The nonlinear bilevel programming problem:formulations,regularity and optimality conditions

In this paper, we study regularity and optimality conditions for the BLPP by using a marginal function formulation, where the marginal function is defined by the optimal value function of the lower problem. We address the regularity issue by exploring the structure of the tangent cones of the feasible set of the BLPP. These regularity results indicate that the nonlinear/nonlinear BLPP is most likely degenerate and a class of nonlinear/linear BLPP is regular in the conventional sense. Existence of exact penalty function is proved for a class of nonlinear/linear BLPP. Fritz-John type optimality conditions are derived for nonlinear BLPP, while KKT type conditions are obtained for a class of nonlinear/linear BLPP in the framework of nonsmooth analysis. A typical example is examined for these conditions and some applications of these conditions are pointed out     

Regularity for nonlinear elliptic systems with dini coefficients under natural growth condition for the case: 1 < m < 2

In this article, we consider interior regularity for weak solutions to nonlinear elliptic systems of divergence type with Dini continuous coefficients under natural growth condition for the case 1 < m < 2. All estimates in the case of m ≥ 2 is no longer suitable, and we can't obtain the Caccioppoli's second inequality by using these techniques developed in the case of m ≥ 2. But the Caccioppoli's second inequality is the key to use A-harmonic approximation method. Thus, we adopt another technique introduced by Acerbi and Fcsco to overcome the difficulty and we also overcome those difficulties due to Dini condition. And then we apply the A-harmonic approximation method to prove partial regularity of weak solutions.     

Radial basis functions method for solving of a non-local boundary value problem with Neumann’s boundary conditions

In this paper, the problem of solving the two-dimensional diffusion equation subject to a non-local condition involving a double integral in a rectangular region is considered. The solution of this type of problems are complicated. Therefore, a simple meshless method using the radial basis functions is constructed for the non-local boundary value problem with Neumann’s boundary conditions. Numerical examples are included to demonstrate the reliability and efficiency of this method. Also N_e and root mean square errors are obtained to show the convergence of the method.     

Self-assembly of 1-D, 2-D,and 3-D transition-metal coordination polymers based on a T-shaped tripodal ligand 4-(4,5-dicarboxy-1H-imidazol-2-yl)pyridine 1-oxide

Three coordination polymers, {[Co(C₁₀H₅N₃O₅)(H₂O)₂]·H₂O}_n (1), {[Mn₃(C₁₀H₅N₃O₅)₂Cl₂(H₂O)₆]·2H₂O}_n (2), and {[Cu₃(C₁₀H₄N₃O₅)₂(H₂O)₃]·4H₂O}_n (3), based on a T-shaped tripodal ligand 4-(4,5-dicarboxy-1H-imidazol-2-yl)pyridine 1-oxide (H₃DCImPyO), were synthesized under hydrothermal conditions. The polymers showed diverse coordination modes, being characterized by elemental analysis, infrared spectroscopy, and single-crystal X-ray structure analysis. In 1, the HDCImPyO^2? generated a 1-D chain by adopting a μ₂-kN, O : kN′, O′ coordination mode to bridge two Co(II) ions in two bis-N,O-chelating modes. In 2, the HDCImPyO^2? adopted a μ₃-kN, O : kO′, O′′ : O′′′ coordination mode to bridge two crystallographically independent Mn(II) ions, forming a 2-D hcb network with {6³} topology. In 3, by adopting μ₄-kN, O : kO′, O′′ : kN′′, O′′′ : O′′′′ coordination, DCImPyO^3? bridged three crystallographically independent Cu(II) ions to form a 3-D framework having the stb topology.     

A modified conjugate gradient algorithm with cyclic Barzilai–Borwein steplength for unconstrained optimization

For solving large-scale unconstrained minimization problems, the nonlinear conjugate gradient method is welcome due to its simplicity, low storage, efficiency and nice convergence properties. Among all the methods in the framework, the conjugate gradient descent algorithm — CG_DESCENT is very popular, in which the generated directions descend automatically, and this nice property is independent of any line search used. In this paper, we generalize CG_DESCENT with two Barzilai–Borwein steplength reused cyclically. We show that the resulting algorithm owns attractive sufficient descent property and converges globally under some mild conditions. We test the proposed algorithm by using a large set of unconstrained problems with high dimensions in CUTEr library. The numerical comparisons with the state-of-the-art algorithm CG_DESCENT illustrate that the proposed method is effective, competitive, and promising.     

Blow-up of Solutions to Porous Medium Equations with a Nonlocal Boundary Condition and a Moving Localized Source

This paper is devoted to the blow-up properties of solutions to the porous medium equations with a nonlocal boundary condition and a moving localized source.Conditions for the existence of global or bl...