A New Method for Solving Pareto Eigenvalues Complementarity Problems - PowerPoint PPT Presentation

A New Method for Solving Pareto Eigenvalues Complementarity Problems 1 Samir ADLY University of Limoges, France ADGO 2013 Playa Blanca, october 16, 2013 1 A joint work with H. Rammal

Complementarity problems Let F : R n → R n be a map and K ⊂ R n be a closed convex cone. The Nonlinear Complementarity Problem (NCP) is defined by  Find z ∈ K such that  NCP ( F , K ) F ( z ) ∈ K ∗ and � z , F ( z ) = 0 .  K ∗ is the positive polar of K , defined by K ∗ = � p ∈ R n : � p , x � ≥ 0 , ∀ x ∈ K � . K ∋ z ⊥ F ( z ) ∈ K ∗ . - Other formulation as a variational inequality:  Find z ∈ K such that  VI ( F , K ) � F ( z ) , y − z � ≥ 0 , ∀ y ∈ K 

Linear Complementarity Problems on the positive orthant - K = R n + and F ( z ) = Mz + q with M ∈ R n × n LCP( M , q ) : 0 ≤ z ⊥ Mz + q ≥ 0 . - Existence result: LCP( M , q ) has a unique solution for all q ∈ R n if and only M is a P-matrix, i.e. all its principal minors are positive. - Numerical solvers: ◮ Lemke’s algorithm ◮ PATH Solver ◮ Quadratic Programming with bound constraints (if M is symmetric): 1 2 z T Mz + q T z . min z ≥ 0

PATH Solver 0 ≤ x ⊥ F ( x ) ≥ 0 . - Nonsmooth Normal map (S. Robinson): Φ( x ) = F ( x + ) + x − x + = 0 . 0 ≤ x + ⊥ F ( x + ) = x + − x ≥ 0 . - Nonsmooth Newton Method applied to Φ. - Merit function: 1 2 � F ( x + ) � 2 2 with Armijo line search. - Generalization to a closed convex cone K x + → P K .

Applications of Complementarity ◮ Mechanical engineering: unilateral contact problems with friction ◮ Electrical engineering: electrical circuits with diodes ◮ Pricing electricity markets and options ◮ Electricity market deregulation ◮ Congestion in Networks ◮ Structural engineering ◮ Economic equilibria ◮ Game Theory (Nash equilibria) ◮ transportation planning ◮ Crack propoagation ◮ Video games

Unconstrained eigenvalue problems Let A , B , C ∈ R n × n be given. ◮ The standard eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that Ax = λ x ◮ The generalized eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that Ax = λ Bx ◮ The quadratic eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that Q ( λ ) x = 0 with Q ( λ ) = λ 2 A + λ B + C .

Pencil applications ◮ Mechanical systems: M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) = f . ◮ Electrical systems: Ld 2 i dt ( t ) + R di dt ( t ) + 1 C i ( t ) = u ′ ( t ) . � � � � � � � ⇒ ( M λ 2 + C λ + K ) x = 0 . q ( t ) = e λ t x =

Constrained eigenvalue problems Let A , B , C ∈ R n × n be given and K be a closed convex cone of R n . We denote by K + its positive polar. ◮ The constrained eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that K ∋ x ⊥ ( Ax − λ x ) ∈ K + ◮ The constrained generalized eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that K ∋ x ⊥ ( Ax − λ Bx ) ∈ K + ◮ The constrained quadratic eigenvalue problem is: � Find λ ∈ R and x ∈ R n \ { 0 } such that K ∋ x ⊥ Q ( λ ) x ∈ K + , with Q ( λ ) = λ 2 A + λ B + C .

Pencil applications ◮ Mechanical systems with impact and/or friction M ¨ q ( t ) + C ˙ q ( t ) + Kq ( t ) ∈ − N K ( q ( t )) . F . . x x x x Before impact After impact

Buck Converter as a piecewise-smooth system v r V ref - a v co + i L S V in v D C R � V in di − v ( t ) L , S is conducting = + 0 , S is blocking dt L dv C i ( t ) − 1 1 = RC v ( t ) , dt

Applications of Constrained Eigenvalue Problem CEiP A wide variety of applications require the solution of CEiP: ◮ Dynamic analysis of structural mechanical systems ◮ Vibro-acoustic systems ◮ Electrical circuit simulation ◮ Signal processing ◮ fluid dynamics How can instability and unwanted resonance be avoided for a given system? Eigenvalues corresponding to unstable modes or yielding large vibrations can be relocated or damped.

Pareto eigenvalue problems An important particular case is given when K = R n + (pareto eigenvalue problem): ◮ 0 ≤ x ⊥ ( Ax − λ x ) ≥ 0 . ◮ 0 ≤ x ⊥ ( Ax − λ Bx ) ≥ 0 . ◮ 0 ≤ x ⊥ Q ( λ ) x ≥ 0 , with Q ( λ ) = λ 2 A + λ B + C .

Stability Analysis of Finite Dimensional Elastic Systems with Frictional Contact. - A. Costa, J. Martins, I. Figueiredo and J. J´ udice, “The directional instability in systems with frictional contacts”, Com- put. Methods Appl. Mech. En- grg., 193 (2004)357–384. A necessary and sufficient condition for the occurrence of divergence instability along a constant admissible direction, is to find λ 2 ≥ 0 and ( x , y ) ∈ R n × R n with x � = 0 such that  ( λ 2 M + K ) x = y , � x f � � y f �  y f = 0 x = , y = x c y c 0 ≤ x c ⊥ y c ≥ 0 

Applications in mechanics ◮ P. Quittner (1986): Spectral analysis of variational inequalities. ◮ J. A. C. Martins and A. Pinto da Costa (2001): Computation of Bifurcations and Instabilities in Some Frictional Contact Problems. ◮ A. Pinto da Costa, J.A.C. Martins, I.N. Figueiredo and J.J. Judice (2004): The directional instability problem in systems with frictional contacts. ◮ J. A. C. Martins and A. Pinto da Costa (2004): Bifurcations and Instabilities in Frictional Contact Problems: Theoretical Relations, Computational Methods and Numerical Results.

Pareto Eigenvalue Complementarity Problem EiCP Definition Let A ∈ M n ( R ). Find λ > 0 and x ∈ R n \ { 0 } , such that � ( EiCP ) x ≥ 0 , λ x − Ax ≥ 0 , � x , λ x − Ax � = 0 . σ ( A ) = { λ > 0 : ∃ x ∈ R n \ { 0 } , 0 ≤ x ⊥ ( λ x − Ax ) ≥ 0 } . Let π n = max A ∈ R n × n card [ σ R n + ( A )] . ◮ We have 3(2 n − 1 − 1) ≤ π n ≤ n 2 n − 1 − ( n − 1) , ◮ We have π 1 = 1, π 2 = 3 and that π 3 = 9 or 10. We note that e.g. π 20 ≥ 1 572 861

Definitions Let Φ : R n → R n be a locally Lipschitz function. ◮ The B-subdifferential de Φ at z ∈ R n is defined by � M ∈ R n × n : ∃ ( z k ) ⊂ D Φ : z k → z , � ∂ B Φ( z ) = k → + ∞ ∇ Φ( z k ) = M lim where D φ is the set of differentiability points of Φ. ◮ The Clarke generalized Jacobian of Φ is given by ∂ Φ( z ) = co ∂ B Φ( z ) , ◮ The function Φ is said to be semismooth at z ∈ R n if it is locally Lipschitz around z , directionally differentiable at z and satisfies the following condition sup � Φ( z + h ) − Φ( z ) − Mz � = o ( � h � ) . M ∈ ∂ Φ( z + h )

Example 1.4 1.2 1 0.8 0.6 0.4 0.2 0 � 2 � 1.5 � 1 � 0.5 0 0.5 1 1.5 2 f ( x ) = ln(1 + | x | ) is semismooth but not differentiable at 0.

Example 0.25 0.2 0.15 0.1 0.05 0 � 0.05 � 0.1 � 0.15 � 0.2 � 0.25 � 0.5 0 0.5 � x 2 sin(1 / x ) if x � = 0 f ( x ) = 0 if x = 0 is not semismooth but locally Lipschitz.

Examples of semismooth functions ◮ The Euclidean norm: � · � 2 . ◮ The Fischer–Burmeister function: √ ϕ FB : R 2 → R , ( a , b ) �→ ϕ FB ( a , b ) = a 2 + b 2 − ( a + b ). ◮ Piecewise continuously differentiable functions: ( a , b ) ∈ R 2 �→ min( a , b ) or max( a , b ).

SNM algorithm 1. Initialization: Choose an initial point z 0 and set k = 0 . 2. Iteration: One has a current point z k , If � Φ( z k ) � � 10 − 8 , then STOP. 3. Else choose M k ∈ ∂ Φ( z k ) and compute h k by solving the linear system M k h k = − Φ( z k ) . Then, set z k +1 = z k + h k , k = k + 1 and go to STEP 2.

Convergence Theorem Theorem Let z ∗ be a zero of the function Φ . Suppose the following ◮ Φ is semismooth (resp. strongly semismooth) at z ∗ ; ◮ all matrices in ∂ Φ( z ∗ ) are nonsingular. Then, there exists a neighborhood V of z ∗ such that the SNM initialized at any z 0 ∈ V generates a sequence ( z k ) k ∈ N that converges superlinearly (resp. quadratically) to z ∗ .

Reformulation Reformulation of EiCP 2 : by using Nonlinear Complementarity Function NCP. φ : R 2 → R is a NCP-function if and only if φ ( a , b ) = 0 ⇐ ⇒ a ≥ 0 , b ≥ 0 , ab = 0 . We conisder a 2 + b 2 , � φ FB ( a , b ) = a + b − φ min ( a , b ) = min( a , b ) . 2 S. Adly and A. Seeger , A Nonsmooth Algorithm for Cone-Constrained Eigenvalue Problems , Springer, Computational Optimization and Applications 49, 299-318 (2011).

Resolution Let z = ( x , y , λ ) and Φ : R n × R n × R − → R 2 n +1 defined by  U φ ( x , y )   , Φ( z ) = Φ( x , y , λ ) = λ x − Ax − y  � 1 n , x � − 1 where U φ : R n × R n − → R n is given by   φ ( x 1 , y 1 ) . . U φ ( x , y ) =  ,   .  φ ( x n , y n ) with φ = φ min or φ = φ FB . ( x , y , λ ) is a solution of EiCP if and only if Φ( x , y , λ ) = 0 R 2 n +1 .

Jacobian matrix ∂ Φ( z )   U φ ( x , y )  . λ x − Ax − y Φ( z ) = Φ( x , y , λ ) =  � 1 n , x � − 1 Lemma The function Φ is semismooth. Moreover, its Clarke generalized Jacobian at z = ( x , y , λ ) is given by     E F 0    : [ E , F ] ∈ ∂ U φ ( x , y ) λ I n − A − I n ∂ Φ( z ) = x  .  1 T 0 0  n