Efficiency for continuous facility location problems A. Jourani Efficiency for continuous facility location problems with attraction and repulsion ∗ Abderrahim Jourani Universit´ e de Dijon Institut de Math´ ematiques de Bourgogne UMR 5584-CNRS jourani@u-bourgogne.fr Journ´ ees Franco-Chiliennes d’Optimisation Toulon, may 2008 ∗ collaboration with C. Michelot and M. Ndiaye ∗ to appear in AOR ∗ http ://math.u-bourgogne.fr/IMB/IMB.html Universit´ e de Dijon France Page 1
Efficiency for continuous facility location problems A. Jourani • ( X, � · � ) normed linear space • a + i ∈ X , ω + i > 0 ∀ i = 1 , · · · , m • a − i ∈ X, ω − i > 0 ∀ i = 1 , · · · , n � ω + i � x − a + i � ( P + ) min x � � ω + i � x − a + i � − i � x − a − i � ( P + − ) min ω − x Universit´ e de Dijon France Page 2
Efficiency for continuous facility location problems A. Jourani M ω set of solutions to ( P + ) ◮ X inner product space ⇒ M ω ⊂ co { a + i : i = 1 , · · · , p } ⇒ M ω ∩ co { a + i : i = 1 , · · · , p } � = ∅ ◮ dim X = 2 = Can we replace co { a + i : i = 1 , · · · , p } ? What happens in the case of ( P + − ) ? Universit´ e de Dijon France Page 3
Efficiency for continuous facility location problems A. Jourani Vector Optimization R p mapping, D ⊂ Y F : Y �→ I ( P ) min y ∈ D F ( y ) • y 0 weakly efficient if R p ∄ y ∈ D ; F ( y ) − F ( y 0 ) ∈ intI − • y 0 strictly efficient if R p ∄ y ∈ D, y � = y 0 ; F ( y ) − F ( y 0 ) ∈ I − • y 0 efficient if R p ∄ y ∈ D ; F ( y ) − F ( y 0 ) ∈ I F ( y ) � = F ( y 0 ) − , Universit´ e de Dijon France Page 4
Efficiency for continuous facility location problems A. Jourani Basic elements • X euclidean • Ω ⊂ X convex and closed • A + ⊂ X, A − ⊂ X compact, A + ∩ A − = ∅ • x ∈ X Universit´ e de Dijon France Page 5
Efficiency for continuous facility location problems A. Jourani Weak efficiency : x ∈ WE ( A + , A − , Ω) ∃ a + ∈ A + , � a + − x � ≤ � a + − y � ∀ y ∈ Ω , ou ∃ a − ∈ A − , � a − − x � ≥ � a − − y � Strict efficiency : x ∈ SE ( A + , A − , Ω) ∃ a + ∈ A + , � a + − x � < � a + − y � ∀ y ∈ Ω , y � = x, ou ∃ a − ∈ A − , � a − − x � > � a − − y � Efficiency : x ∈ E ( A + , A − , Ω) ∃ a + ∈ A + , � a + − x � < � a + − y � ou ∃ a − ∈ A − , � a − − x � > � a − − y � ∀ y ∈ Ω , y � = x, ou ∀ a + ∈ A + , � a + − x � ≤ � a + − y � et ∀ a − ∈ A − , � a − − x � ≥ � a − − y � Universit´ e de Dijon France Page 6
Efficiency for continuous facility location problems A. Jourani NSOC ◮ (Carrizosa-Plastria) A − = ∅ . x weakly efficient ⇐ ⇒ � ∂ ( � · − a + � )( x )] + N (Ω , x ) 0 ∈ co[ a + ∈ A + A + = { a + n } , A − = { a − 1 , · · · , a + 1 , · · · , a − m } ◮ If x is locally weakly efficient, then n m ∃ λ + ≥ 0 , ∃ λ − ≥ 0 , � � λ + λ − i + j = 1 i =1 j =1 [ � m j =1 λ − j ∂ ( � · − a − j � )( x )] � [ � n i =1 λ + i ∂ ( � · − a + i � )( x ) + N (Ω , x )] � = ∅ The last one is also sufficient provided that the norm and Ω are locally polyhedral. Universit´ e de Dijon France Page 7
Efficiency for continuous facility location problems A. Jourani Notations. • Ω = X : SE ( A + , A − ), E ( A + , A − ), WE ( A + , A − ) • SE ( A + ) = SE ( A + , ∅ ) • E ( A + ) = E ( A + , ∅ ) • WE ( A + ) = WE ( A + , ∅ ) Properties ◮ SE ( A + ) ⊂ SE ( A + , A − ), E ( A + ) ⊂ E ( A + , A − ), WE ( A + ) ⊂ WE ( A + , A − ) ◮ A + ∩ Ω ⊂ SE ( A + , A − , Ω) ⊂ E ( A + , A − , Ω) ⊂ WE ( A + , A − , Ω) ◮ SE ( A + , A − ) ∩ Ω ⊂ SE ( A + , A − , Ω) E ( A + , A − ) ∩ Ω ⊂ E ( A + , A − , Ω) WE ( A + , A − ) ∩ Ω ⊂ WE ( A + , A − , Ω) Universit´ e de Dijon France Page 8
Efficiency for continuous facility location problems A. Jourani Weak efficiency Theorem 1. x ∈ WE ( A + , A − ) ⇐ ⇒ co( A + ) ∩ co( { x } ∪ A − ) � = ∅ . Corollary 1. ⇒ A − = ∅ WE ( A + , A − ) compact ⇐ � WE ( A + , A − ) = co( A + ) Proposition 1. co( A + ) ∩ co( A − ) � = ∅ ⇐ ⇒ WE ( A + , A − ) = X Universit´ e de Dijon France Page 9
Efficiency for continuous facility location problems A. Jourani Strict efficiency Theorem 2. • co( A + ) ∩ ri[co( { x } ∪ A − )] � = ∅ = ⇒ x ∈ SE ( A + , A − ) • SE ( A + , A − ) = co( A + ) + cl[cone[co( A + ) − co( A − )]] Corollary 2. • SE ( A + , A − ) closed and convex • co( A + ) et co( A − ) polyhedral : SE ( A + , A − ) = co( A + ) + cone[co( A + ) − co( A − )] Corollary 3. • co( A + ) ∩ co( A − ) � = ∅ = ⇒ co( A − ) ⊂ SE ( A + , A − ) Universit´ e de Dijon France Page 10
Efficiency for continuous facility location problems A. Jourani Theorem 2. x ∈ ri[ SE ( A + , A − )] ⇔ ri[co( A + )] ∩ ri[co( { x }∪ A − )] � = ∅ . Universit´ e de Dijon France Page 11
Efficiency for continuous facility location problems A. Jourani Efficiency Proposition 2. ri[co( A + )] ∩ ri[co( A − )] � = ∅ ⇐ ⇒ E ( A + , A − ) = X Theorem 3. • A + et A − not contained in the same hyperplan : E ( A + , A − ) = SE ( A + , A − ) • A + et A − contained in the same hyperplan H : ri[co( A + )] ∩ ri[co( A − )] = ∅ ⇐ ⇒ E ( A + , A − ) ⊂ H ri[co( A + )] ∩ ri[co( A − )] = ∅ • � E ( A + , A − ) = SE ( A + , A − ) � = X Universit´ e de Dijon France Page 12
Efficiency for continuous facility location problems A. Jourani Coincidence Theorem 5. K = co( A + ) + cl[cone[co( A + ) − co( A − )]] E ( A + , A − ) = WE ( A + , A − ) • ou E ( A + , A − ) = SE ( A + , A − ) co( A + ) ∩ co( A − ) = ∅ • ⇓ SE ( A + , A − ) = E ( A + , A − ) = WE ( A + , A − ) = K Corollary 4. E ( A + , A − ) et WE ( A + , A − ) closed and convex Universit´ e de Dijon France Page 13
Efficiency for continuous facility location problems A. Jourani Constrained efficiency Proj Ω WE ( A + , A − ) ⊂ WE ( A + , A − , Ω) Inner product spaces ( X, � · � ) linear normed space dim X ≥ 3. Theorem 6. i ) X inner product space � ii ) ∀ A + , A − ⊂ X with A + ∩ A − = ∅ , card A + < + ∞ , card A − < + ∞ , we have x ∈ WE ( A + , A − ) ⇐ ⇒ co( A + ) ∩ co( { x } ∪ A − ) � = ∅ . Universit´ e de Dijon France Page 14
Efficiency for continuous facility location problems A. Jourani Complexity Theorem 7. A + , A − ⊂ I R 2 , | A + | = n , | A − | = m , co A + ∩ co A − = ∅ ⇓ SE ( A + , A − ) can be computed in O ( nm ) + O ( n log n ) time Universit´ e de Dijon France Page 15
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