On Eigenvalue Complementarity Problems Alfredo Iusem May 10, 2018 - - PowerPoint PPT Presentation

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

On Eigenvalue Complementarity Problems

Alfredo Iusem May 10, 2018

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

1

The eigenvalue complementarity problem

2

Existence and number of solutions of EiCP

3

Computational methods for EiCP

4

The quadratic and the conic EiCP (QEiCP, CEiCP)

5

The Quadratic Conic EiCP (CQEiCP)

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem I

Given matricesB, C ∈ Rn×n, the Eigenvalue Complementarity Problem (to be denoted EiCP(B, C)) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λBx − Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , with e = (1, 1, . . . , 1)t ∈ Rn.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem I

Given matricesB, C ∈ Rn×n, the Eigenvalue Complementarity Problem (to be denoted EiCP(B, C)) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λBx − Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , with e = (1, 1, . . . , 1)t ∈ Rn. The last constraint has been introduced, WLOG, in order to prevent the x component of a solution to vanish.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem I

Given matricesB, C ∈ Rn×n, the Eigenvalue Complementarity Problem (to be denoted EiCP(B, C)) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λBx − Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , with e = (1, 1, . . . , 1)t ∈ Rn. The last constraint has been introduced, WLOG, in order to prevent the x component of a solution to vanish. Usually, the matrix B is assumed to be positive definite.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem I

Given matricesB, C ∈ Rn×n, the Eigenvalue Complementarity Problem (to be denoted EiCP(B, C)) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λBx − Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , with e = (1, 1, . . . , 1)t ∈ Rn. The last constraint has been introduced, WLOG, in order to prevent the x component of a solution to vanish. Usually, the matrix B is assumed to be positive definite. The problem has many applications in engineering (e.g., contact problems).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem II

EiCP(A, B) generalizes the Generalized Eigenvalue Problem, denoted GEiP, which consists of finding (λ, x) such that 0 = λBx − Cx

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem II

EiCP(A, B) generalizes the Generalized Eigenvalue Problem, denoted GEiP, which consists of finding (λ, x) such that 0 = λBx − Cx A solution (λ, x) of GEiP is just an eigenvalue and eigenvector

• f the matrix B−1C in the usual sense, when B is nonsingular.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem II

EiCP(A, B) generalizes the Generalized Eigenvalue Problem, denoted GEiP, which consists of finding (λ, x) such that 0 = λBx − Cx A solution (λ, x) of GEiP is just an eigenvalue and eigenvector

• f the matrix B−1C in the usual sense, when B is nonsingular.

If a triplet (λ, x, w) solves EiCP, then the scalar λ is called a complementary eigenvalue and x is a complementary eigenvector associated to λ for the pair (B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The eigenvalue complementarity problem II

EiCP(A, B) generalizes the Generalized Eigenvalue Problem, denoted GEiP, which consists of finding (λ, x) such that 0 = λBx − Cx A solution (λ, x) of GEiP is just an eigenvalue and eigenvector

• f the matrix B−1C in the usual sense, when B is nonsingular.

If a triplet (λ, x, w) solves EiCP, then the scalar λ is called a complementary eigenvalue and x is a complementary eigenvector associated to λ for the pair (B, C). The conditions xtw = 0, x ≥ 0, w ≥ 0 imply that either xi = 0 or wi = 0 for 1 ≤ i ≤ n. These two variables are called complementary.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP I

Given an operator T : Rn → Rn and a closed and convex set U ⊂ Rn the Variational Inequality Problem VIP(T, U) consists of finding ¯ x ∈ U such that T(¯ x), x − ¯ x ≥ 0 for all x ∈ U .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP I

Given an operator T : Rn → Rn and a closed and convex set U ⊂ Rn the Variational Inequality Problem VIP(T, U) consists of finding ¯ x ∈ U such that T(¯ x), x − ¯ x ≥ 0 for all x ∈ U . When T = ∇g for a convex and differentiable g : Rn → R, then VIP(T, U) is equivalent to the problem of minimizing g(x) subject to x ∈ U.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP I

Given an operator T : Rn → Rn and a closed and convex set U ⊂ Rn the Variational Inequality Problem VIP(T, U) consists of finding ¯ x ∈ U such that T(¯ x), x − ¯ x ≥ 0 for all x ∈ U . When T = ∇g for a convex and differentiable g : Rn → R, then VIP(T, U) is equivalent to the problem of minimizing g(x) subject to x ∈ U. A classical result, in the line of Weierstrass’ Theorem, states than if T is continuous and U is compact then VIP(T, U) has solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP II

It is easy to prove that EiCP(B, C) can be reformulated as VIP( ¯ F, Ω) with feasible set Ω = {x ∈ Rn : etx = 1, x ≥ 0} and operator ¯ F : Ω → Rn given by ¯ F(x) =

• xtCx

xtBx

• Bx − Cx .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP II

It is easy to prove that EiCP(B, C) can be reformulated as VIP( ¯ F, Ω) with feasible set Ω = {x ∈ Rn : etx = 1, x ≥ 0} and operator ¯ F : Ω → Rn given by ¯ F(x) =

• xtCx

xtBx

• Bx − Cx .

B is strictly copositive if xtBx > 0 for all x ∈ Rn

+, x = 0.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP II

It is easy to prove that EiCP(B, C) can be reformulated as VIP( ¯ F, Ω) with feasible set Ω = {x ∈ Rn : etx = 1, x ≥ 0} and operator ¯ F : Ω → Rn given by ¯ F(x) =

• xtCx

xtBx

• Bx − Cx .

B is strictly copositive if xtBx > 0 for all x ∈ Rn

+, x = 0.

It is immediate that if B is strictly copositive then ¯ F is continuous in Ω. Also, Ω is always convex and compact.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of EiCP II

It is easy to prove that EiCP(B, C) can be reformulated as VIP( ¯ F, Ω) with feasible set Ω = {x ∈ Rn : etx = 1, x ≥ 0} and operator ¯ F : Ω → Rn given by ¯ F(x) =

• xtCx

xtBx

• Bx − Cx .

B is strictly copositive if xtBx > 0 for all x ∈ Rn

+, x = 0.

It is immediate that if B is strictly copositive then ¯ F is continuous in Ω. Also, Ω is always convex and compact. It follows from the existence result for VIP that EiCP(B, C) always has solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

On the number of solutions of EiCP

Let (λ, x) be a solution of EiCP(B, C). Let J = {i : xi > 0}, define r as the cardinality of J and consider the matrices BJJ, C JJ ∈ Rr×r with BJJ

ik = Bik, C JJ ik = Cik(i, k ∈ J).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

On the number of solutions of EiCP

Let (λ, x) be a solution of EiCP(B, C). Let J = {i : xi > 0}, define r as the cardinality of J and consider the matrices BJJ, C JJ ∈ Rr×r with BJJ

ik = Bik, C JJ ik = Cik(i, k ∈ J).

It is easy to check λ is an eigenvalue of (BJJ)−1C JJ with eigenvector xJ ∈ Rr, defined as xJ

i = xi(i ∈ J).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

On the number of solutions of EiCP

Let (λ, x) be a solution of EiCP(B, C). Let J = {i : xi > 0}, define r as the cardinality of J and consider the matrices BJJ, C JJ ∈ Rr×r with BJJ

ik = Bik, C JJ ik = Cik(i, k ∈ J).

It is easy to check λ is an eigenvalue of (BJJ)−1C JJ with eigenvector xJ ∈ Rr, defined as xJ

i = xi(i ∈ J).

Hence, every complementary eigenvalue of (B, C) is a regular eigenvalue of a matrix of size r ≤ n associated to submatrices

• f B, C. Since matrices of size r have at most r real

eigenvalues, it follows easily that EiCP(B, C) has at most n2n complementary eigenvalues. See Seeger (2011).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

On the number of solutions of EiCP

Let (λ, x) be a solution of EiCP(B, C). Let J = {i : xi > 0}, define r as the cardinality of J and consider the matrices BJJ, C JJ ∈ Rr×r with BJJ

ik = Bik, C JJ ik = Cik(i, k ∈ J).

It is easy to check λ is an eigenvalue of (BJJ)−1C JJ with eigenvector xJ ∈ Rr, defined as xJ

i = xi(i ∈ J).

Hence, every complementary eigenvalue of (B, C) is a regular eigenvalue of a matrix of size r ≤ n associated to submatrices

• f B, C. Since matrices of size r have at most r real

eigenvalues, it follows easily that EiCP(B, C) has at most n2n complementary eigenvalues. See Seeger (2011). Examples of EiCPs of size n with 3(2n−1 − 2) complementary eigenvalues were given in Seeger-Vicente P´ erez (2011).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of symmetric EiCP’s

When B, C are both symmetric, the EiCP is called symmetric and reduces to the problem of finding a Stationary Point (SP)

• f the so-called Rayleigh Quotient on the simplex Ω, meaning

a SP of the following Standard Quadratic Fractional Program (SQFP): min xtCx

xtBx s.t. x ∈ Ω .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of symmetric EiCP’s

When B, C are both symmetric, the EiCP is called symmetric and reduces to the problem of finding a Stationary Point (SP)

• f the so-called Rayleigh Quotient on the simplex Ω, meaning

a SP of the following Standard Quadratic Fractional Program (SQFP): min xtCx

xtBx s.t. x ∈ Ω .

A spectral projected-gradient method was proposed in J´ udice-Raydan (2008) for solving SQFP. The structure of SQFP is fully exploited for computing the gradient and the projection in each iteration. The stepsize is found with an exact line-search, requiring the solution of a binomial equation.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of symmetric EiCP’s

When B, C are both symmetric, the EiCP is called symmetric and reduces to the problem of finding a Stationary Point (SP)

• f the so-called Rayleigh Quotient on the simplex Ω, meaning

a SP of the following Standard Quadratic Fractional Program (SQFP): min xtCx

xtBx s.t. x ∈ Ω .

A spectral projected-gradient method was proposed in J´ udice-Raydan (2008) for solving SQFP. The structure of SQFP is fully exploited for computing the gradient and the projection in each iteration. The stepsize is found with an exact line-search, requiring the solution of a binomial equation. This is the most efficient known algorithm for solving the symmetric EiCP.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s I

The best alternatives for the numerical solution of the nonsymmetric EICP work through the reduction of EiCP to some nonlinear optimization problems.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s I

The best alternatives for the numerical solution of the nonsymmetric EICP work through the reduction of EiCP to some nonlinear optimization problems. It is easy to check that EiCP(B, C) is equivalent to Problem P, defined as min f (x, y, w, λ) := y − λx2 + xtw s.t. w = By − Cx, etx = 1, ety = λ, x, w ≥ 0 .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s I

The best alternatives for the numerical solution of the nonsymmetric EICP work through the reduction of EiCP to some nonlinear optimization problems. It is easy to check that EiCP(B, C) is equivalent to Problem P, defined as min f (x, y, w, λ) := y − λx2 + xtw s.t. w = By − Cx, etx = 1, ety = λ, x, w ≥ 0 . It is required to find a global solution of P, namely one with

• bjective value equal to 0.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s II

J´ udice, Sherali et al (2007, 2009) developed an enumerative method for solving P, which can find a complementary eigenvalue in a given interval. Using a branch and bound technique, the method dynamically adjusts the intervals, and is guaranteed to find all complementary eigenvalues, but in most cases it is rather slow.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s II

J´ udice, Sherali et al (2007, 2009) developed an enumerative method for solving P, which can find a complementary eigenvalue in a given interval. Using a branch and bound technique, the method dynamically adjusts the intervals, and is guaranteed to find all complementary eigenvalues, but in most cases it is rather slow. Another alternative consists of incorporating the complementarity condition xtw = 0 to the objective through the Fischer-Burmeister function φ : R2 → R defined as φ(a, b) = a + b − √ a2 + b2.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s II

J´ udice, Sherali et al (2007, 2009) developed an enumerative method for solving P, which can find a complementary eigenvalue in a given interval. Using a branch and bound technique, the method dynamically adjusts the intervals, and is guaranteed to find all complementary eigenvalues, but in most cases it is rather slow. Another alternative consists of incorporating the complementarity condition xtw = 0 to the objective through the Fischer-Burmeister function φ : R2 → R defined as φ(a, b) = a + b − √ a2 + b2. It is easy to check that φ(a, b) = 0 iff a ≥ 0, b ≥ 0, ab = 0 .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s III

Define Φ(x, w) = (φ(x1, w1), . . . φ(xn, wn)) , Ψ(x, w, λ) = (Φ(x, w), (λB − C)x − w, etx − 1) .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s III

Define Φ(x, w) = (φ(x1, w1), . . . φ(xn, wn)) , Ψ(x, w, λ) = (Φ(x, w), (λB − C)x − w, etx − 1) . It turns out that EiCP(B, C) is equivalent to solving the system of equations Ψ(x, w, λ) = 0.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s III

Define Φ(x, w) = (φ(x1, w1), . . . φ(xn, wn)) , Ψ(x, w, λ) = (Φ(x, w), (λB − C)x − w, etx − 1) . It turns out that EiCP(B, C) is equivalent to solving the system of equations Ψ(x, w, λ) = 0. Due to the square root in the definition of φ, Ψ is not differentiable, but it possible to solve the system of equations using the semi-smooth Newton method, which substitutes subgradients for the gradients.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s IV

This method was suggested for EiCP in Seeger (2011). As all Newton-type methods, it has fast local convergence, but it is not guaranteed to work when it starts far from the solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Numerical solution of nonsymmetric EiCP’s IV

This method was suggested for EiCP in Seeger (2011). As all Newton-type methods, it has fast local convergence, but it is not guaranteed to work when it starts far from the solutions. In J´ udice, Sherali et al. (2014) a hybrid method for EiCP was

• proposed. It starts with the enumerative method, and when it

is close to a solution, (i.e., when f (x, w, λ) is close to 0), it switches to the semi-smooth Newton method. If this is not succesful, the algorithm goes back to the enumerative method. The hybrid method is the most efficient known procedure for solving the nonsymmetric EiCP.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP I

In 2001 A. Seeger introduced an extension of EiCP called Quadratic Eigenvalue Complementarity Problem, denoted as QEiCP(A, B, C)

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP I

In 2001 A. Seeger introduced an extension of EiCP called Quadratic Eigenvalue Complementarity Problem, denoted as QEiCP(A, B, C) Compared to EiCP, it has an additional quadratic term on λ.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP I

In 2001 A. Seeger introduced an extension of EiCP called Quadratic Eigenvalue Complementarity Problem, denoted as QEiCP(A, B, C) Compared to EiCP, it has an additional quadratic term on λ. Given A, B, C ∈ Rn×n, QEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λ2Ax + λBx + Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , where, as before, e = (1, 1, . . . , 1)t ∈ Rn.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP I

In 2001 A. Seeger introduced an extension of EiCP called Quadratic Eigenvalue Complementarity Problem, denoted as QEiCP(A, B, C) Compared to EiCP, it has an additional quadratic term on λ. Given A, B, C ∈ Rn×n, QEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λ2Ax + λBx + Cx , w ≥ 0, x ≥ 0 , xtw = 0 , etx = 1 , where, as before, e = (1, 1, . . . , 1)t ∈ Rn. QEiCP also has significant engeneering applications.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP II

As before, the condition etx = 1 prevents the x component of a solution from vanishing.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP II

As before, the condition etx = 1 prevents the x component of a solution from vanishing. Note that when A = 0, QEiCP(A, B, C) reduces to EiCP(B, −C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP II

As before, the condition etx = 1 prevents the x component of a solution from vanishing. Note that when A = 0, QEiCP(A, B, C) reduces to EiCP(B, −C). The λ component of a solution of QEiCP(A, B, C) is called a quadratic complementary eigenvalue for A, B, C, and the x component a quadratic complementary eigenvector forA, B, C associated to λ.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

QEiCP: the quadratic EiCP II

As before, the condition etx = 1 prevents the x component of a solution from vanishing. Note that when A = 0, QEiCP(A, B, C) reduces to EiCP(B, −C). The λ component of a solution of QEiCP(A, B, C) is called a quadratic complementary eigenvalue for A, B, C, and the x component a quadratic complementary eigenvector forA, B, C associated to λ. xi and wi are still called complementary variables, because xiwi = 0 for all i ∈ {1, . . . , n}.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP I

We consider now a natural generalization of EiCP and QEiCP, where the nonnegative orthant of Rn is replaced by a rather general cone in Rn.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP I

We consider now a natural generalization of EiCP and QEiCP, where the nonnegative orthant of Rn is replaced by a rather general cone in Rn. We recall that a set K ⊂ Rn is a cone when it is closed by multiplication by nonnegative scalars. It is easy to see that the convex cones are precisely the subsets of Rn which are closed by linear combinations with nonnegative scalars.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP I

We consider now a natural generalization of EiCP and QEiCP, where the nonnegative orthant of Rn is replaced by a rather general cone in Rn. We recall that a set K ⊂ Rn is a cone when it is closed by multiplication by nonnegative scalars. It is easy to see that the convex cones are precisely the subsets of Rn which are closed by linear combinations with nonnegative scalars. Here we consider only closed convex cones, i.e. those convex cones which are closed in Rn with respect to the topology induced by any norm.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP I

We consider now a natural generalization of EiCP and QEiCP, where the nonnegative orthant of Rn is replaced by a rather general cone in Rn. We recall that a set K ⊂ Rn is a cone when it is closed by multiplication by nonnegative scalars. It is easy to see that the convex cones are precisely the subsets of Rn which are closed by linear combinations with nonnegative scalars. Here we consider only closed convex cones, i.e. those convex cones which are closed in Rn with respect to the topology induced by any norm. We recall that a cone K is pointed if it does not contain lines,

• r equivalently, if there exists no nonzero x ∈ K such that

−x ∈ K.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP II

Given a cone K, its dual cone (or positive polar cone) K ∗ is defined as K ∗ = {x ∈ Rn : xty ≥ 0 ∀y ∈ K} .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP II

Given a cone K, its dual cone (or positive polar cone) K ∗ is defined as K ∗ = {x ∈ Rn : xty ≥ 0 ∀y ∈ K} . It is easy to check that K is pointed if and only if K ∗ has nonempty interior.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP II

Given a cone K, its dual cone (or positive polar cone) K ∗ is defined as K ∗ = {x ∈ Rn : xty ≥ 0 ∀y ∈ K} . It is easy to check that K is pointed if and only if K ∗ has nonempty interior. We define now the Conic Eigenvalue Complementary Problem CEiCP(B, C). Let K ⊂ Rn be a closed, convex and pointed

• cone. Take any a ∈ int(K ∗). Given B, C ∈ Rn×n,

CEiCP(B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λBx − Cx , x ∈ K, w ∈ K ∗ , xtw = 0 , atx = 1 .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP III

If (λ, x, w) solves CEiCP(B, C), then λ is said to be a conic complementary eigenvalue and x a conic complementary eigenvector.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP III

If (λ, x, w) solves CEiCP(B, C), then λ is said to be a conic complementary eigenvalue and x a conic complementary eigenvector. As in the case of EiCP, the condition atx = 1 is included just to ensure that conic complementary eigenvectors are nonzero.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP III

If (λ, x, w) solves CEiCP(B, C), then λ is said to be a conic complementary eigenvalue and x a conic complementary eigenvector. As in the case of EiCP, the condition atx = 1 is included just to ensure that conic complementary eigenvectors are nonzero. It is easy to check that changing the vector a ∈ int(K ∗) does not alter the set of conic complementary eigenvalues, and that each conic complementary eigenvector is replaced by a positive multiple of itself.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

CEiCP: the conic EiCP III

If (λ, x, w) solves CEiCP(B, C), then λ is said to be a conic complementary eigenvalue and x a conic complementary eigenvector. As in the case of EiCP, the condition atx = 1 is included just to ensure that conic complementary eigenvectors are nonzero. It is easy to check that changing the vector a ∈ int(K ∗) does not alter the set of conic complementary eigenvalues, and that each conic complementary eigenvector is replaced by a positive multiple of itself. Note that when K = Rn

+ (i.e., the nonnegative orthant of Rn,

in which case K ∗ = K), and a = e, CEiCP(B, C) reduces to EiCP(B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of CEiCP

It has been proved in Seeger-Torky (2003) that if K is closed, convex and pointed, and xtBx = 0 for all nonzero x ∈ K, then CEiCP(B, C) has solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of CEiCP

It has been proved in Seeger-Torky (2003) that if K is closed, convex and pointed, and xtBx = 0 for all nonzero x ∈ K, then CEiCP(B, C) has solutions. The proof works through the reduction of CEiCP(B, C) to VIP(F, ∆), with F(x) =

• xtCx

xtBx

• Bx − Cx ,

and ∆ = {x ∈ K : atx = 1}.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of CEiCP

It has been proved in Seeger-Torky (2003) that if K is closed, convex and pointed, and xtBx = 0 for all nonzero x ∈ K, then CEiCP(B, C) has solutions. The proof works through the reduction of CEiCP(B, C) to VIP(F, ∆), with F(x) =

• xtCx

xtBx

• Bx − Cx ,

and ∆ = {x ∈ K : atx = 1}. Pointedness of K is a key factor in the proof, because it ensures that int(K ∗) = ∅, and the fact that a ∈ int(K ∗) is essential for establishing compactness of ∆, which in turn is a critical ingredient in the proof of existence of solutions of VIP(F, ∆).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The Quadratic Conic Eigenvalue Complementary Problem (QCEiCP)

We define the Quadratic Conic Eigenvalue Complementary

• Problem. Given A, B, C ∈ Rn×n, a closed, convex and pointed

cone K ⊂ Rn and a vector a ∈ int(K ∗), QCEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λ2Ax + λBx + Cx , x ∈ K, w ∈ K ∗, xtw = 0 , atx = 1 .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The Quadratic Conic Eigenvalue Complementary Problem (QCEiCP)

We define the Quadratic Conic Eigenvalue Complementary

• Problem. Given A, B, C ∈ Rn×n, a closed, convex and pointed

cone K ⊂ Rn and a vector a ∈ int(K ∗), QCEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λ2Ax + λBx + Cx , x ∈ K, w ∈ K ∗, xtw = 0 , atx = 1 . If (λ, x, w) solves CEiCP(B, C), then λ is a quadratic complementary eigenvalue and x a quadratic complementary eigenvector.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

The Quadratic Conic Eigenvalue Complementary Problem (QCEiCP)

We define the Quadratic Conic Eigenvalue Complementary

• Problem. Given A, B, C ∈ Rn×n, a closed, convex and pointed

cone K ⊂ Rn and a vector a ∈ int(K ∗), QCEiCP(A, B, C) consists of finding (λ, x, w) ∈ R × Rn × Rn such that w = λ2Ax + λBx + Cx , x ∈ K, w ∈ K ∗, xtw = 0 , atx = 1 . If (λ, x, w) solves CEiCP(B, C), then λ is a quadratic complementary eigenvalue and x a quadratic complementary eigenvector. When K = Rn

+, QCEiCP(A, B, C) reduces to QEiCP(A, B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP I

Contrary to CEiCP, QCEiCP may lack solutions, even when A is positive definite.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP I

Contrary to CEiCP, QCEiCP may lack solutions, even when A is positive definite. Consider QEICP(I, 0, I) with an arbitrary cone K. It is easy to check that 0 = (λ2 + 1) x2, wich has no solution (λ, x) ∈ R × Rn because atx = 1 implies that x = 0.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP I

Contrary to CEiCP, QCEiCP may lack solutions, even when A is positive definite. Consider QEICP(I, 0, I) with an arbitrary cone K. It is easy to check that 0 = (λ2 + 1) x2, wich has no solution (λ, x) ∈ R × Rn because atx = 1 implies that x = 0. This difference between CEiCP and QCEiCP mirrors the fact that linear equations in one real variable always have solutions, while quadratic equations may fail to have them.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP I

Contrary to CEiCP, QCEiCP may lack solutions, even when A is positive definite. Consider QEICP(I, 0, I) with an arbitrary cone K. It is easy to check that 0 = (λ2 + 1) x2, wich has no solution (λ, x) ∈ R × Rn because atx = 1 implies that x = 0. This difference between CEiCP and QCEiCP mirrors the fact that linear equations in one real variable always have solutions, while quadratic equations may fail to have them. We present two independent sets of sufficient conditions for existence of solutions. First the so called co-regularity and co-hyperbolicity properties introduced in Seeger (2011).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP I

Contrary to CEiCP, QCEiCP may lack solutions, even when A is positive definite. Consider QEICP(I, 0, I) with an arbitrary cone K. It is easy to check that 0 = (λ2 + 1) x2, wich has no solution (λ, x) ∈ R × Rn because atx = 1 implies that x = 0. This difference between CEiCP and QCEiCP mirrors the fact that linear equations in one real variable always have solutions, while quadratic equations may fail to have them. We present two independent sets of sufficient conditions for existence of solutions. First the so called co-regularity and co-hyperbolicity properties introduced in Seeger (2011). Then, a set of conditions introduced in Br´ as-Iusem-Judice (2008) for QEiCP.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP II

Definition Given a cone K ⊂ Rn,

i) A matrix A ∈ Rn×n is K-regular if xtAx = 0 for all nonzero x ∈ K. ii) A triplet (A, B, C), with A, B, C ∈ Rn×n is K-hyperbolic if (xtBx)2 ≥ 4(xtAx)(xtCx) for all nonzero x ∈ K.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP II

Definition Given a cone K ⊂ Rn,

i) A matrix A ∈ Rn×n is K-regular if xtAx = 0 for all nonzero x ∈ K. ii) A triplet (A, B, C), with A, B, C ∈ Rn×n is K-hyperbolic if (xtBx)2 ≥ 4(xtAx)(xtCx) for all nonzero x ∈ K.

Definition Given a set ∆ ⊂ Rn and an operator F : ∆ → Rn the Variational Inequality Problem VIP(F, ∆) consists of finding ¯ x ∈ ∆ such that F(¯ x)t(x − ¯ x) ≥ 0 for all x ∈ ∆.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP III

We present next the equivalence between QCEiCP and a variational inequality problem. Consider a closed, convex and pointed cone K ⊂ Rn, fix a ∈ int(K ∗), and define ∆ = {x ∈ K : atx = 1}.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP III

We present next the equivalence between QCEiCP and a variational inequality problem. Consider a closed, convex and pointed cone K ⊂ Rn, fix a ∈ int(K ∗), and define ∆ = {x ∈ K : atx = 1}. Assume that A is K-regular and (A, B, C) is K-hyperbolic, and define λ1, λ2 : ∆ → R and F1, F2 : ∆ → Rn as λ1(x) = −xtBx+√

(xtBx)2−4(xtAx)(xtCx) 2xtAx

, λ2(x) = −xtBx−√

(xtBx)2−4(xtAx)(xtCx) 2xtAx

, Fi(x) = λi(x)2Ax + λi(x)Bx + Cx, (i = 1, 2) .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP IV

We have the following connection between QEiCP(A, B, C) and VIP(F1, ∆), VIP(F2, ∆):

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP IV

We have the following connection between QEiCP(A, B, C) and VIP(F1, ∆), VIP(F2, ∆): Proposition If (¯ λ, ¯ x) solves QCEiCP(A, B, C) then ¯ x solves either VIP(F1, ∆)

• r VIP(F2, ∆). If ¯

x is a solution of VIP(Fi, ∆) (i = 1, 2) then (¯ λ, ¯ x) solves QCEiCP(A, B, C) with ¯ λ = λi(¯ x) (i = 1, 2),

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP IV

We have the following connection between QEiCP(A, B, C) and VIP(F1, ∆), VIP(F2, ∆): Proposition If (¯ λ, ¯ x) solves QCEiCP(A, B, C) then ¯ x solves either VIP(F1, ∆)

• r VIP(F2, ∆). If ¯

x is a solution of VIP(Fi, ∆) (i = 1, 2) then (¯ λ, ¯ x) solves QCEiCP(A, B, C) with ¯ λ = λi(¯ x) (i = 1, 2), Now we invoke the following classical existence result for VIP. Proposition If ∆ is compact and convex and F is continuous then VIP(F, ∆) has solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP V

We present next the existence result in Seeger (2011), based upon the previous two propositions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP V

We present next the existence result in Seeger (2011), based upon the previous two propositions. Theorem Assume that K is a closed, convex and pointed cone, that A is K-regular and that (A, B, C) is K-hyperbolic. Then QCEiCP(A, B, C) has solutions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VI

Our second existence result is based upon the relation between an arbitrary n-dimensional QCEiCP and two specific instances of CEiCP with matrices in R2n×2n.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VI

Our second existence result is based upon the relation between an arbitrary n-dimensional QCEiCP and two specific instances of CEiCP with matrices in R2n×2n. Consider now QCEiCP(A, B, C) with A, B, C ∈ Rn×n and define D, G, H ∈ R2n×2n as: D = A I

• ,

G = −B −C I

• ,

H = B −C I

• .

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VII

Consider now QCEiCP(A, B, C) with A, B, C ∈ Rn×n. Given the cone K ⊂ Rn, we define the cone ˜ K ⊂ R2n as ˜ K = K × K; also, given a ∈int(K ∗), we define ˜ a ∈ R2n as ˜ a = (a, a). Note that ˜ a belongs to int( ˜ K). We consider CEiCP(D, G) and CEiCP(D, H) with cone ˜ K and vector ˜ a.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VII

Consider now QCEiCP(A, B, C) with A, B, C ∈ Rn×n. Given the cone K ⊂ Rn, we define the cone ˜ K ⊂ R2n as ˜ K = K × K; also, given a ∈int(K ∗), we define ˜ a ∈ R2n as ˜ a = (a, a). Note that ˜ a belongs to int( ˜ K). We consider CEiCP(D, G) and CEiCP(D, H) with cone ˜ K and vector ˜ a. Next we prove a relation between the solutions of QCEiCP(A, B, C) and those of CEiCP(D, G) and CEiCP(D, H).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VII

Consider now QCEiCP(A, B, C) with A, B, C ∈ Rn×n. Given the cone K ⊂ Rn, we define the cone ˜ K ⊂ R2n as ˜ K = K × K; also, given a ∈int(K ∗), we define ˜ a ∈ R2n as ˜ a = (a, a). Note that ˜ a belongs to int( ˜ K). We consider CEiCP(D, G) and CEiCP(D, H) with cone ˜ K and vector ˜ a. Next we prove a relation between the solutions of QCEiCP(A, B, C) and those of CEiCP(D, G) and CEiCP(D, H). We emphasize that the following result holds without making any additional hypotheses on A, B, C.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP VIII

Proposition a) Assume that (λ, x) solves QCEiCP(A, B, C) and consider D, G, H as above.

i) If λ = 0 then (λ, z) = (0, z) solves both CEiCP(D, G) and CEiCP(D, H), with z = (0, x). ii) If λ > 0 then (λ, z) solves EiCP(D, G), with z = (1 + λ)−1(λx, x). iii) If λ < 0 then the pair (−λ, z) solves EiCP(D, H) with z = (1 − λ)−1(−λx, x).

b) Consider D, G, H as above.

i) If (λ, z) solves CEiCP(D, G) with z = (y, x) ∈ Rn × Rn and λ = 0, then λ > 0 and (λ, (1 + λ)x) solves QCEiCP(A, B, C). ii) If (λ, z) solves CEiCP(D, H) with z = (y, x) ∈ Rn × Rn and λ = 0, then λ > 0 and (−λ, (1+ λ)x) solves QCEiCP(A, B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP IX

Now we rewrite this result in terms of complementary eigenvalues. Corollary Consider QCEiCP(A, B, C) with A, B, C ∈ Rn×n and the matrices D, G, H ∈ R2n×2n as defined above. Then, i) all quadratic complementary eigenvuales for (A, B, C) are complementary eigenvalues for either (D, G), (D, H) or both, ii) all nonzero complementary eigenvalues for (D, G) are positive, and are quadratic complementary eigenvalues for (A, B, C), iii) all nonzero complementary eigenvalues for (D, H) are positive, and their additive inverses are quadratic complementary eigenvalues for (A, B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP X

Corollary 1 signals a clear path for obtaining a sufficient condition for existence of solutions of QEiCP(A, B, C): we must first find a sufficient condition for solvability of CEiCP(D, G) or CEiCP(D, H) and then impose conditions ensuring that either 0 is a quadratic complementary eigenvalue for (A, B, C), or that 0 is not a complementary eigenvalue of CEiCP(D, G), CEiCP(D, H)

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP X

Corollary 1 signals a clear path for obtaining a sufficient condition for existence of solutions of QEiCP(A, B, C): we must first find a sufficient condition for solvability of CEiCP(D, G) or CEiCP(D, H) and then impose conditions ensuring that either 0 is a quadratic complementary eigenvalue for (A, B, C), or that 0 is not a complementary eigenvalue of CEiCP(D, G), CEiCP(D, H) We present next some classes of matrices needed for our sufficient conditions.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XI

Definition Given a cone K ⊂ Rn

i) A matrix M ∈ Rn×n is said to be strictly K-copositive if xtMx > 0 for all 0 = x ∈ K. ii) The class R′

0(K) ⊂ Rn×n consists of those matrices M ∈ Rn×n such

that xtMx = 0 for all x ∈ K such that Mx ∈ K ∗. iii) The class S′

0(K) ⊂ Rn×n consists of those matrices M ∈ Rn×n such

that there exists no nonzero x ∈ K such that Mx ∈ K ∗.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XI

Definition Given a cone K ⊂ Rn

i) A matrix M ∈ Rn×n is said to be strictly K-copositive if xtMx > 0 for all 0 = x ∈ K. ii) The class R′

0(K) ⊂ Rn×n consists of those matrices M ∈ Rn×n such

that xtMx = 0 for all x ∈ K such that Mx ∈ K ∗. iii) The class S′

0(K) ⊂ Rn×n consists of those matrices M ∈ Rn×n such

that there exists no nonzero x ∈ K such that Mx ∈ K ∗.

We comment that for K = Rn

+, the complements of classes

R′

0(K), S′ 0(K) are the well known clases R0, S0.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XII

The next proposition exhibits the relation between the classes R′

0(K), S′ 0(K) and the existence of solutions of CEiCP,

CQEiCP.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XII

The next proposition exhibits the relation between the classes R′

0(K), S′ 0(K) and the existence of solutions of CEiCP,

CQEiCP. Proposition

i) If M ∈ Rn×n is strictly K-copositive then CEiCP(M, C) has solutions for any C ∈ Rn×n. ii) If C ∈ R′

0(K) then 0 is a quadratic complementary eigenvalue for

(A, B, C) for any A, B, C ∈ Rn×n. iii) If C ∈ S′

0(K) then 0 is not a complementary eigenvalue for either

(D, G) or (D, H) with D, G, H as above.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XIII

Now, all the pieces are in place for stating and proving our existence result for QCEiCP.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XIII

Now, all the pieces are in place for stating and proving our existence result for QCEiCP. Theorem Consider QCEiCP(A, B, C) and assume that either

i) C ∈ R′

0(K), or

ii) C ∈ S′

0(K) and A is strictly K-copositive.

Then QCEiCP(A, B, C) has solutions. Additionally, under assumption (i) 0 is a quadratic complementary eigenvalue for (A, B, C), and under assumption (ii) there exist at least one positive and one negative quadratic complementary eigenvalue for (A, B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XIV

The next corollary states that the roles of A and C in item (ii)

• f Theorem 2 can be interchanged.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XIV

The next corollary states that the roles of A and C in item (ii)

• f Theorem 2 can be interchanged.

Corollary Consider QCEiCP(A, B, C) and assume that A ∈ S′

0(K) and C is

strictly K-copositive. Then there exist at least one positive and

• ne negative quadratic complementary eigenvalue for (A, B, C).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XIV

The next corollary states that the roles of A and C in item (ii)

• f Theorem 2 can be interchanged.

Corollary Consider QCEiCP(A, B, C) and assume that A ∈ S′

0(K) and C is

strictly K-copositive. Then there exist at least one positive and

• ne negative quadratic complementary eigenvalue for (A, B, C).

We compare now the two sets of sufficient conditions for existence of solutions of QCEiCP(A, B, C) given by Theorems 1 and 2, which are indeed independent.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XV

We say that a triplet (A, B, C) satisfies (P) when either C ∈ S′

0(K) and A is strictly K-copositive, or C ∈ R′ 0(K), and

that it satisfies (P’) when A is K-regular and (A, B, C) is K-hyperbolic.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XV

We say that a triplet (A, B, C) satisfies (P) when either C ∈ S′

0(K) and A is strictly K-copositive, or C ∈ R′ 0(K), and

that it satisfies (P’) when A is K-regular and (A, B, C) is K-hyperbolic. First we give an example for which (P) but not (P’) Consider any pointed cone K which is not a halfline (i.e., it contains at least two linearly independent vectors, say (c, d)), take a ∈ int(K ∗), find a vector b ∈ Rn such that btc < 0, btd > 0, and define C ∈ Rn×n as C = bat. It is easy to check that if A is positive definite then (A, 0, C) satisfies (P) but not (P’).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XVI

The above defined C has rank 1. If we take K = Rn

+, then it

is easy to check that if A is positive definite and C has at least one positive diagonal element and at least one fully negative row, then (A, 0, C) satisfies (P) but not (P’).

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XVI

The above defined C has rank 1. If we take K = Rn

+, then it

is easy to check that if A is positive definite and C has at least one positive diagonal element and at least one fully negative row, then (A, 0, C) satisfies (P) but not (P’). There are many instances of QCEiCP for which (P’) holds but not (P). For instance, the triplet (I, 2I, I) in an arbitrary closed and convex cone K.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Existence of solutions of QCEiCP XVI

The above defined C has rank 1. If we take K = Rn

+, then it

is easy to check that if A is positive definite and C has at least one positive diagonal element and at least one fully negative row, then (A, 0, C) satisfies (P) but not (P’). There are many instances of QCEiCP for which (P’) holds but not (P). For instance, the triplet (I, 2I, I) in an arbitrary closed and convex cone K. We mention that (P) depends only upon the matrices A and C, while (P’) also involves the matrix B.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

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Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

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Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

J´ udice, J., Raydan, M., Rosa, S., Santos, S. On the solution of the symmetric complementarity problem by the spectral projected gradient method. Numerical Algorithms 44 (2008) 391-407. J´ udice, J, Sherali, H.D., Ribeiro, I. The eigenvalue complementarity problem. Computational Optimization and Applications 37 (2007) 139-156. J´ udice, J., Sherali, D.H., Ribeiro, I., Rosa, S. On the asymmetric eigenvalue complementarity problem. Optimization Methods and Software 24 (2009) 549-586. Le Thi, H., Moeini, M., Pham Dinh, T., J´ udice, J. A DC programming approach for solving the symmetric eigenvalue complementarity problem. Computational Optimization and Applications 5 (2012) 1097-1117.

Alfredo Iusem On Eigenvalue Complementarity Problems

Outline The eigenvalue complementarity problem Existence and number of solutions of EiCP Computational methods for EiCP The quadratic and the conic EiCP (QEiCP, CEiCP) The Quadratic Conic EiCP (CQEiCP)

Pinto da Costa, A., Seeger, A. Cone constrained eigenvalue problems, theory and algorithms. Computational Optimization and Applications 45 (2010) 25-57. Queiroz, M., J´ udice, J., Humes, C. The symmetric eigenvalue complementarity problem. Mathematics of Computation 73 (2003) 1849-1863. Seeger, A. Eigenvalue analysis of equilibrium processes defined by linear complementarity conditions. Linear Algrebra and Its Applications 294 (1999) 1-14. Seeger, A. Quadratic eigenvalue problems under cone constraints. SIAM Journal on Matrix Analysis and Applications 32 (2011) 700-721. Seeger, A., Torky, M. On eigenvalues induced by a cone constraint. Linear Algebra and Its Applications 372 (2003) 181-206.

Alfredo Iusem On Eigenvalue Complementarity Problems