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### Open Access Journals

JIMO

An alternating linearization method with inexact data, for the bilevel problem
of minimizing a nonsmooth convex function over the optimal solution set of another
nonsmooth convex problem, is presented in this paper. The problem is first approximately
transformed into an unconstrained optimization with the help of a penalty
function and we prove that the penalty function admits exact penalization under some mild conditions. The
objective function of this unconstrained problem is the sum of two nonsmooth convex
functions and in the algorithm each iteration involves solving two easily solved subproblems.
It is shown that the iterative sequence converges to a solution of the original problem by
parametric minimization theorem. Numerical experiments validate the theoretical convergence
analysis and illustrate the
implementation
of the alternating linearization algorithm
for this bilevel program.

DCDS-B

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source

$\begin{eqnarray*}\left\{\begin{array}{llll}n_t+u·\nabla n = \nabla·(D(n)\nabla n)-\nabla·(n \mathcal{S}(x, n, c)·\nabla c)\\ +ξ n-μ n^{2}, &x∈ Ω, &t>0, \\c_{t}+u·\nabla c = Δ c-c+n, &x∈Ω, &t>0, \\u_{t}+\nabla P = Δ u+n\nablaφ, &x∈Ω, &t>0, \\\nabla· u = 0, &x∈Ω, &t>0\end{array}\right.\end{eqnarray*}$ |

in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills

$ \begin{equation*}{\label{1.3}}\begin{split}|\mathcal{S}(x, n, c)|\leq C_{\mathcal{S}}(1+n)^{-α}\end{split}\end{equation*}$ |

with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.

keywords:
Chemotaxis- Stokes
,
nonlinear diffusion
,
tensor-valued sensitivity
,
boundedness
,
logistic source

JIMO

In this paper, we
consider using the inexact nonsmooth Newton method to efficiently
solve the symmetric cone constrained variational inequality (VISCC)
problem. It red provides a unified framework for dealing
with the variational inequality with nonlinear constraints,
variational inequality with the second-order cone constraints, and
the variational inequality with semidefinite cone constraints. We
get convergence of the above method
and apply the results to three special types symmetric cones.

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