fadsdfasadfs Ordered Weighted Average Optimization in - PowerPoint PPT Presentation

fadsdfasadfs Ordered Weighted Average Optimization in Multiobjective Spanning Tree Problems andez 1 Elena Fern´ Miguel Pozo 2 Justo Puerto 2 1 Universitat Polit` ecnica de Catalunya - BarcelonaTech 2 Universidad de Sevilla

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

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

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

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • f i : Q → R , i ∈ P = { 1 , . . . , p } objective functions. • ω ∈ R p + : non-negative weights. • y = f ( x ) ∈ R p , with x ∈ Q . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ f σ i ( x ) ≥ f σ i +1 ( x )). • Ordered Weighted Average Operator (OWA): OWA ( f ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( f ,ω ) ( x )

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • C i : Q → R , i ∈ P = { 1 , . . . , p } f i ( x ) = C i x , C i ∈ R n linear • ω ∈ R p + : non-negative weights. • y = f ( x ) ∈ R p , with x ∈ Q . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ f σ i ( x ) ≥ f σ i +1 ( x )). • Ordered Weighted Average Operator (OWA): OWA ( f ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( f ,ω ) ( x )

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • C i : Q → R , i ∈ P = { 1 , . . . , p } f i ( x ) = C i x , C i ∈ R n linear • ω ∈ R p + : non-negative weights. • y = Cx , with x ∈ Q , C ∈ R p × n . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ f σ i ( x ) ≥ f σ i +1 ( x )). • Ordered Weighted Average Operator (OWA): OWA ( f ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( f ,ω ) ( x )

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • C i : Q → R , i ∈ P = { 1 , . . . , p } f i ( x ) = C i x , C i ∈ R n linear • ω ∈ R p + : non-negative weights. • y = Cx , with x ∈ Q , C ∈ R p × n . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ C σ i x ≥ C σ i +1 x ). • Ordered Weighted Average Operator (OWA): OWA ( f ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( f ,ω ) ( x )

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • C i : Q → R , i ∈ P = { 1 , . . . , p } f i ( x ) = C i x , C i ∈ R n linear • ω ∈ R p + : non-negative weights. • y = Cx , with x ∈ Q , C ∈ R p × n . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ C σ i x ≥ C σ i +1 x ). • Ordered Weighted Average Operator (OWA): OWA ( C ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( f ,ω ) ( x )

The Ordered Weighted Average Operator (OWA) • Feasible domain Q ⊆ R n ← − Combinatorial Object • C i : Q → R , i ∈ P = { 1 , . . . , p } f i ( x ) = C i x , C i ∈ R n linear • ω ∈ R p + : non-negative weights. • y = Cx , with x ∈ Q , C ∈ R p × n . • σ : y σ 1 ≥ . . . ≥ y σ p ( ⇔ C σ i x ≥ C σ i +1 x ). • Ordered Weighted Average Operator (OWA): OWA ( C ,ω ) ( x ) = ω ′ y σ • OWA optimization Problem (OWAP): min x ∈ Q OWA ( C ,ω ) ( x )

Ordered Median Operator Q ⊆ R n , d ∈ R n cost vector, ω ∈ R n weights. For x ∈ Q , σ : permutation s.t. d σ j x σ j ≥ d σ j +1 x σ j +1 � OM ( d ,ω ) ( x ) = ω j d σ j x σ j j ∈ P

Ordered Median Operator Q ⊆ R n , d ∈ R n cost vector, ω ∈ R n weights. For x ∈ Q , σ : permutation s.t. d σ j x σ j ≥ d σ j +1 x σ j +1 � OM ( d ,ω ) ( x ) = ω j d σ j x σ j j ∈ P ( C i ) ′ = d i e i , i ∈ { 1 , . . . , n } . C = Diag ( d ) OM ( d ,ω ) ( x ) = OWA ( Diag ( d ) ,ω ) ( x )

Ordered Median Operator Q ⊆ R n , d ∈ R n cost vector, ω ∈ R n weights. For x ∈ Q , σ : permutation s.t. d σ j x σ j ≥ d σ j +1 x σ j +1 � OM ( d ,ω ) ( x ) = ω j d σ j x σ j j ∈ P ( C i ) ′ = d i e i , 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 fadsafds ω ′ = (0 . 8 , 0 , 0 . 2) (3 , 4 , 1) 2 3 (1 , 2 , 1) (1 , 3 , 4) (4 , 1 , 2) 1 (1 , 2 , 1) (3 , 3 , 2) 4 (2 , 3 , 4) (4 , 2 , 1) 6 5 (1 , 2 , 3)

Example 2 3 fadsafds ω ′ = (0 . 8 , 0 , 0 . 2) 1 4 (3 , 4 , 1) 6 5 2 3 (1 , 2 , 1) (1 , 3 , 4) (4 , 1 , 2) 1 (1 , 2 , 1) (3 , 3 , 2) 4 fadsafds Cx = (10 , 11 , 8) fadsafdabs σ = (2 , 1 , 3) (2 , 3 , 4) (4 , 2 , 1) 6 5 fadsafdsValue (1 , 2 , 3) 0 . 8( C 2 x ) + 0( C 1 x ) + 0 . 2( C 3 x ) = = 0 . 8 × 11 + 0 . 2 × 8 Value = 10 . 4

Our Formulation of the OWAP x : Defines solutions in the domain Q . ( y i = C i x : Value of objective function i ) F., Pozo, Puerto (2014)

Our Formulation of the OWAP x : Defines solutions in the domain Q . ( y i = C i x : Value of objective function i ) � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise. F., Pozo, Puerto (2014)

Our Formulation of the OWAP x : Defines solutions in the domain Q . ( y i = C i x : Value of objective function i ) � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise. θ j : Value of the objective occupying position j in the ordering. F., Pozo, Puerto (2014)

Our Formulation of the OWAP F θ : � V = min ω j θ j j ∈ P � z ij = 1 j ∈ P s . t . i ∈ P � z ij = 1 i ∈ P j ∈ P θ j ≥ θ j +1 j ∈ P : j < p � z ij ( C i x ) θ j = i , j ∈ P i ∈ P x ∈ Q θ j ≥ 0 j ∈ P z ∈ { 0 , 1 } p × p

Our Formulation of the OWAP F θ : � V = min ω j θ j j ∈ P � s . t . z ij = 1 j ∈ P i ∈ P � z ij = 1 i ∈ P j ∈ P θ j ≥ θ j +1 j ∈ P : j < p C i x ≤ θ j + M (1 − z ij ) i , j ∈ P x ∈ Q θ j ≥ 0 j ∈ P z ∈ { 0 , 1 } p × p

Our Formulation of the OWAP F θ : � V = min ω j θ j j ∈ P � s . t . z ij = 1 j ∈ P i ∈ P � z ij = 1 i ∈ P j ∈ P θ j ≥ θ j +1 j ∈ P : j < p � C i x ≤ θ j + M (1 − z ik ) i , j ∈ P k ≥ j x ∈ Q θ j ≥ 0 j ∈ P z ∈ { 0 , 1 } p × p

Galand-Spanjaard [Galand, Spanjaard, 2012] x : Defines solutions in the domain Q . � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise.

Galand-Spanjaard [Galand, Spanjaard, 2012] x : Defines solutions in the domain Q . � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise. y ij θ j 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 . � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise. y ij θ j Value of objective placed Value of objective i if in position j it occupies position j � θ j = y ij i ∈ P

Galand-Spanjaard [Galand, Spanjaard, 2012] x : Defines solutions in the domain Q . � 1 if cost function i occupies position j in the ordering, z ij = 0 otherwise. y ij θ j Value of objective placed Value of objective i if in position j it occupies position j � θ j = y ij i ∈ P y ij = z ij C i x

Galand-Spanjaard [Galand, Spanjaard, 2012] F GS : � � V = min ω j y ij j ∈ P i ∈ P � z ij = 1 j ∈ P s . t . i ∈ P � z ij = 1 i ∈ P j ∈ P � � y ij ≥ y ij +1 j ∈ P : j < p i ∈ P i ∈ P y ij ≤ Mz ij i , j ∈ P � y ij = C i x i ∈ P j ∈ P x ∈ Q y ij ≥ 0 i , j ∈ P z ∈ { 0 , 1 } p × p

Comparison of Formulations Ω GS LP ⊂ Ω θ LP but ...

Comparison of Formulations Ω GS LP ⊂ Ω θ LP but ... ... but F θ has O ( p ) variables θ whereas F GS has O ( p 2 ) variables y