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Random perturbation of the variable metric method for unconstrained nonsmooth nonconvex optimization

El Mouatasim A., Ellaia R., Souza de Cursi J. E.

International Journal of Applied Mathematics and Computer Science

2006

Vol. 16, no 4

463-474

We consider the global optimization of a nonsmooth (nondifferentiable) nonconvex real function. We introduce a variable metric descent method adapted to nonsmooth situations, which is modified by the incorporation of suitable random perturbations. Convergence to a global minimum is established and a simple method for the generation of suitable perturbations is introduced. An algorithm is proposed and numerical results are presented, showing that the method is computationally effective and stable.

Existence theorems for the Dirichlet elliptic inclusion involving exponential-growth-type multivalued right-hand side

Nguyêñ H. T., Pączka D.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

Vol. 53, no 4

361-375

We present two existence results for the Dirichlet elliptic inclusion with an upper semicontinuous multivalued right-hand side in exponential-type Orlicz spaces involving a vector Laplacian, subject to Dirichlet boundary conditions on a domain Ω ⊂ R2. The first result is obtained via the multivalued version of the Leray–Schauder principle together with the Nakano–Dieudonné sequential weak compactness criterion. The second result is obtained by using the nonsmooth variational technique together with a formula for Clarke's subgradient for Lipschitz integral functionals on “nonregular” Orlicz spaces.