Multi-step nonlinear conjugate gradient methods for unconstrained minimization

Conjugate gradient methods are appealing for large scale nonlinear optimization problems, because they avoid the storage of matrices. Recently, seeking fast convergence of these methods, Dai and Liao (Appl. Math. Optim. 43:87–101, 2001) proposed a conjugate gradient method based on the secant condition of quasi-Newton methods, and later Yabe and Takano (Comput. Optim. Appl. 28:203–225, 2004) proposed another conjugate gradient method based on the modified secant condition. In this paper, we make use of a multi-step secant condition given by Ford and Moghrabi (Optim. Methods Softw. 2:357–370, 1993; J. Comput. Appl. Math. 50:305–323, 1994) and propose two new conjugate gradient methods based on this condition. The methods are shown to be globally convergent under certain assumptions. Numerical results are reported.     

Sequential optimality conditions for composed convex optimization problems

Using a general approach which provides sequential optimality conditions for a general convex optimization problem, we derive necessary and sufficient optimality conditions for composed convex optimization problems. Further, we give sequential characterizations for a subgradient of the precomposition of a K-increasing lower semicontinuous convex function with a K-convex and K-epi-closed (continuous) function, where K is a nonempty convex cone. We prove that several results from the literature dealing with sequential characterizations of subgradients are obtained as particular cases of our results. We also improve the above mentioned statements.     

Convergence of Conjugate Gradient Methods with a Closed-Form Stepsize Formula

Conjugate gradient methods are efficient methods for minimizing differentiable objective functions in large dimension spaces. However, converging line search strategies are usually not easy to choose, nor to implement. Sun and colleagues (Ann. Oper. Res. 103:161–173, 2001; J. Comput. Appl. Math. 146:37–45, 2002) introduced a simple stepsize formula. However, the associated convergence domain happens to be overrestrictive, since it precludes the optimal stepsize in the convex quadratic case. Here, we identify this stepsize formula with one iteration of the Weiszfeld algorithm in the scalar case. More generally, we propose to make use of a finite number of iterates of such an algorithm to compute the stepsize. In this framework, we establish a new convergence domain, that incorporates the optimal stepsize in the convex quadratic case. The authors thank the associate editor and the reviewer for helpful comments and suggestions. C. Labat is now in postdoctoral position, Johns Hopkins University, Baltimore, MD, United States.     

加细函数正交性的一个判别准则

In this note, a criterion for orthonormality of refinable functions via characteristicpolynomial of a matrix is given.     

A first high enantiocontrol of an asymmetric tertiary carbon center attached with a fluoroalkyl group via Rh(I)-catalyzed conjugate addition reaction

Treatment of fluoroalkylated electron-deficient olefins with various boronic acids in the presence of a catalytic amount of Rh(I) coordinated with (S)-BINAP in toluene/H₂O at the reflux temperature for 3 h gave the corresponding conjugate addition products with high enantioselectivity in high yields.     

On strong and total Lagrange duality for convex optimization problems

We give some necessary and sufficient conditions which completely characterize the strong and total Lagrange duality, respectively, for convex optimization problems in separated locally convex spaces. We also prove similar statements for the problems obtained by perturbing the objective functions of the primal problems by arbitrary linear functionals. In the particular case when we deal with convex optimization problems having infinitely many convex inequalities as constraints the conditions we work with turn into the so-called Farkas-Minkowski and locally Farkas-Minkowski conditions for systems of convex inequalities, recently used in the literature. Moreover, we show that our new results extend some existing ones in the literature.     

Implicit finite element methods for two-phase flow in oil reservoirs

The equations governing immiscible, incompressible, two-phase, porous media flow are discretized by generalized streamline diffusion Petrov–Galerkin methods in space and by implicit differences in time. Systems of non-linear algebraic equations are solved by Newton–Raphson iteration employing ILU-preconditioned conjugate-gradient-like methods to the non-symmetric matrix system in each iteration. The resulting solution methods are robust, enable complex grids with irregular nodal orderings and allow capillary effects. Several numerical formulations are tested and compared for one-, two- and three-dimensional flow cases, with emphasis on problems involving saturation shocks, heterogeneous media and curved boundaries. For reservoirs consisting of multiple rock types with differing capillary pressure properties, it is shown that traditional Bubnov-Galerkin methods give poor results and the new Petrov–Galerkin formulations are required. Investigations regarding the behaviour of several preconditioned conjugate-gradient-like methods in these type of problems are also reported.     

DABCO-mediated aza-Michael addition of 4-aryl-1H-1,2,3-triazoles to cycloalkenones. Regioselective synthesis of disubstituted 1,2,3-triazoles

Aza-Michael addition of 4-aryl-1H-1,2,3-triazoles to 2-cycloalken-1-ones has been studied in the presence of DABCO as organic base. The reactions were carried out in acetonitrile at room temperature to provide 2,4-disubstituted 2H-1,2,3-triazoles as major adducts and 1,4-disubstituted 1H-1,2,3-triazoles as minor adducts. Though the reaction times are longer (4–8 days), the two regioisomers were separated by using column chromatography and the adducts were obtained in very good to excellent combined chemical yields. The electron-rich and electron-poor substituents on aryl moiety of 4-aryl-triazoles could tolerate the reaction conditions to afford the title adducts.     

Nucleophilic addition to nitroacrylates: application towards the synthesis of 2,3-dehydroamino acids and 2,3-diamino acids

The addition of the N-pronucleophiles to 2- or 3-nitro-2-alkenoates in the presence of base provided Michael addition products. In the case of 3-nitro compounds, reaction occurred via the formation of α-adducts and the subsequent elimination of nitrous acid to produce olefins with high Z stereoselectivity. 3-Phthalimido-2-nitrocinnamate adduct 8a was converted into 2,3-diamino ester 11a.     

苯丙炔酸低聚物的合成及性能

极性-共轭双取代乙炔-苯丙炔酸在Pd(PPh3)2Cl2催化下直接通过易位催化机理聚合为低聚物.单体苯丙炔酸由两步制得,先通过液溴和肉桂酸的加成反应得到二溴代苯丙酸,然后消去两分子溴化氢得到单体.通过红外光谱(IR)、核磁共振(NMR)对单体及低聚物的结构进行了表征.凝胶渗透色谱(GPC)结果显示,聚合反应以甲苯作溶剂,75℃下效果最好,产率中等.这种共轭低聚物可以溶于四氢呋喃(THF)等极性溶剂,TGA结果表明低聚物热稳定性高.紫外可见吸收光谱(UV-Vis)显示,由于聚合后共轭主链大大增长,苯丙炔酸低聚物骨架上电子的跃迁使得低聚物的四氢呋喃溶液在波长高于310 nm光谱的区域中有吸收.荧光光谱(FS)显示,低聚物的四氢呋喃溶液在342 nm激发光下发蓝光,420 nm处有特征荧光发射峰.