UTA-splines and UTADIS-splines Olivier Sobrie 1 , 2 - Nicolas Gillis - PowerPoint PPT Presentation

UTA-splines and UTADIS-splines Olivier Sobrie 1 , 2 - Nicolas Gillis 2 - Vincent Mousseau 1 - Marc Pirlot 2 1 École Centrale de Paris - Laboratoire de Génie Industriel 2 University of Mons - Faculty of engineering July 5, 2016 University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 1 / 28

1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 2 / 28

Additive value function model 1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 3 / 28

Additive value function model Additive value function model I Principle ◮ A score is computed for each alternative ◮ The score is used to rank or to sort alternatives University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 4 / 28

Additive value function model Additive value function model II ◮ A marginal value function u j is associated to each criterion j ◮ Marginal value functions are monotonic ◮ Marginal value functions are normalized between 0 and 1, s.t. u j ( a j ) = 0 and u j ( a j ) = 1 ◮ A weight w j is associated to each criterion j , s.t. � n j = 1 w j = 1 u j 1 ◮ Utility of an alternative a : u j ( a j ) n � U ( a ) = w j · u j ( a j ) j = 1 0 a j j + a j a j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 5 / 28

Additive value function model Additive value function model II ◮ A marginal value function u j is associated to each criterion j ◮ Marginal value functions are monotonic ◮ Marginal value functions are normalized between 0 and 1, s.t. u ∗ j ( a j ) = 0 and u ∗ j ( a j ) = w j ◮ A weight w j is associated to each criterion j , s.t. � n j = 1 w j = 1 u ∗ j w j ◮ We also have : u ∗ j ( a ) = w j · u j ( a j ) and u ∗ j ( a j ) = w j ◮ Utility of an alternative a : u ∗ j ( a j ) n n � � u ∗ U ( a ) = w j · u j ( a j ) = j ( a j ) j = 1 j = 1 0 a j j + a j a j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 5 / 28

Additive value function model Example price rating dist. beach dist. center size 0 . 2 0 . 2 0 . 2 0 . 2 0 . 2 j ( a j ) 0 . 1 0 . 1 0 . 1 0 . 1 0 . 1 u ∗ 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 600 0 800 0 200 5 45 1 5 m m ⋆ e m 2 Ranking Plaza Miramar Hilton Hotel W ≻ ≻ ≻ 0.53 0.51 0.43 0.41 Sorting 0.5 Good Bad Plaza Miramar Hilton Hotel W 0.53 0.51 0.43 0.41 University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 6 / 28

Learning an AVF model 1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 7 / 28

Learning an AVF model Existing methods for learning an AVF model ◮ UTA : LP for learning the parameters of an AVF-ranking model ◮ UTADIS : LP for learning the parameters of an AVF-sorting model ◮ Other methods : UTA * , ACUTA , . . . ◮ Monotonicity of the marginals is ensured ◮ Marginals are modeled with piecewise linear functions u ∗ j w j + u ∗ 3 + j u ∗ 2 + j u ∗ j ( a j ) u ∗ 1 + j 0 a j j + + + + + a j a j g 1 g 2 g 3 j j j University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 8 / 28

UTA(DIS)-poly 1 Additive value function model 2 Learning an AVF model 3 UTA(DIS)-poly 4 UTA(DIS)-splines 5 Experiments 6 Conclusion and further research University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 9 / 28

UTA(DIS)-poly UTA(DIS)-poly I Principle ◮ Use of polynomials for the marginal value functions u ∗ u ∗ j j w j w j + u ∗ 3 + j u ∗ 2 ⇒ + j u ∗ j ( a j ) u ∗ j ( a j ) u ∗ 1 + j 0 a j j + + + + + 0 a j j a j + g 1 g 2 g 3 a j a j a j j j j Motivations ◮ Improve the flexibility of the model ◮ Improve the interpretability of the model University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 10 / 28

UTA(DIS)-poly UTA(DIS)-poly II ◮ Use of semi-definite programming (SDP) ◮ Based on interior point methods ◮ Possibility to impose the nonnegativity of a symmetric matrix   q 1 , 1 q 1 , 2 q 1 , 3 · · · q 1 , n q 2 , 2 q 2 , 3 · · · q 2 , n     q 3 , 3 · · · q 3 , n Q = ≥ 0    .  ... .   .   (symmetric) q n , n ◮ Monotonicity of the marginals guaranteed ◮ Ensured by imposing the nonnegativity of the derivative ◮ Use of Hilbert’s theorems University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 11 / 28

UTA(DIS)-poly UTA(DIS)-poly III Theorem (Hilbert) A polynomial F : R n → R is nonnegative if it is possible to decompose it as a sum of squares (SOS) : � f 2 with z ∈ R n . F ( z ) = s ( z ) s Theorem (Hilbert) A non-negative polynomial in one variable is always a SOS. Theorem (Hilbert) A polynomial p ( x ) in one variable x is non-negative in the interval [ v 1 , v 2 ] , if and only if p ( x ) = ( x − v 1 ) · q ( x ) + ( v 2 − x ) · r ( x ) where q ( x ) and r ( x ) are SOS. University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 12 / 28

UTA(DIS)-poly UTA-poly - Example I x y a 1 10 7 a 1 ≻ a 2 ≻ a 3 a 2 6 8 a 3 7 5 ◮ We define u ∗ 1 ( x ) and u ∗ 2 ( y ) as third degree polynomials : 1 ( x ) = p x , 0 + p x , 1 · x + p x , 2 · x 2 + p x , 3 · x 3 , u ∗ 2 ( y ) = p y , 0 + p y , 1 · y + p y , 2 · y 2 + p y , 3 · y 3 . u ∗ ◮ Scores of a 1 , a 2 and a 3 are given by : U ( a 1 ) = p x , 0 + 10 p x , 1 + 100 p x , 2 + 1000 p x , 3 + p y , 0 + 7 p y , 1 + 49 p y , 2 + 343 p y , 3 , U ( a 2 ) = p x , 0 + 6 p x , 1 + 36 p x , 2 + 324 p x , 3 + p y , 0 + 8 p y , 1 + 64 p y , 2 + 512 p y , 3 , U ( a 3 ) = p x , 0 + 7 p x , 1 + 49 p x , 2 + 343 p x , 3 + p y , 0 + 5 p y , 1 + 25 p y , 2 + 125 p y , 3 . University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 13 / 28

UTA(DIS)-poly UTA-poly - Example II ◮ Scores of a 1 , a 2 and a 3 are given by : U ( a 1 ) = p x , 0 + 10 p x , 1 + 100 p x , 2 + 1000 p x , 3 + p y , 0 + 7 p y , 1 + 49 p y , 2 + 343 p y , 3 , U ( a 2 ) = p x , 0 + 6 p x , 1 + 36 p x , 2 + 324 p x , 3 + p y , 0 + 8 p y , 1 + 64 p y , 2 + 512 p y , 3 , U ( a 3 ) = p x , 0 + 7 p x , 1 + 49 p x , 2 + 343 p x , 3 + p y , 0 + 5 p y , 1 + 25 p y , 2 + 125 p y , 3 . ◮ We have a 1 ≻ a 2 and a 2 ≻ a 3 , which implies : � U ( a 1 ) − U ( a 2 ) + σ + ( a 1 ) − σ − ( a 1 ) − σ + ( a 2 ) + σ − ( a 2 ) > 0 , U ( a 2 ) − U ( a 3 ) + σ + ( a 2 ) − σ − ( a 2 ) − σ + ( a 1 ) + σ − ( a 1 ) > 0 . ◮ By replacing U ( a 1 ) , U ( a 2 ) and U ( a 3 ) , we have :  4 p x , 1 + 64 p x , 2 + 776 p x , 3 − p y , 1 − 15 p y , 2 − 231 p y , 3 + σ + ( a 1 ) − σ − ( a 1 )   − σ + ( a 2 ) + σ − ( a 2 ) > 0 ,  − p x , 1 − 13 p x , 2 − 19 p x , 3 + 3 p y , 1 + 39 p y , 2 + 387 p y , 3 + σ + ( a 2 ) − σ − ( a 2 )   − σ + ( a 3 ) + σ − ( a 3 ) > 0 .  University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 14 / 28

UTA(DIS)-poly UTA-poly - Example III ◮ We impose the derivative of u ∗ 1 and u ∗ 2 to be SOS : 1 = x T Qx u ∗ ′ � T � q 0 , 0 � 1 � � 1 � q 0 , 1 = x q 1 , 0 q 1 , 1 x = q 0 , 0 + ( q 0 , 1 + q 0 , 1 ) x + q 1 , 1 x 2 , 2 = y T Ry u ∗ ′ = r 0 , 0 + ( r 0 , 1 + r 1 , 0 ) y + r 1 , 1 y 2 . ◮ Q and R have to be semi-definite positive , in conjunction with :   p x , 1 = q 0 , 0 , p y , 1 = r 0 , 0 ,     2 p x , 2 = q 0 , 1 + q 1 , 0 , and 2 p y , 2 = r 0 , 1 + r 1 , 0 ,   3 p x , 3 = q 1 , 1 , 3 p y , 3 = r 1 , 1 .   University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 15 / 28

UTA(DIS)-poly UTA-poly - Example IV ◮ We add normalization constraints :  = 0 , p x , 0  p y , 0 = 0 , 10 p x , 1 + 100 p x , 2 + 1000 p x , 3 + 10 p y , 1 + 100 p y , 2 + 1000 p y , 3 = 1 .  University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 16 / 28

UTA(DIS)-poly UTA-poly - Example V min σ + ( a 1 ) + σ − ( a 1 ) + σ + ( a 2 ) + σ − ( a 2 ) + σ + ( a 3 ) + σ − ( a 3 ) . such that :  4 p x , 1 + 64 p x , 2 + 776 p x , 3 − p y , 1 − 15 p y , 2 − 231 p y , 3  + σ + ( a 1 ) − σ − ( a 1 ) − σ + ( a 2 ) + σ − ( a 2 )  > 0 ,     − p x , 1 − 13 p x , 2 − 19 p x , 3 + 3 p y , 1 + 39 p y , 2 + 387 p y , 3     + σ + ( a 2 ) − σ − ( a 2 ) − σ + ( a 3 ) + σ − ( a 3 ) > 0 ,     p x , 0 = 0 ,     p y , 0 = 0 ,    10 p x , 1 + 100 p x , 2 + 1000 p x , 3 + 10 p y , 1 + 100 p y , 2 + 1000 p y , 3 = 1 , p x , 1 = q 0 , 0 ,     2 p x , 2 = q 0 , 1 + q 1 , 0 ,      3 p x , 3 = q 1 , 1 ,     p y , 1 = r 0 , 0 ,     2 p y , 2 = r 0 , 1 + r 1 , 0 ,    =  3 p y , 3 r 1 , 1 , with : � Q , R PSD , σ + ( a 1 ) , σ − ( a 1 ) , σ + ( a 2 ) , σ − ( a 2 ) , σ + ( a 3 ) , σ − ( a 3 ) 0 . ≥ University of Mons O. Sobrie - N. Gillis - V. Mousseau - M. Pirlot - July 5, 2016 17 / 28