range ( 1, 0,, 0, 1) max i min 1 i n x i x (i) x (i) x (i) 1 i n x i Mutifacility Location Problems MINLP 2014 6

Our program starts about 2006... Mutifacility Location Problems MINLP 2014 7

Previous Works Discrete Case: (Nickel & P., 2005; Boland, Marin, Nickel, P., 2006; Marín, Nickel, P., 2009). Networks: ( Nickel & P., 1999; Kalcsics, Nickel, P., 2002; Kalcsics, Nickel, P., Tamir, 2006; P. & Tamir, 2005); Continuous: Planar Weber Problem: (Weiszfeld, 1937; and extensions...) Planar Euclidean: BTST (Drezner, 2007); D.C. (Drezner& Nickel, 2009). Planar Convex OM with l p norms: (Espejo, et. al, 2009). Planar Euclidean k-centrum: (Rodríguez-Chía et. al, 2010). OM 1-Facility Location, any dimension, any l p or polyhedral norm: (Blanco, P. & ElHaj-BenAli, 2013, 2014). Mutifacility Location Problems MINLP 2014 8

7 minutes for the DOMP: An INLP A = {a 1,..., a n} C = (c ij ) i,j=1,...,n X A with X = p p c i (X ) := min k X c ik σ X a permutation of {1,..., n} c σx (1)(X ) c σx (2)(X ) c σx (n)(x ) Discrete ordered median problem (DOpP) min X A, X =p λ i c σx (i)(x ). with λ = (λ 1,..., λ n) y λ i 0, i = 1,..., n. Mutifacility Location Problems MINLP 2014 9

Sorting as an integer program (IP) INPUT: real numbers r 1,..., r n. OUTPUT: r σ(1) r σ(2)... r σ(n), σ Π({1,..., n}). Decision variables: s ki = { 1 if σ(k) = i 0 otherwise i, k = 1,..., n. Mutifacility Location Problems MINLP 2014 10

r σ(k) := s ki r i, k = 1,..., n Desired output obtained Sorting as an IP: (SORT) minimize 1 s.t. s ki = 1 k=1 s ki = 1 s ki r i i = 1,..., n k = 1,..., n s k+1,i r i k= 1,..., n 1 s ki {0, 1} i, k = 1,..., n. Mutifacility Location Problems MINLP 2014 11

p-median formulation Decision variables: { 1 a new facility is built in aj y j = 0 otherwise { 1 site ai is served by facility a j and x ij = 0 otherwise minimize c ij x ij j=1 y j = p j=1 x ij = 1 j=1 y j x ij i = 1,..., n j, i = 1,..., n y j {0, 1}, x ij 0 j, i = 1,..., n Mutifacility Location Problems MINLP 2014 12

Quadratic formulation for DOMP: Constraints to sort Setting r i = minimize c ij x ij, i = 1,..., n. (SORT+ p-median) (DOMP) j=1 n k=1 λ k s ki n s n ki j=1 c ijx ij s ki = 1 k=1 s ki = 1 c ij x ij j=1 s k+1,i i = 1,..., n k = 1,..., n c ij x ij k = 1,..., n 1 j=1 Mutifacility Location Problems MINLP 2014 13

Quadratic formulation for DOMP: Constraints to sort Constraints corresponding to p-median problem y j = p j=1 x ij = 1 j=1 y j x ij x ij 0 s ki {0, 1} y j {0, 1} i = 1,..., n j, i = 1,..., n j, i = 1,..., n i, k = 1,..., n j = 1,..., n Mutifacility Location Problems MINLP 2014 14

A first linearization: DOMP1 Constraints to sort. Adding artificial variables x k (L1) minimize n k=1 λ k n n j=1 c ij s ki x ij }{{} x k ij ij j=1 k=1 s ki }{{} j xk ij = 1 k = 1,..., n s ki = 1 k = 1,..., n }{{} j xk ij c ij s ki x ij k = 1,..., n 1 }{{} x k ij j=1 c ij s k+1,i x ij }{{} x k+1 ij Mutifacility Location Problems MINLP 2014 15

Linearization Constraints corresponding to p-median problem y j = p j ( ) x ij = 1 i = 1,..., n }{{} j k xk ij y j x ij j, i = 1,..., n }{{} k xk ij xij k {0, 1} y j {0, 1} i, k, j = 1,..., n j = 1,..., n Mutifacility Location Problems MINLP 2014 16

A new formulation for DOMP: DOMP3 Example j=1 c ij x k ij j=1 c ij x k+1 ij k = 1,..., n 1 0 2 7 4 C = 1 0 5 5 3 6 0 2 9 4 1 0 11 22 33 44 21 43 12 34 31 14 42 23 24 32 13 41 Mutifacility Location Problems MINLP 2014 17

A new formulation for DOMP: DOMP3 Example j=1 c ij x k ij j=1 c ij x k+1 ij k = 1,..., n 1 0 2 7 4 C = 1 0 5 5 3 6 0 2 9 4 1 0 11 22 33 44 21 43 12 34 31 14 42 23 24 32 13 41 x k ij = y j = 1 if facility i allocated to server j and C ij is the k-th smallest cost 0 otherwise { 1 if j J 0 if j / J Mutifacility Location Problems MINLP 2014 17

The rationale: Incompatibilities k k 1 Mutifacility Location Problems MINLP 2014 18

The rationale: Incompatibilities k k 1 k k 1 Mutifacility Location Problems MINLP 2014 18

DOMP 3 min s.t. îĵ ij j=1 k=1 j=1 k=1 k=1 x k 1 îĵ j=1 λ k C ij x k ij x k ij = 1 x k ij = 1 x k ij y j y j = p j=1 + îĵ ij x k ij {0, 1} x k îĵ 1 i = 1,..., n k = 2,..., n i, j = 1,..., n i, j = 1,, n k = 2,, n i, j, k = 1,..., n Mutifacility Location Problems MINLP 2014 19

DOMP 4 Formulation min j=1 k=1 λ k C ij x k ij s.t. j=1 k=1 x k ij = 1 i = 1,..., n j=1 x k ij = 1 k = 2,..., n x k ij y j k=1 y j = p j=1 i, j = 1,..., n i x k i j + j i =1 :i j j =1 ij i =1 :i j x k 1 j i j n2 =1 ij k = 2,, n x k ij {0, 1} i, j, k = 1,..., n Mutifacility Location Problems MINLP 2014 20

Comparing formulations Notation We denote by P l the polytope defining the feasible set of the linear relaxation of formulation DOMP l, by z l ( ) the value of the objective function of DOMP l evaluated at the point ( ) and by Pl I the convex hull of the integer solutions within that polytope. Theorem 1 g(p 2) P 1. 2 f (P 3) P 2. 3 P 3 P 1. 4 z LP 3 z LP 2 z LP 1. Mutifacility Location Problems MINLP 2014 21

Formulation GAP DOMP 1 2.79% DOMP 2 6.18% DOMP 3 0.92% DOMP 4 2.57% DOMP 4 1 2.48% Table: Average integrality gaps Mutifacility Location Problems MINLP 2014 22

Formulation GAP DOMP 1 2.79% DOMP 2 6.18% DOMP 3 0.92% DOMP 4 2.57% DOMP 4 1 2.48% Table: Average integrality gaps Solution approach Time (s) #nodes DOMP 3(B&B) 463.24(8) 46.20 DOMP 4 1(B&B) 18.64 1389.51 DOMP 4 1(B&C 3) 52.25 24.94 DOMP 4(B&C 3) 39.39 29.37 Table: CPU-Time and Number of nodes of the different formulations for n = 20, 30, 50. Mutifacility Location Problems MINLP 2014 22

Continuous Multifacility Ordered Median Problems in 10 minutes Demand points : A = {a 1,..., a n } R d ( d ) 1 τ Norm : l τ ( x τ = x i τ, for all x R d ) Feasible Domain : R d Facilities : p facilities to be located. Goal : Find p points x1,..., xp R d minimizing some globalizing function of the distances to the set A. Mutifacility Location Problems MINLP 2014 23

An easy model: Multiple allocation MF OMP Rodríguez-Chía, Nickel, P., and Fernández. (2000). 10 8 6 4 2 2 4 6 8 10 Mutifacility Location Problems MINLP 2014 24

Multiple allocation MF OMP where, µ jj 0. f NI λ (x 1, x 2,..., x p) = p j=1 p 1 λ ij d (i) (x j ) + X = {x 1,..., x p}, d i (x) = a i x τ, p j=1 j =j+1 d (1) (x) d (2) (x)... d (n) (x), λ 11 λ 21... λ n1 0, λ 12 λ 22... λ n2 0,... λ 1p λ 2p... λ np 0, µ jj x j x j τ, (1) Mutifacility Location Problems MINLP 2014 25

Decision variables v ij and w lj continuous variables modelling the ordered weighted average in the o.f., x j = coordinates in R d of the j-th facility, y ijk = absolute value of the k-th coordinate of the vector x j a i, u ij = x j a i τ τ. z jj k = absolute value of the k-th coordinate of the vector x j x j, t jj = x j x j τ τ. Mutifacility Location Problems MINLP 2014 26

Multiple allocation MF OMP Theorem Let τ = r s be such that r, s N \ {0}, r s and gcd(r, s) = 1. For any set of lambda weights satisfying λ 1j... λ nj 0 for all j = 1,..., p, Problem (LOCOMF NI) is equivalent to p p ρ NI λ = min p 1 p v ij + w lj + t jj (2) j=1 l=1 j=1 j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p y ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d y ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, y r ijk ςs ijk ur s ij, i = 1,..., n, j = 1,..., p, k = 1,..., d, r ω s i d ς ijk u ij, k=1 i = 1,..., n, j = 1,..., p z jj k x jk + x j k 0, j, j = 1,..., p, k = 1,..., d, z jj k + x jk x j k 0, j, j = 1,..., p, k = 1,..., d, z r jj k ξs jj k tr s jj, j, j = 1,..., p, k = 1,..., d, r µ s jj d ξ jj k t jj, j, j = 1,..., p, k=1 ς ijk 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, ξ jj k 0 j, j = 1,..., p, k = 1,..., d. Mutifacility Location Problems MINLP 2014 27

Sorting with permutations: λ 1... λ n p j=1 λ ij u (i)j = max σ P n p λ ij u σ(i)j. The permutations in P n can be represented by the binary variables s ijk {0, 1} Next, combining the two sets of variables the objective function is: p p λ ij u (i)j = max λ ij u ij s ijk j=1 s.t j=1 k=1 j=1 s ijk = 1, j = 1,..., p, k = 1,..., n, s ijk = 1, i = 1,..., n, j = 1,..., p, k=1 s ijk {0, 1}. (3) Mutifacility Location Problems MINLP 2014 28

Sorting with a linear program For fixed j {1,..., p} and for any vector u j R n, by using the dual of the assignment problem (7) we obtain the following expression λ ij u (i)j = min v ij + w lj (4) l=1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n. Finally, we replace (4) for the objective function and we get min p v ij + j=1 l=1 p j=1 p 1 w lj + p j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p, u ij ω i x j a i τ, i = 1,..., n, j = 1,..., p, t jj µ jj x j x j τ, j, j = 1,..., p. t jj (5) Mutifacility Location Problems MINLP 2014 29

Second Order Cone Constraints Lemma (Blanco, P., ElHaj-BenAli, 2014) Let τ = r s be such that r, s N \ {0}, r > s, gcd(r, s) = 1 and k = log 2 (r) 1. Let x, u and t be non negative and satisfying x r u s t r s. (6) Then, there exists w such that, (6) is equivalent to: w 2 1 A 1B 1,.. w 2 m A mb m, x 2 A m+1b m+1, (7) where A i, B i {w i 1, u, t, x} for i = 1,..., m and m = 1 + 2#{i : α i + β i + γ i 2, 1 i < k 1} + #{i : α i + β i + γ i 1, 1 i < k 1} O(log(r)) with: α i, β i, γ i {0, 1}. s = α k 1 2 k 1 + α k 2 2 k 2 +... + α 12 1 + α 02 0, r s = β k 1 2 k 1 + β k 2 2 k 2 +... + β 12 1 + β 02 0, 2 k r = γ k 1 2 k 1 + γ k 2 2 k 2 +... + γ 12 1 + γ 02 0, 2 k = (α k 1 + β k 1 + γ k 1 )2 k 1 + + (α 0 + β 0 + γ 0)2 0, Mutifacility Location Problems MINLP 2014 30

Corollary (sequel) Moreover, Problem (2) satisfies Slater condition and it can be represented as a second order cone (or semidefinite) program with (np + p 2 )(2d + 1) + p 2 linear inequalities and at most 4(p 2 d + npd) log r second order cone (or linear matrix) inequalities. Remark As consequence of the previous theorem, problem (2) is convex and can be solved, up to any given accuracy, in polynomial time for any dimension d. Mutifacility Location Problems MINLP 2014 31

Experiments Coded in: Gurobi 5.6. Intel Core i7 processor; 2x 2.40 GHz; 4 GB of RAM Instance: 50-points data set in Eilon, Watson-Gandy & Christofides (1971). 10 8 6 4 2 0 0 2 4 6 8 10 CPU-Times (secs.): τ 1.5 2 3 p 2 2.51 2.12 3.75 5 12.78 6.52 9.81 10 29.19 10.57 19.55 15 49.49 19.11 40.45 30 148.75 40.56 85.57 1 Mutifacility Location Problems MINLP 2014 32

The program continues with some extensions... Mutifacility Location Problems MINLP 2014 33

A hard MF problem: Single allocation MF OMP Miehle (1958), Cooper (1963) C. Pey-Chun, P. Hansen, B. Jaumard and T. Hoang (1998). 10 8 6 4 2 0 2 4 6 8 10 Mutifacility Location Problems MINLP 2014 34

Single allocation MF OMP Assume that: f i (x) : R pd R x = (x 1,..., x p) t i := min j a i τ }. j=1..p K R d. τ := r 1, r, s N with gcd(r, s) = 1. s λ l 0 for all l = 1,..., n. Consider the following problem ρ λ := min x { λ i f(i) (x) : x = (x 1,..., x p), x j K, j = 1,..., p}, (LOCOMF) Mutifacility Location Problems MINLP 2014 35

Single allocation MF OMP ˆρ λ = min λ l θ l l=1 (MFOMP λ ) s.t. h il 1 := t i θ l + UB i (1 w il ), i = 1,..., n, l = 1,..., n, h l 2 := θ l θ l+1, l = 1,..., n 1, h 3 ij := u ij t i + UB i (1 z ij ), i = 1,..., n, j = 1,..., p, hijk 4 := v ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, h ijk 5 := v ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,.., d, h ijk 6 := vr ijk ζs ijk ur s, i = 1,..., n, j = 1,..., p, k = 1,..., d, ij hij 7 d := ζ ijk u ij, i = 1,..., n, j = 1,..., p, k=1 hi 8 p := z ij = 1, i = 1,..., n, j=1 hl 9 := w il = 1, l = 1,..., n, h 10 i := w il = 1, i = 1,..., n, l=1 w il {0, 1}, i, l = 1,..., n, z ij {0, 1}, i = 1,..., n, j = 1,..., p, θ l, t i, v ijk, ζ ijk, u ij R +, i, l = 1,..., n, j = 1,..., p, k = 1,..., d, x j K, j = 1,..., p. Mutifacility Location Problems MINLP 2014 36

Theorem Let x be a feasible solution of LOCOMF then there exists a solution (x, z, u, v, ζ, w, t, θ) for MFOMP λ such that their objective values are equal. Conversely, if (x, z, u, v, ζ, w, t, θ) is a feasible solution for MFOMP λ then x is a feasible solution for LOCOMF. Furthermore, if K satisfies Slater condition then the feasible region of the continuous relaxation of MFOMP λ also satisfies Slater condition and ρ λ = ˆρ λ. Mutifacility Location Problems MINLP 2014 37

Experiments Problem p τ CPU Time f Problem p τ CPU Time f Problem p τ CPU Time f p-median 2 5 10 15 30 1.5 22.31 150.955 1.5 1.03 4.9452 1.5 10.08 100.8474 2 2 2 1.13 135.5222 2 0.28 4.8209 2 0.38 95.0892 3 23.68 130.856 3 13.51 4.788 3 139.03 89.0238 1.5 55.28 78.6074 1.5 3.73 2.8831 1.5 33.09 53.4995 5 5 2 12.49 72.2369 2 5.37 2.661 2 7.61 49.6932 3 125.1 68.1791 3 2.87 2.5094 3 18.23 46.9844 1.5 5.36 45.0525 1.5 2.66 1.6929 1.5 68.36 30.7137 10 10 2 2.31 41.6851 2 5.3 1.6113 2 17.93 28.9017 p-center 3 4.76 39.7222 3 55.76 1.595 3 225.64 27.5376 1.5 6.7 30.0543 1.5 9.44 1.1139 1.5 49.92 22.4165 15 15 2 43.91 27.6282 2 0.62 1.0717 2 11.26 20.6536 3 150.99 26.6047 3 50.08 1.053 3 244.59 20.8544 1.5 14.45 9.9488 1.5 74.43 1.008 1.5 202.54 9.0806 30 30 2 4.81 8.7963 2 1.53 0.9192 2 5.29 8.521662 3 198.78 8.6995 3 57.37 0.8508 3 287.90 8.001695 p-25-centrum Table: Computational results for p-median, p-center and p-25-centrum problems for the 50-points in Eilon, Watson-Gandy & Christofides data set. Mutifacility Location Problems MINLP 2014 38

Constrained MF Ordered Median Problem A SDP relaxation of the LOCOMF Let h 0(θ) := m λ l θ l, and denote ξ j := (deg g j )/2 and ν j := (deg h j )/2, where l=1 {g 1,..., g nk }, and {h 1,..., h nc1} are, respectively, the polynomial constraints that define K and ˆK \ K in MFOMP λ. For r r 0 := max{ max ξ k, max ν j}, introduce the k=1,...,n K j=0,...,nc1 hierarchy of semidefinite programs: min y L y(p λ ) s.t. M r (y) 0, M r ξk (g k, y) 0, k = 1,..., n K, M r νj (h j, y) 0, j = 1,..., nc1, y 0 = 1, (Q1 r ) Mutifacility Location Problems MINLP 2014 39

Theorem Let ˆK R nv 1 (compact) be the feasible domain of Problem MFOMP λ. Let inf Q1 r be the optimal value of the semidefinite program Q1 r. Then, with the notation above: (a) inf Q1 r ρ λ as r. (b) Let y r be an optimal solution of the SDP relaxation Q1 r. If rank M r (y r ) = rank M r r0 (y r ) = ϕ then min Q1 r = ρ λ and one may extract ϕ points (x 1 (k),..., x p (k), z (k), u (k), v (k), ζ (k), w (k), t (k), θ (k)) ϕ k=1 ˆK, all global minimizers of the MFOMP λ problem. Mutifacility Location Problems MINLP 2014 40

The size of SDP relaxations grows rapidly with the original problem size. In view of the present status of SDP-solvers, only small to medium size problems can be solved. We found in our model patterns to exploit: Sparsity. Symmetry. Mutifacility Location Problems MINLP 2014 41

Thank you! puerto@us.es http://arxiv.org/abs/1401.0817 Mutifacility Location Problems MINLP 2014 42