Multivalued complementarity problems with asymptotically bounded - PowerPoint PPT Presentation

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Multivalued complementarity problems with asymptotically bounded multifunctions Fabián Flores-Bazán 1 1 Departamento de Ingeniería Matemática, Universidad de Concepción VI Journées Franco-Chiliennes d’Optimisation U. Sud Toulon-Var, 19 - 21 Mai, 2008, Toulon - France. Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Asymptotically well-behaved ... Sensitivity Results Contents Formulation 1 Examples Asymptotic bounded multifunctions 2 Asymptotic bounded multifunctions Examples Preliminaries The Basic Lemma 3 Some Notations The Basic Lemma Asymptotically well-behaved ... 4 Asymptotically well-behaved: Existence Thms. Asymptotically Regular-type mappings-Robustness property Sensitivity Results 5 Sensitivity Results Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Sensitivity Results Motivation Consider m« x ≥ 0 h ( x ) , ın ( 1 ) h differentiable on R n + . The KKT optimality condition says: if ¯ x ≥ 0 is a solution to (1), then there exists λ ≥ 0 such that ∇ h (¯ x ) − λ = 0 � λ, ¯ x � = 0 . Moreover, it is sufficient when h is pseudoconvex. By replacing ∇ h ( x ) by F ( x ) , the KKT condition leads ¯ x ≥ 0 : F (¯ x ) ≥ 0 , � F (¯ x ) , ¯ x � = 0 . Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Sensitivity Results Formulation → R n be given: Let q ∈ R n , F : R n + ֒ find ¯ x ≥ 0 , ¯ y ∈ F (¯ x ) : ¯ y + q ≥ 0 , � ¯ y + q , ¯ x � = 0 ( MCP ( q , F )) find ¯ x ≥ 0 , ¯ y ∈ F (¯ x ) : � ¯ y + q , x − ¯ x � ≥ 0 ∀ x ≥ 0 ( VIP ( q , F )) S ( q , F ) : set of solutions to ( MCP ( q , F )) . F ( x ) � = ∅ ∀ x ≥ 0; F ( x ) is convex, compact ∀ x ≥ 0; F is upper semicontinuous on R n + . Karamardian S. 1972, 1976; Moré J.J 1974; García C.B. 1973; Saigal R. 1976; Isac G. 1989,...; Gowda S.M., Pang J.S. 1992; Crouzeix 1997; .... Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Sensitivity Results References Cottle R. W., Pang J. S., Stone R. E., The linear complementarity problem, Academic Press, Boston (1992). Harker P .T., Pang J.S., Finite-dimensional variational inequality and nonlinear compl. prob.... Mathematical Prog. (1990); Ferris M.C., Pang J.S., Engineering and economic appl. of compl. prob. SIAM, Review (1997); Facchinei F ., Pang J.S., Finite-dimensional var. ineq. and comp. prob. (two volumes) Springer-Verlag (2003). Flores-Bazán F ., López R., The linear complementarity problem under asymptotic analysis, Mathematics of Operations Research (2005); Flores-Bazán F ., López R., Asymptotic analysis, existence and sensitivity results for a class of multivalued complementarity problems, ESAIM, Control Opt and Cal of Var (2006); Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Sensitivity Results Example 1. M ∈ R n × n , A , B ∈ R m × n , q ∈ R n . Set F ( x ) = { Mu : Au + Bx ≤ 0 } , x ≥ 0 . The problem is to find ( x , y ) ∈ R n × R n , u ∈ R n such that x ≥ 0 , y = Mu + q ≥ 0 , � Mu + q , x � = 0 , Au + Bx ≤ 0 . Example 2. F ( x ) = Mx + ∂ h ( x ) , M ∈ R n × n , � x , u � , h ( x ) = sup u ∈ C C � = ∅ , convex, compact. If M is symmetric, the MCP is the stationary point problem of the following minimax problem � 1 � m« x ≥ 0 sup ın 2 � x , Mx � + � x , q + u � u ∈ C Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Sensitivity Results Set R ++ = ] 0 , + ∞ [ . Let us consider C . � � = c : R ++ → R ++ : t → + ∞ c ( t ) = + ∞ l« ım c ( λ t ) C 0 . � � = c ∈ C : l« ım c ( t ) ∈ R for all λ > 0 . t → + ∞ For c ∈ C 0 , set c ( λ t ) c ∞ ( λ ) . c ( t ) ; c ∞ ( 1 ) = 1 ; c ∞ ( ts ) = c ∞ ( t ) c ∞ ( s ) , t , s > 0 . = l« ım t → + ∞ Examples c 1 ( t ) = t p ( p > 0 ) , c ∞ 1 ( t ) = t p , t p c 2 ( t ) = t p ln ( t γ + 1 ) , c 2 ( t ) = ln ( t γ + 1 ) , ( p > 0 , γ ≥ 1 ) , c ∞ 2 ( t ) = t p . Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results Asymptotic bounded multifunctions: Given a sequence of multifunctions F k : R n → R n , k ∈ N and + ֒ c ∈ C , the c -asymptotic multifunction associated to { F k } is defined by w ∈ R n : λ k m ↑ + ∞ , x k m ≥ 0 , x k m ∞ , c F k ( v ) . � l« ım sup = → v , λ k m k y k m y k m ∈ F k m ( x k m ) , � c ( λ k m ) → w . Indeed, ∞ , c F k ) = l« ∞ , c ( gph F k ) , gph ( l« ım sup ım sup k k where, Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results ∞ , c ( gph F k ) . � + × R n : λ k m ↑ + ∞ , ( v , w ) ∈ R n l« ım sup = k ( x k m , y k m ) ∈ gph F k m , ( x k m y k m � , c ( λ k m )) → ( v , w ) . λ k m c -asymptotically bounded property The property of c - asymptotically bounded at v ( c -AB) means: for every k m → + ∞ , every λ k m ↑ + ∞ , as m → + ∞ , every x k m ≥ 0, every y k m ∈ F k m ( x k m ) , such that { x k m /λ k m } converges to v , the sequence { y k m / c ( λ k m ) } is bounded. When F k = F , c ( t ) = t γ , γ > 0 it was discussed by Gowda-Pang, 92. Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results → R n and c ∈ C , we set Given F : R n + ֒ A F ( x ) . = sup | y | . y ∈ F ( x ) The above property holds iff 1 ∞ A F k , c ( v ) . � c ( λ k m ) A F km ( λ k m x k m ) : l« ım sup = sup l« ım sup m → + ∞ k � k m ↑ + ∞ , λ k m ↑ + ∞ , x k m → v < + ∞ . As before, in case F k = F for all k , we set c ( v ) . F , c ( v ) . F ∞ ∞ , c F k ( v ) , A ∞ ∞ A F k , c ( v ) . = l« ım sup = l« ım sup k k Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results More precisely, one has the following lemma. Lemma: Flores-Bazán, 2007 Let F k : R n → R n with nonempty compact values, c ∈ C and + ֒ v ≥ 0. Then, the following assertions are equivalent: ım sup ∞ ( a ) l« k A F k , c ( v ) < + ∞ ; ( b ) { F k } satisfies the c -asymptotically bounded property at v . Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results Examples: Take M , M k ∈ R n × n satisfying M k → M , and ρ ∈ R : F k ( x ) = | x | γ M k x − ρ x ( γ ≥ 0), c ( t ) = t γ + 1 = c ∞ ( t ) , then ∞ A F k , c ( v ) = | v | γ | Mv | , l« ∞ , c F k ( v ) = | v | γ Mv . l« ım sup ım sup k k F k ( x ) = ln ( | x | γ + 1 ) M k x − ρ x ( γ ≥ 1), c ( t ) = t ln ( t γ + 1 ) , then c ∞ ( t ) = t . ∞ A F k , c ( v ) = | Mv | , l« ∞ , c F k ( v ) = Mv . l« ım sup ım sup k k Take γ > 0 and ∅ � = C k ⊆ R n , closed, converging (in the sense of Painlevé-Kuratowski) to a closed ∅ � = C ⊆ R n . F k ( x ) = | x | γ C k , l« ∞ , c F k ( v ) = | v | γ C . ım sup k Flores-Bazán Asymptotically bounded multifucntions and the MCP

Formulation Asymptotic bounded multifunctions Asymptotic bounded multifunctions The Basic Lemma Examples Asymptotically well-behaved ... Preliminaries Sensitivity Results Examples: Continued ... Take F ( x ) = ln ( | x | γ + 1 ) H ( x ) or F ( x ) = 1 ln ( | x | γ + 1 ) H ( x ) ( γ ≥ 1), with H being positive homogeneous of degree p > 0, that is H ( tx ) = t p H ( x ) ∀ t > 0, x ≥ 0. Consider c ( t ) = t p ln ( t γ + 1 ) in the first case, and c ( t ) = t p ln ( t γ + 1 ) in the second case. Here c ∞ ( t ) = t p . F ( x ) = M 1 x + ln ( | x | + 1 ) M 2 x , where M i ∈ R n × n Here we use c ( t ) = t ln ( t + 1 ) . Here c ∞ ( t ) = t . Take F ( x ) = ( � M 1 x , x � , ln ( | x | + 1 ) � M 2 x , x � ) ∈ R 2 , x ≥ 0. Here c ( t ) = t 2 ln ( t + 1 ) is useful. Flores-Bazán Asymptotically bounded multifucntions and the MCP