We present a random perturbation of the projected variable metric method for solving linearly constrained nonsmooth (i.e., nondifferentiable) nonconvex optimization problems, and we establish the convergence to a global minimum for a locally Lipschitz continuous objective function which may be nondifferentiable on a countable set of points. Numerical results show the effectiveness of the proposed approach.

The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier–Stokes equations, semilinear wave equations and reaction diffusion systems are given.

The receding horizon control strategy for dynamical systems posed in infinite dimensional spaces is analysed. Its stabilising property is verified provided control Lyapunov functionals are used as terminal penalty functions. For closed loop dissipative systems the terminal penalty can be chosen as quadratic functional. Applications to the Navier–Stokes equations, semilinear wave equations and reaction diffusion systems are given.

In this report we propose a new recursive matrix formulation of limited memory variable metric methods. This approach can be used for an arbitrary update from the Broyden class (and some other updates) and also for the approximation of both the Hessian matrix and its inverse. The new recursive formulation requires approximately $4mn$ multiplications and additions per iteration, so it is comparable with other efficient limited memory variable metric methods. Numerical experiments concerning Algorithm...

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter α we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter α, based on the so-called U-curve....

We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems$$$$where $\Omega \subset {𝐑}^n$ is a bounded Lipschitz domain, $F_s$ are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that $WZ$ is the zero level set for an integral functional with the integrand $Qℱ$ being the $𝐀$-quasiconvex envelope for a certain function $ℱ$ and the operator $𝐀={\left(\text{curl,div}\right)}^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda$ (defined by the operator $𝐀$) $Qℱ$ coincides...

We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$\begin{array}{c}\text{div}\left(\sum _{s=1}^{s_0}{\sigma }_s\left(x\right)F_s^{\text{'}}(\nabla u\left(x\right)+g\left(x\right))-f\left(x\right)\right)=0\text{in}\Omega ,u=(u_1,\cdots ,u_m)\in H_0^1(\Omega ;{𝐑}^m),\sigma =({\sigma }_1,\cdots ,{\sigma }_{s_0})\in S,\end{array}$$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Qℱ$ being the A-quasiconvex envelope for a certain function $ℱ$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone...

The contribution is devoted to computations of the limit load for a perfectly plastic model with the von Mises yield criterion. The limit factor of a prescribed load is defined by a specific variational problem, the so-called limit analysis problem. This problem is solved in terms of deformation fields by a penalization, the finite element and the semismooth Newton methods. From the numerical solution, we derive a guaranteed upper bound of the limit factor. To achieve more accurate results, a local...

This paper introduces the application of asynchronous iterations theory within the framework of the primal Schur domain decomposition method. A suitable relaxation scheme is designed, whose asynchronous convergence is established under classical spectral radius conditions. For the usual case where local Schur complement matrices are not constructed, suitable splittings based only on explicitly generated matrices are provided. Numerical experiments are conducted on a supercomputer for both Poisson's...