A Modified Lagrange Method for Solving Convex Quadratic Optimization Problems

In this paper, a modified version of the Classical Lagrange Multiplier method is developed for convex quadratic optimization problems. The method, which is evolved from the first order derivative test for optimality of the Lagrangian function with respect to the primary variables of the problem, decomposes the solution process into two independent ones, in which the primary variables are solved for independently, and then the secondary variables, which are the Lagrange multipliers, are solved for, afterward. This is an innovation that leads to solving independently two simpler systems of equations involving the primary variables only, on one hand, and the secondary ones on the other. Solutions obtained for small sized problems (as preliminary test of the method) demonstrate that the new method is generally effective in producing the required solutions.

Optimization of Extraction Conditions of Pomegranate Peel Polyphenols and Its Protective Effect on Acute Alcoholic Liver Injury in Mice

[Objectives]This study was conducted to explore the optimization of ultrasonic-assisted organic solvent extraction of pomegranate peel polyphenols(PPPs),and to study the protective effect of PPPs on acute alcoholic liver injury in mice.[Methods]The optimal extraction conditions of PPPs were determined by single factor and orthogonal experiments,and an acute alcoholic liver injury model in mice was established.Bifendate was used as the positive control group to investigate the protective effect of low,medium and high doses of PPPs on acute alcoholic liver injury.[Results]The optimum extraction process parameters were followed as 60%ethanol concentration,solid-liquid ratio of 1:40(w/v),extraction temperature of 50℃,and extraction time of 1.5 h,and the yield was 1.42%.The results of animal experiments showed that PPPs could effectively reduce the degree of alcoholic liver injury in mice,reduce the levels of serum alanine aminotransferase(ALT)and aspartate aminotransferase(AST),and reduce the inflammation and necrosis of liver tissue in mice.Meanwhile,the total polyphenols from pomegranate peel also significantly reduced the expression levels of malondialdehyde(MDA),tumor necrosis factor(TNF-α)and interleukin-6(IL-6)in mice,and increased the levels of superoxide dismutase(SOD)and reduced glutathione(GSH)in liver tissue of mice,indicating its antioxidant and anti-inflammatory effects,further illustrating its protective effect on alcoholic liver injury.[Conclusions]PPPs could reduce the expression levels of TNF-α,IL-6 and MDA in mice,and increase the expression levels of SOD and GSH to achieve the protective effect on acute alcoholic liver injury in mice.This study will provide new ideas for the development of new anti-alcoholic liver injury drug resources.

Optimization of Extraction Conditions for Total Flavonoids from Fructus Aurantii Immaturus and Its Anti-UVB Radiation Activity

[Objectives]Optimum extraction conditions of total flavonoids from Fructus Aurantii Immaturus(TFFAI)and its resistance activity to ultraviolet radiation were investigated in present research.[Methods]The optimal extraction conditions of TFFAI were determined by single factor and orthogonal experiments,and the survival rate of TFFAI on HaCaT cells irradiated with UVB rays was investigated.It s antioxidant capacity was determined by ABTS.[Results]The results showed that the highest yield of TFFAI was obtained with 70%ethanol at a solid-to-liquid ratio of 1:50(w/v)and 40℃for 1.5 h by single-factor and orthogonal experiments.Total flavonoids(0.25-1.00 mg/ml)could significantly improve the survival rate of HaCaT cell line.Meanwhile,the maximum absorption peak of TFFAI was found at 283 nm,and in-vitro antioxidant experiment identified that TFFAI had a good clearance rate to ABTS.It suggestes that TFFAI could protect the cells from UVB damage by absorption of UVB rays and anti-oxidation.[Conclusions]TFFAI played a protective role on UVB irradiated cells through UVB ultraviolet absorption and antioxidant pathways.

AN OPTIMAL CONTROL PROBLEM FOR A LOTKA-VOLTERRA COMPETITION MODEL WITH CHEMO-REPULSION

In this paper we study a bilinear optimal control problem for a diffusive Lotka-Volterra competition model with chemo-repulsion in a bounded domain of ℝ^(ℕ),N=2,3.This model describes the competition of two species in which one of them avoid encounters with rivals through a chemo-repulsion mechanism.We prove the existence and uniqueness of weak-strong solutions,and then we analyze the existence of a global optimal solution for a related bilinear optimal control problem,where the control is acting on the chemical signal.Posteriorly,we derive first-order optimality conditions for local optimal solutions using the Lagrange multipliers theory.Finally,we propose a discrete approximation scheme of the optimality system based on the gradient method,which is validated with some computational experiments.

Application of the Conditional Nonlinear Optimal Perturbation Method to the Predictability Study of the Kuroshio Large Meander

A reduced-gravity barotropic shallow-water model was used to simulate the Kuroshio path variations. The results show that the model was able to capture the essential features of these path variations. We used one simulation of the model as the reference state and investigated the effects of errors in model parameters on the prediction of the transition to the Kuroshio large meander （KLM） state using the conditional nonlinear optimal parameter perturbation （CNOP-P） method. Because of their relatively large uncertainties, three model parameters were considered： the interracial friction coefficient, the wind-stress amplitude, and the lateral friction coefficient. We determined the CNOP-Ps optimized for each of these three parameters independently, and we optimized all three parameters simultaneously using the Spectral Projected Gradient 2 （SPG2） algorithm. Similarly, the impacts caused by errors in initial conditions were examined using the conditional nonlinear optimal initial perturbation （CNOP-I） method. Both the CNOP-I and CNOP-Ps can result in significant prediction errors of the KLM over a lead time of 240 days. But the prediction error caused by CNOP-I is greater than that caused by CNOP-P. The results of this study indicate not only that initial condition errors have greater effects on the prediction of the KLM than errors in model parameters but also that the latter cannot be ignored. Hence, to enhance the forecast skill of the KLM in this model, the initial conditions should first be improved, the model parameters should use the best possible estimates.

Optimization of Conditions for the Extraction of Melanin from Walnut Husk

[Objective] This study was conducted optimize the conditions for the ex- traction of melanin from walnut husk. [Method] Melanin was isolated from walnut husk through alkali extraction and acid precipitation, and the factors that may influ- ence the extraction efficiency： temperature, duration, NaOH concentration and solid- liquid ratio were set at different levels. [Result] The optimal conditions for the ex- traction of melanin from walhut husk were temperature of 80 ℃, extraction duration of 120 min, NaOH concentration of 70 mol/L, and solid-liquid ratio of 1：20. [Conclu- sion] The optimal conditions for the extraction of melanin from walnut husk deter- mined in the present study will provide references for the development and utiliza- tion of natural melanin.

CONE-DIRECTED CONTINGENT DERIVATIVES AND GENERALIZED PREINVEX SET-VALUED OPTIMIZATION

By using cone-directed contingent derivatives, the unified necessary and sufficient optimality conditions are given for weakly and strongly minimal elements respectively in generalized preinvex set-valued optimization.

A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations

A variant constrained genetic algorithm （VCGA） for effective tracking of conditional nonlinear optimal perturbations （CNOPs） is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite pararneters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has ＂on off＂ switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method （ADJ）, and a GA using tournament selection operation and the niching technique （GA-DEB） were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of ~CGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.

ε-strongly Efficient Solutions for Vector Optimization with Set-valued Maps

In locally convex Hausdorff topological vector spaces,ε-strongly efficient solutions for vector optimization with set-valued maps are discussed.Firstly,ε-strongly efficient point of set is introduced.Secondly,under the nearly cone-subconvexlike set-valued maps,the theorem of scalarization for vector optimization is obtained.Finally,optimality conditions of ε-strongly efficient solutions for vector optimization with generalized inequality constraints and equality constraints are obtained.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

Optimization of elastolysis conditions and elastolytic kinetic analysis with elastase from Bacillus licheniformis ZJUEL31410

The solubilization of elastin by Bacillus licheniformis elastase cannot be analyzed by conventional kinetic methods because the biologically relevant substrate is insoluble and the concentration of enzyme-substrate complex has no physical meaning. In this paper we report the optimization of elastolysis conditions and analysis of elastolytic kinetics. Our results indicated that the hydrolyzing temperature and time are very important factors affecting elastolysis rate. The optimized conditions using central composite design were as follows： elastolysis temperature 50 ℃, elastase concentration 1 × 10^4 U/ml, elastin 80 mg, elastolytic time 4 h. Investigation of the effects of substrate content, elastase concentration and pH was also revealed that low or high elastin content inhibits the elastolysis process. Increasingelastase improves elastin degradation, but high elastase may change the kinetics characterization. Alkaline environment can decrease elastin degradation rate and pH may affect elastolysis by changing elastase reaction pH. To further elucidate the elastolysis process, the logistic model was used to elastolysis kinetics study showing clearly that the logistic model can reasonably explain the elastolysis process, especially under lower elastase concentration. However, there is still need for more investigations with the aid of other methods, such as biochemical and molecular methods.

A New Approach for Parameter Optimization in Land Surface Model

In this study,a new parameter optimization method was used to investigate the expansion of conditional nonlinear optimal perturbation （CNOP） in a land surface model （LSM） using long-term enhanced field observations at Tongyu station in Jilin Province,China,combined with a sophisticated LSM （common land model,CoLM）.Tongyu station is a reference site of the international Coordinated Energy and Water Cycle Observations Project （CEOP） that has studied semiarid regions that have undergone desertification,salination,and degradation since late 1960s.In this study,three key land-surface parameters,namely,soil color,proportion of sand or clay in soil,and leaf-area index were chosen as parameters to be optimized.Our study comprised three experiments：First,a single-parameter optimization was performed,while the second and third experiments performed triple-and six-parameter optimizations,respectively.Notable improvements in simulating sensible heat flux （SH）,latent heat flux （LH）,soil temperature （TS）,and moisture （MS） at shallow layers were achieved using the optimized parameters.The multiple-parameter optimization experiments performed better than the single-parameter experminent.All results demonstrate that the CNOP method can be used to optimize expanded parameters in an LSM.Moreover,clear mathematical meaning,simple design structure,and rapid computability give this method great potential for further application to parameter optimization in LSMs.

Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP)

The conditional nonlinear optimal perturbation （CNOP）, which is a nonlinear generalization of the linear singular vector （LSV）, is applied in important problems of atmospheric and oceanic sciences, including ENSO predictability, targeted observations, and ensemble forecast. In this study, we investigate the computational cost of obtaining the CNOP by several methods. Differences and similarities, in terms of the computational error and cost in obtaining the CNOP, are compared among the sequential quadratic programming （SQP） algorithm, the limited memory Broyden-Fletcher-Goldfarb-Shanno （L-BFGS） algorithm, and the spectral projected gradients （SPG2） algorithm. A theoretical grassland ecosystem model and the classical Lorenz model are used as examples. Numerical results demonstrate that the computational error is acceptable with all three algorithms. The computational cost to obtain the CNOP is reduced by using the SQP algorithm. The experimental results also reveal that the L-BFGS algorithm is the most effective algorithm among the three optimization algorithms for obtaining the CNOP. The numerical results suggest a new approach and algorithm for obtaining the CNOP for a large-scale optimization problem.

KUHN-TUCKER CONDITION AND WOLFE DUALITY OF PREINVEX SET-VALUED OPTIMIZATION

The optimality Kuhn-Tucker condition and the wolfe duality for the preinvex set-valued optimization are investigated. Firstly, the concepts of alpha-order G-invex set and the alpha-order S-preinvex set-valued function were introduced, from which the properties of the corresponding contingent cone and the alpha-order contingent derivative were studied. Finally, the optimality Kuhn-Tucker condition and the Wolfe duality theorem for the alpha-order S-preinvex set-valued optimization were presented with the help of the alpha-order contingent derivative.

Application of the Conditional Nonlinear Optimal Perturbations Method in a Theoretical Grassland Ecosystem

Using a simplified nonlinearly theoretical grassland ecosystem proposed by Zeng et al.,we study the sensitivity and nonlinear instability of the grassland ecosystem to finiteamplitude initial perturbations with the approach of conditional nonlinear optimal perturbation （CNOP）.The results show that the linearly stable grassland （desert or latent desert） states can turn to be nonlinearly unstable with finite amplitude initial perturbations.When the precipitation is between the two bifurcation points,a large enough finite amplitude initial perturbation can induce a transition between the grassland statethe desert state or the latent desert.

THE OPTIMALITY CONDITIONS OF NONCONVEXSET-VALUED VECTOR OPTIMIZATION

The concepts of alpha-order Clarke's derivative, alpha-order Adjacent derivative and alpha-order G.Bouligand derivative of set-valued mappings are introduced, their properties are studied, with which the Fritz John optimality condition of set-valued vector optimization is established. Finally, under the assumption of pseudoconvexity, the optimality condition is proved to be sufficient.

Raoultella terrigena RtZG1 Electrical Performance Appraisal and System Optimization

The aims of this study are to provide a strain and its optimal application in electro-producting performance for revealing the electrical generating mechanism in the future.We constructed microbial fuel cells(MFCs)device that was a bipolar chamber MFC,using metal ion media(MIM)and water as anode and cathode reaction substation fluid,respectively.In this study,we identified an isolate as Raoultella terrigena,named RtZG1,which could produce electron.Also,we optimized the conditions of electrical energy generation.The continuous output current could reach about 200μA within 3 h when the ratio of electro-bacterial fluid to matrix fluid was 1∶4,the temperature was 37℃,the carbon-nitrogen ratio of the inorganic salt medium was 10∶1,as well as the concentration of MIM was 1.Based on the optimization,it is clear that the most suitable conditions of electricity production for this strain lay the foundation for the application of this strain.

Applications of Conditional Nonlinear Optimal Perturbation to the Study of the Stability and Sensitivity of the Jovian Atmosphere

A two-layer quasi-geostrophic model is used to study the stability and sensitivity of motions on smallscale vortices in Jupiter＇s atmosphere. Conditional nonlinear optimal perturbations （CNOPs） and linear singular vectors （LSVs） are both obtained numerically and compared in this paper. The results show that CNOPs can capture the nonlinear characteristics of motions in small-scale vortices in Jupiter＇s atmosphere and show great difference from LSVs under the condition that the initial constraint condition is large or the optimization time is not very short or both. Besides, in some basic states, local CNOPs are found. The pattern of LSV is more similar to local CNOP than global CNOP in some cases. The elementary application of the method of CNOP to the Jovian atmosphere helps us to explore the stability of variousscale motions of Jupiter＇s atmosphere and to compare the stability of motions in Jupiter＇s atmosphere and Earth＇s atmosphere further.

Generalized cone-subconvexlike set-valued maps and applications to vector optimization

The definitions of cone-subconvexlike set-valued maps and generalized cone-subconvexlike set-valued maps in topological vector spaces are defined by using the relative interiors of ordering cone. The relationships between the two classes of set-valued maps are investigated, and some properties of them are shown. A Gordan type alternative theorem under the assumption of generalized cone-subconvexlikeness of set-valued maps is proved by applying convex separation theorems involving the relative interiors in infinite dimensional spaces. Finally a necessary optimality condition theorem is shown for a general kind of set-valued vector optimization in a sense of weak E-minimizer.

Optimality of Vector Optimization Involving Generalized Type-I Functions in Banach

In this note,new classes of generalized type-I functions are introduced for functions between Banach spaces.These generalized type-I functions are then utilized to establish sufficient optimality conditions and duality results for a vector optimization problem with functions defined on a Banach space.