Optimization with some uncontrollable variables: a min-equilibrium approach
Jianxin Zhou
Journal of Industrial & Management Optimization 2007, 3(1): 129-138 doi: 10.3934/jimo.2007.3.129
Motivated by instability analysis of unstable (excited state) solutions in computational physics/chemistry, in this paper, the minimax method for solving an optimal control problem with partially uncontrollable variables is embedded into a more general min-equilibrium problem. Results in saddle critical point analysis and computation are modified to provide more information on the minimized objective values and their corresponding riskiness for one to choose in decision making. A numerical algorithm to compute such minimized objective values and their corresponding riskiness is devised. Some convergence results of the algorithm are also established.
keywords: saddle critical point analysis and computation. riskiness index Minimization with some uncontrollable variables
A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems
Xianjin Chen Jianxin Zhou
Conference Publications 2009, 2009(Special): 151-160 doi: 10.3934/proc.2009.2009.151
The aim of this paper is to numerically investigate multiple solutions of semilinear elliptic systems with zero Dirichlet boundary conditions

-$\Delta u=F_u(x;u,v),$   $x\in\Omega,
-$\Delta v=F_v(x;u,v),$   $x\in\Omega,

where $\Omega \subset \mathbb{R}^{N}$ ($N\ge 1$) is a bounded domain. A strongly coupled case where the potential $F(x;u,v)$ takes the form $|u|^{\alpha_1}|v|^{\alpha_2}$ with $\alpha_1, \alpha_2>1$ is specially studied. By using a local min-orthogonal method, both positive and sign-changing solutions are found and displayed.

keywords: min-orthogonal method Cooperative systems multiple solutions

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