Global Krylov subspace methods are the most efficient and robust methods to solve generalized coupled Sylvester matrix equation. In this paper, we propose the nested splitting conjugate gradient process for solving this equation. This method has inner and outer iterations, which employs the generalized conjugate gradient method as an inner iteration to approximate each outer iterate, while each...

We prove a saddle point theorem for locally Lipschitz functionals with arguments based on a version of the mountain pass theorem for such kind of functionals. This abstract result is applied to solve two diierent types of multivalued semilinear elliptic boundary value problems with a Laplace{Beltrami operator on a smooth compact Riemannian manifold. The mountain pass theorem of Ambrosetti and R...

An inflation of an algebra is formed by adding a set of new elements to each element in the original or base algebra, with the stipulation that in forming products each new element behaves exactly like the element in the base algebra to which it is attached. Clarke and Monzo have defined the generalized inflation of a semigroup, in which a set of new elements is again added to each base element...

The convergence theory of generalized pattern search algorithms for unconstrained optimization guarantees under mild conditions that the method produces a limit point satisfying first order optimality conditions related to the local differentiability of the objective function. By exploiting the flexibility allowed by the algorithm, we derive six small dimensional examples showing that the conve...

Abstract In this paper, we are concerned with a Kirchhoff problem in the presence of strongly-singular term perturbed by discontinuous nonlinearity Heaviside type setting Orlicz–Sobolev space. The both and non-continuous terms brings up difficulties associating differentiable functional to finite energy whole space $$W_0^{1,\Phi }(\Omega )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/Math...

We develop a notion of derivative of a real-valued function on a Banach space, called the L-derivative, which is constructed by introducing a generalization of Lipschitz constant of a map. As with the Clarke gradient, the values of the L-derivative of a function are non-empty weak* compact and convex subsets of the dual of the Banach space. The L-derivative, however, is shown to be upper semi c...

Recent characterizations of various types of diierentiable generalized monotone maps by Karamardian{ Schaible{Crouzeix and their strengthened versions by Crouzeix{Ferland are extended to the nonsmooth case. For nondiierentiable locally Lipschitz maps necessary and/or suucient conditions for quasimono-tonicity, pseudomonotonicity and strict/ strong pseudomonotonicity are derived. To accomplish t...

Recent characterizations of various types of differentiable generalized monotone maps by Karamardian– Schaible–Crouzeix and their strengthened versions by Crouzeix–Ferland are extended to the nonsmooth case. For nondifferentiable locally Lipschitz maps necessary and/or sufficient conditions for quasimonotonicity, pseudomonotonicity and strict/ strong pseudomonotonicity are derived. To accomplis...

This paper focuses on Clarke generalized Jacobian of the projection onto symmetric cones. First, we recall some basic concepts. Let ̥ : Ω ⊆ X → Y be a locally Lipschitz function on an open set Ω, where X and Y are two finite dimensional inner product spaces over the field R. Let ∇̥(x) denote the derivative of ̥ at x if ̥ is differentiable at x. The Clarke generalized Jacobian of ̥ at x is defined by...

A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X, 1 (X) 2 (X) : : : n (X), and hence may be written f(1 (X); 2 (X); : : :; n (X)) for some symmetric function f. Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is diierentiable at X if and only if the function f is diier...