fadsdfasadfs Ordered Weighted Average Optimization in - - PowerPoint PPT Presentation

fadsdfasadfs Ordered Weighted Average Optimization in Multiobjective Spanning Tree Problems

Elena Fern´ andez1 Miguel Pozo2 Justo Puerto2

1 Universitat Polit`

ecnica de Catalunya - BarcelonaTech

2 Universidad de Sevilla

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn
• f i : Q → R, i ∈ P = {1, . . . , p} objective functions.
• ω ∈ Rp+: non-negative weights.
• y = f(x) ∈ Rp, with x ∈ Q.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn
• f i : Q → R, i ∈ P = {1, . . . , p} objective functions.
• ω ∈ Rp+: non-negative weights.
• y = f(x) ∈ Rp, with x ∈ Q.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn
• f i : Q → R, i ∈ P = {1, . . . , p} objective functions.
• ω ∈ Rp+: non-negative weights.
• y = f(x) ∈ Rp, with x ∈ Q.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• f i : Q → R, i ∈ P = {1, . . . , p} objective functions.
• ω ∈ Rp+: non-negative weights.
• y = f(x) ∈ Rp, with x ∈ Q.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• C i : Q → R, i ∈ P = {1, . . . , p} f i(x) = C ix, C i ∈ Rn linear
• ω ∈ Rp+: non-negative weights.
• y = f(x) ∈ Rp, with x ∈ Q.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• C i : Q → R, i ∈ P = {1, . . . , p} f i(x) = C ix, C i ∈ Rn linear
• ω ∈ Rp+: non-negative weights.
• y = Cx, with x ∈ Q, C ∈ Rp×n.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ f σi(x) ≥ f σi+1(x)).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• C i : Q → R, i ∈ P = {1, . . . , p} f i(x) = C ix, C i ∈ Rn linear
• ω ∈ Rp+: non-negative weights.
• y = Cx, with x ∈ Q, C ∈ Rp×n.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ C σix ≥ C σi+1x).
• Ordered Weighted Average Operator (OWA): OWA(f ,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• C i : Q → R, i ∈ P = {1, . . . , p} f i(x) = C ix, C i ∈ Rn linear
• ω ∈ Rp+: non-negative weights.
• y = Cx, with x ∈ Q, C ∈ Rp×n.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ C σix ≥ C σi+1x).
• Ordered Weighted Average Operator (OWA): OWA(C,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(f ,ω)(x)

The Ordered Weighted Average Operator (OWA)

• Feasible domain Q ⊆ Rn

← − Combinatorial Object

• C i : Q → R, i ∈ P = {1, . . . , p} f i(x) = C ix, C i ∈ Rn linear
• ω ∈ Rp+: non-negative weights.
• y = Cx, with x ∈ Q, C ∈ Rp×n.
• σ: yσ1 ≥ . . . ≥ yσp (⇔ C σix ≥ C σi+1x).
• Ordered Weighted Average Operator (OWA): OWA(C,ω)(x) = ω′yσ
• OWA optimization Problem (OWAP): minx∈Q OWA(C,ω)(x)

Ordered Median Operator

Q ⊆ Rn, d ∈ Rn cost vector, ω ∈ Rn weights. For x ∈ Q, σ: permutation s.t. dσjxσj ≥ dσj+1xσj+1 OM(d,ω)(x) =

• j∈P

ωjdσjxσj

Ordered Median Operator

Q ⊆ Rn, d ∈ Rn cost vector, ω ∈ Rn weights. For x ∈ Q, σ: permutation s.t. dσjxσj ≥ dσj+1xσj+1 OM(d,ω)(x) =

• j∈P

ωjdσjxσj (C i)′ = diei, i ∈ {1, . . . , n}. C = Diag(d) OM(d,ω)(x) = OWA(Diag(d),ω)(x)

Ordered Median Operator

Q ⊆ Rn, d ∈ Rn cost vector, ω ∈ Rn weights. For x ∈ Q, σ: permutation s.t. dσjxσj ≥ dσj+1xσj+1 OM(d,ω)(x) =

• j∈P

ωjdσjxσj (C i)′ = diei, i ∈ {1, . . . , n}. C = Diag(d) OM(d,ω)(x) = OWA(Diag(d),ω)(x) Also generalizes the Vector Assignment Ordered Median Given cost d: fractions of sorted elements of solutions.

In this presentation ... The OWAP for Spanning Trees Q = {x : defines a spanning tree}

Example

fadsafdsω′ = (0.8, 0, 0.2)

1 2 3 4 5 6

(1, 2, 1) (2, 3, 4) (3, 4, 1) (4, 1, 2) (1, 2, 1) (1, 3, 4) (3, 3, 2) (4, 2, 1) (1, 2, 3)

Example

fadsafdsω′ = (0.8, 0, 0.2)

1 2 3 4 5 6

(1, 2, 1) (2, 3, 4) (3, 4, 1) (4, 1, 2) (1, 2, 1) (1, 3, 4) (3, 3, 2) (4, 2, 1) (1, 2, 3)

1 2 3 4 5 6

fadsafdsCx = (10, 11, 8) fadsafdabsσ = (2, 1, 3) fadsafdsValue 0.8(C 2x) + 0(C 1x) + 0.2(C 3x) = = 0.8 × 11 + 0.2 × 8 Value = 10.4

Our Formulation of the OWAP

x: Defines solutions in the domain Q. (yi = C ix: Value of objective function i)

F., Pozo, Puerto (2014)

Our Formulation of the OWAP

x: Defines solutions in the domain Q. (yi = C ix: Value of objective function i) zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

F., Pozo, Puerto (2014)

Our Formulation of the OWAP

x: Defines solutions in the domain Q. (yi = C ix: Value of objective function i) zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

θj: Value of the objective occupying position j in the ordering.

F., Pozo, Puerto (2014)

Our Formulation of the OWAP

F θ : V = min

• j∈P

ωjθj s.t.

• i∈P

zij = 1 j ∈ P

• j∈P

zij = 1 i ∈ P θj ≥ θj+1 j ∈ P : j < p θj =

• i∈P

zij(C ix) i,j ∈ P x ∈ Q θj ≥ 0 j ∈ P z ∈ {0, 1}p×p

Our Formulation of the OWAP

F θ : V = min

• j∈P

ωjθj s.t.

• i∈P

zij = 1 j ∈ P

• j∈P

zij = 1 i ∈ P θj ≥ θj+1 j ∈ P : j < p C ix ≤ θj + M(1 − zij) i, j ∈ P x ∈ Q θj ≥ 0 j ∈ P z ∈ {0, 1}p×p

Our Formulation of the OWAP

F θ : V = min

• j∈P

ωjθj s.t.

• i∈P

zij = 1 j ∈ P

• j∈P

zij = 1 i ∈ P θj ≥ θj+1 j ∈ P : j < p C ix ≤ θj + M(1 −

• k≥j

zik) i, j ∈ P x ∈ Q θj ≥ 0 j ∈ P z ∈ {0, 1}p×p

Galand-Spanjaard [Galand, Spanjaard, 2012]

x: Defines solutions in the domain Q. zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

Galand-Spanjaard [Galand, Spanjaard, 2012]

x: Defines solutions in the domain Q. zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

θj yij Value of objective placed Value of objective i if in position j it occupies position j

Galand-Spanjaard [Galand, Spanjaard, 2012]

x: Defines solutions in the domain Q. zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

θj yij Value of objective placed Value of objective i if in position j it occupies position j θj =

• i∈P

yij

Galand-Spanjaard [Galand, Spanjaard, 2012]

x: Defines solutions in the domain Q. zij =

• 1

if cost function i occupies position j in the ordering,

• therwise.

θj yij Value of objective placed Value of objective i if in position j it occupies position j θj =

• i∈P

yij yij = zijC ix

Galand-Spanjaard [Galand, Spanjaard, 2012]

F GS : V = min

• j∈P

ωj

• i∈P

yij s.t.

• i∈P

zij = 1 j ∈ P

• j∈P

zij = 1 i ∈ P

• i∈P

yij ≥

• i∈P

yij+1 j ∈ P : j < p yij ≤ Mzij i, j ∈ P

• j∈P

yij = C ix i ∈ P x ∈ Q yij ≥ 0 i, j ∈ P z ∈ {0, 1}p×p

Comparison of Formulations

ΩGS

LP ⊂ Ωθ LP

but ...

Comparison of Formulations

ΩGS

LP ⊂ Ωθ LP

but ... ... but F θ has O(p) variables θ whereas F GS has O(p2) variables y

Formulations for the Spanning Tree

Constraints Vars Const. Int. Subtour Subtour [Edmonds 70] (Di‐ cut) [MW 95] x((S))|S|‐1 O(|E|) Exp(n) Y [MW 95] Flow [Gavish 83] n‐1, u=k (+(u))‐(‐(u))= ‐1 uk k fixed O(|E|) O(|E|) N ‐1 uk Miller‐Tucker‐ Zemlin [60] ujui‐(n‐1)(1‐xij) O(|E|) O(|E|) N Kipp Martin [Martin 91] 0, u=k qk(+(u))  1, uk For all k O(n.|E|) O(n.|E|) Y Kipp Martin Relax. 0, u=k q (+(u))  1, uk k fixed O(|E|) O(|E|) N Kipp Martin Relax. 0, u=k q (+(u))  1, uk k f d O(|E|) Exp(n) Y Relax. Reinforced 1, uk + cut constraints fixed O(|E|) Exp(n) Y

Comparison of ST formulations

P(T sub) = P(T km

LP ) = P(T dc LP ) ⊂

• P(T mtz

LP )

= P(T flow

LP )

Comparison of ST formulations when embed in F θ

Pθ(Ωsub

LP ) = Pθ(Ωkm LP) = Pθ(Ωdc LP) ⊂

• Pθ(Ωmtz

LP )

= Pθ(Ωflow

LP )

Computational experiments

◮ Find out empirically the best ST formulation for F θ ◮ Compare it empirically with F GS

(where the flow formulation is used for the ST)

Computational experiments

◮ Find out empirically the best ST formulation for F θ

The KP-RR formulation outperformed the other ones

◮ Compare it empirically with F GS

(where the flow formulation is used for the ST)

Computational experiments

◮ Find out empirically the best ST formulation for F θ

The KP-RR formulation outperformed the other ones

◮ Compare it empirically with F GS

(where the flow formulation is used for the ST) Formulation F θ with KP-RR outperformed F GS

Computational experiments

◮ Complete graphs with|V | ∈ {40, 50, 60} ◮ Cost vectors randomly drawn from U[1, 100]. ◮ |P| ∈ {5, 8, 10} ◮ 10 instances, for each combination of parameters (|V |, p, α)

Computational experiments

◮ Complete graphs with|V | ∈ {40, 50, 60} ◮ Cost vectors randomly drawn from U[1, 100]. ◮ |P| ∈ {5, 8, 10} ◮ 10 instances, for each combination of parameters (|V |, p, α) ◮ Hurwicz criterion [Hurwicz(1951)]: α maxi∈P yi + (1 − α) mini∈P yi

(ω1 = α, ωp = 1 − α, ωi = 0, i = 1, p.)

• Non-monotonic and non-convex
• Considered by other authors [Galand & Spanjaard 2012]

◮ α ∈ {0.4, 0.6, 0.8}.

Computational experiments

◮ Complete graphs with|V | ∈ {40, 50, 60} ◮ Cost vectors randomly drawn from U[1, 100]. ◮ |P| ∈ {5, 8, 10} ◮ 10 instances, for each combination of parameters (|V |, p, α) ◮ Hurwicz criterion [Hurwicz(1951)]: α maxi∈P yi + (1 − α) mini∈P yi

(ω1 = α, ωp = 1 − α, ωi = 0, i = 1, p.)

• Non-monotonic and non-convex
• Considered by other authors [Galand & Spanjaard 2012]

◮ α ∈ {0.4, 0.6, 0.8}. ◮ 180 instances in total.

Computational experiments

◮ Complete graphs with|V | ∈ {40, 50, 60} ◮ Cost vectors randomly drawn from U[1, 100]. ◮ |P| ∈ {5, 8, 10} ◮ 10 instances, for each combination of parameters (|V |, p, α) ◮ Hurwicz criterion [Hurwicz(1951)]: α maxi∈P yi + (1 − α) mini∈P yi

(ω1 = α, ωp = 1 − α, ωi = 0, i = 1, p.)

• Non-monotonic and non-convex
• Considered by other authors [Galand & Spanjaard 2012]

◮ α ∈ {0.4, 0.6, 0.8}. ◮ 180 instances in total. ◮ MIP Xpress 7.5 optimizer, under a Windows 7 environment in an

Intel(R) Core(TM)i7 CPU 2.93 GHz processor and 8 GB RAM.

◮ Callbacks with separation of subtours with Gomory-Hu tree

Computational experiments

◮ Complete graphs with|V | ∈ {40, 50, 60} ◮ Cost vectors randomly drawn from U[1, 100]. ◮ |P| ∈ {5, 8, 10} ◮ 10 instances, for each combination of parameters (|V |, p, α) ◮ Hurwicz criterion [Hurwicz(1951)]: α maxi∈P yi + (1 − α) mini∈P yi

(ω1 = α, ωp = 1 − α, ωi = 0, i = 1, p.)

• Non-monotonic and non-convex
• Considered by other authors [Galand & Spanjaard 2012]

◮ α ∈ {0.4, 0.6, 0.8}. ◮ 180 instances in total. ◮ MIP Xpress 7.5 optimizer, under a Windows 7 environment in an

Intel(R) Core(TM)i7 CPU 2.93 GHz processor and 8 GB RAM.

◮ Callbacks with separation of subtours with Gomory-Hu tree ◮ CPU time limit of 3600 seconds.

Computational experiments

Computational experiments

GS KP-RR |P| |V | α t∗ t t∗ t∗ t t∗ 5 20 0.4 0.3 1.1 2.3 1.1 1.6 3.6 5 20 0.6 0.6 1.7 2.8 1 1.4 2.4 5 20 0.8 0.5 1.5 3.1 0.9 1.2 1.8 5 30 0.4 2.2 4.6 10.1 2.8 5.8 16.1 5 30 0.6 1.6 9.1 43.8 2.4 14.4 72.9 5 30 0.8 0.5 18.4 90.5 1.9 15.5 75 5 40 0.4 6.6 18.1 45.3 5.4 12.1 40.1 5 40 0.6 7.7 46.1 155.6 5.1 12.2 33 5 40 0.8 4.1 51.7 226 4.2 8.7 17.2 5 50 0.4 21.5 124.6 335.8 8.3 273.4 2564.3 5 50 0.6 26 368 2394.3 7.6 42.3 267.4 5 50 0.8 14.5 225.3 1978.9 10 21.4 63.1 5 60 0.4 40.5 461.1 4092.9 18.8 37.4 116.1 5 60 0.6

• 12.7

270.8 2019.6 5 60 0.8 2.6 1586.2 29575.1 18.7 529.3 3253

Averages over 30 instances for each combination (|P|, |V |,α)

• : average execution time > 15min (1800s).

Computational experiments KP-RR

|P| |V | α t(#) t∗/gap∗ gapLR nodes 8 40 0.4 36.3 103.4 38.59 12104 8 40 0.6 23.6 45.2 29.25 8565 8 40 0.8 41.4 78.4 18.58 19360 8 50 0.4 89.7 271.5 39.6 27126 8 50 0.6 438.2 (9) 0.4% 30.34 129181 8 50 0.8 147.4 370.9 18.97 52554 8 60 0.4 249 942.9 38.44 65411 8 60 0.6 351.3 853.8 29.54 97160 8 60 0.8 489 1090.4 18.74 160118 8 80 0.4 2176.4 (5) 1.02% 38.59 152486 8 80 0.6 2355.2 (5) 0.5% 29.65 316561 8 80 0.8 2010.7 (8) 0.98% 18.85 299480 8 100 0.4 2903.9 (4) 1.11% 38.82 119588 8 100 0.6 2998.1 (3) 0.58% 29.73 153248 8 100 0.8 3396.8 (1) 0.93% 19 232543 10 40 0.4 263.8 837.5 35.82 128102 10 40 0.6 986.5 (8) 1% 27.25 435060 10 40 0.8 1054.3 (9) 1.34% 17.55 589547 10 50 0.4 1679.6 (6) 1.12% 36.01 494558 10 60 0.4 2604.4 (4) 0.85% 27.53 863090 10 60 0.6 2798.2 (4) 0.9% 18.08 1097300 10 60 0.8 2135.2 (7) 0.44% 35.72 600445 10 80 0.4 2625.9 (7) 0.58% 27.17 934221 10 80 0.6 3514.7 (1) 0.54% 17.74 996344 10 80 0.8

• (0)

0.91% 34.99 467169 10 100 0.4

• (0)

0.47% 27.03 458488 10 100 0.6

• (0)

0.9 % 17.72 492075 10 100 0.8

• (0)

0.84% 35.01 222124

To conclude

◮ OWA generalizes several aggregating operators for multiobjective

• ptimization

◮ We have studied the OWAP with linear objective functions when

feasible solutions are spanning trees

◮ The spanning tree OWAP is a MINLP ◮ Several modeling alternatives have been considered for both the

OWAP and the combinatorial object

◮ Our best formulations outperform existing ones ◮ We can solve problems with up to 10 objective functions and

(nearly) 100 nodes in one hour of computing time

Thank you for your attention!!!