An alternating linearization method with inexact data for bilevel nonsmooth convex optimization
Dan Li Li-Ping Pang Fang-Fang Guo Zun-Quan Xia
Journal of Industrial & Management Optimization 2014, 10(3): 859-869 doi: 10.3934/jimo.2014.10.859
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.
keywords: bilevel programming. exact penalty function convex programming nonsmooth optimization Proximal point
Boundedness in a three-dimensional Keller-Segel- Stokes system involving tensor-valued sensitivity with saturation
Dan Li Chunlai Mu Pan Zheng Ke Lin
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-19 doi: 10.3934/dcdsb.2018209
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
An inexact semismooth Newton method for variational inequality with symmetric cone constraints
Shuang Chen Li-Ping Pang Dan Li
Journal of Industrial & Management Optimization 2015, 11(3): 733-746 doi: 10.3934/jimo.2015.11.733
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.
keywords: Jordan algebra. Variational inequality Newton method semismooth symmetric cone
Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion
Dan Li Jing'an Cui Yan Zhang
Discrete & Continuous Dynamical Systems - B 2015, 20(7): 2069-2088 doi: 10.3934/dcdsb.2015.20.2069
Using stochastic differential equations with Lévy jumps, this paper studies the effect of environmental stochasticity and random catastrophes on the permanence of Lotka-Volterra facultative systems. Under certain simple assumptions, we establish the sufficient conditions for weak permanence in the mean and extinction of the non-autonomous system, respectively. In particular, a necessary and sufficient condition for permanence and extinction of autonomous system with jump-diffusion are obtained. We generalize some former results under weaker assumptions. Finally, we discuss the biological implications of the main results.
keywords: Lévy jump random catastrophe. permanence environmental stochasticity Stochastic differential equation facultative system

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