In many practical application problems, the existence of uncertainty has an impact on the performance of the optimal solution to an optimization problem. When solving optimization problems in uncertain environments, it is often necessary to consider the robustness of the solution. The definition of robustness of an optimal solution usually considers the performance of all solutions in its local neighborhood. In the context of multiobjective optimization, it is a very challenging task to approximate the robust Pareto fronts. Existing robust multi-objective evolutionary algorithms (MOEA) can handle low-dimensional robust multi-objective optimization problems (MOPs), i.e., the dimensionality of the decision variables is less than 10, but often perform poorly for high-dimensional robust MOPs. In this paper, we propose an MOEA, called MOEA/D-AECC (Decomposition-based Multiobjective Evolutionary \mbox{Algorithm} Assisted by Autoencoder and Cooperative Coevolution, MOEA/D-AECC), which combines \mbox{autoencoder} as well as co-evolutionary methods, for solving high-dimensional robust MOPs with low eff-ective dimension. The algorithm utilizes two different populations to optimize the original MOPs and the corresponding robust MOPs, respectively. To improve the ability to handle high-dimensional problems, the algorithm utilizes an autoencoder for dimensionality reduction in order to extract the low-dimensional features of the high-dimensional data. The descent directions are learned by reconstructing these low-dimensional features, then new \mbox{solutions} are generated by sampling along these descent directions. Finally, the performance of MOEA/D-AECC is tested on a set of high-dimensional robust MOPs with low effective dimensions in this paper. The experimental results show that the performance of MOEA/D-AECC is significantly better than several other representative robust MOEA.

As a special kind of linear system, the three-order block saddle point problem has challenging to study its iterative solution. Based on the classical generalized successive over relaxation (GSOR) method, the centered preconditioned GSOR method with three parameters for a class of three-order block large sparse saddle point problem is established and the convergence condition is discussed in this paper. Moreover, experimental results show that the new method has an advantage of computational cost over the centered preconditioned Uzawa-Low method. In addition, an extended one of the new method is provided, implementation details and analyses of corresponding framework about $i$-order block systems are shown, the blocking for saddle point problems are preliminarily proposed by some numerical results.

Tunneling function method is an effective method for global optimization problems. Its ability to jump out of the local minimizer is deeply affected by the properties of the tunneling function. With the complexity of practical optimization problems, the corresponding forms of tunneling functions become more complex. Therefore, constructing tunneling functions with simple form and good properties is one of the main research objectives of the tunneling function method. In order to improve the efficiency of the tunneling function method for solving multi-modal functions, a new tunneling function is constructed. Its minimizers are not only the better feasible points of the objective function than the current local minimizer, but also the better local minimizers, i.e., the  tunneling function and the objective function have the same local minimizers. Thus, the better local minimizer of the objective function can be obtained directly by minimizing the tunneling function. Based on this feature, a new tunneling function algorithm is designed. It changes the frame of conventional tunneling function methods that objective function and tunneling function are minimized alternately, and can effectively reduce the iterations of local optimization and accelerate the speed of global optimization. Theoretical analysis and numerical experiments exhibit the feasibility and effectiveness of the algorithm.

With the rapid economic development of China, the usage of mineral resources continues to grow and pollution of the environment is also increasing. As a result, it is an important measure for China to develop wind power generation so as to realize low-carbon transformation. However, due to the strong instability of wind power generation, it brings greater uncertainty to the operation of the power grid. Therefore, this paper takes into account the uncertainty in the wind power generation process and models them to carry out the power grid planning. Firstly, this paper establishes a mathematical model of wind turbine output by modeling the uncertainty. Secondly, it proposes an optimal power flow model considering the uncertainty of wind power with the objective to simultaneously minimize the total cost and the total network loss. At the same time, an improved particle swarm optimization algorithm is proposed to solve the problem by using a local model and introducing dynamic inertia weight coefficients. By comparing it with the traditional particle swarm optimization algorithm with some real-world data, the novel algorithm is verified to have better performance in terms of solving speed, convergence and robustness.

With the extensive utilization of wind energy resources, more and more attention has been paid to the efficiency and safety of wind penetration. Using statistical methods to analyze the characteristics of wind speed, we can know the change rule of wind energy, so as to make reasonable distribution and scheduling of electricity, and to maximize the reduction of losses and risks. At present, there are a large number of literatures on the use of energy storage system to suppress wind farm output, but there are few researches on the introduction of conditional VaR (Value-at-Risk) as an evaluation index into wind energy resource assessment. Aiming at sloving the instability of wind speed and the serious problem of wind curtailment and power limitation, this paper proposes to configure energy storage devices with a certain capacity at the grid-connected wind farms. In the proposed method, the conditional risk value of the system is taken as the optimization objective, and energy storage devices are adopted to control the charging and discharging of wind farms, so as to achieve the purpose of stabilizing wind power fluctuations and reducing wind curtailment. We analyze the calming effect of energy storage system on wind power fluctuation through real-world examples. The quantitative and visual results show that the proposed method is of great significance in wind power peaking and valley filling.

Drug controlled release system is a kind of controlled release system which can achieve specific drug release target by adjusting some design parameters. In order to optimize the initial concentration of drug controlled release system based on time-fractional order diff-usion equation, B-spline wavelet method is used to solve the forward problem. The niche cuckoo search algorithm, which combines the niche strategy with the cuckoo search strategy, is applied to optimize the initial drug concentrations at different fractions, thus approximately achieving the three expected drug release targets. For the solution of the forward problem, an iterative solution scheme of B-spline wavelet method is established by combining the definition of the Caputo derivative and the cubic B-spline scaling function. The problem of optimal design of drug controlled release system is reduced to the problem of parameter identification of fractional diffusion equation based on the inverse problem solution scheme. In order to realize parameter inversion control, the niche cuckoo search optimization algorithm is introduced to inverse the initial concentration in the controlled release system, which effectively solves the problem. The niche cuckoo search algorithm is easily trapped in local extremum. The optimal control parameters are given based on the proposed algorithm for three kinds of release targets: constant velocity release, linear reduced release and nonlinear release. Numerical results show the effectiveness of the proposed method.

As for a two-dimensional convection-diffusion equation in the singular perturbation, a novel multiscale finite element method based on the optimally graded meshes is proposed. The multiscale finite element method just solves the sub-problems on coarse meshes, and the data mapping relationship for related scales is provided in details and the microscopic information is inherited to the macroscopic level. Then the matrix is reduced and its matrix equation is ready for solving efficiently. Based on the perturbed parameter, an adaptively graded mesh is constructed from its iterative formula, and the meshes are capable of approximating the boundary layers effectively. Through mathematical analyses and numerical experiments, to contrast the computational cost and execution time, the multiscale strategy on the graded mesh is validated to be the stable, high-order and short-time uniform convergence. Its computational efficiency and application advantage are prominent.

Firstly, the extended Stroh formulism is used to study the Green's function of an infinite octagonal two-dimensional quasicrystals plate with an elliptical hole. And the fundamental solution of the boundary element method for solving the finite octagonal two-dimensional quasicrystals plate with an elliptical hole is obtained. Secondly, the boundary integral equation is established by using the weighted residual method. The boundary integral equation with unknown quantity is discretized by linear interpolation function and Gauss integral, and the discrete format is reorganized to form a linear system of equations with unified variables. Finally, the hole edge stress of the elliptical hole is numerically solved, and the numerical results of the finite plate are compared with the analytical solution of the infinite plate to verify the effectiveness of the boundary element method. The influence of the size of the plate, the size of the hole and the inclination angle on the stress at the edge of the hole under vertical tension are further analyzed. The results of numerical examples show that the stress concentration at the edge of the hole is more obvious with the increase of the size of the elliptical hole. The inclination of the elliptical hole aggravates the stress concentration at the edge of the hole. The stress intensity factor at the crack tip increases with the growth of the crack.

This paper investigates an optimal reinsurance-investment problem with HARA utility. In order to avoid claim risks, the insurer is allowed to purchase reinsurance, and is assumed to invest in one risk-free asset and one risky asset whose instantaneous rate is governed by an Ornstein-Uhlenbeck (O-U) process, which could describe the features of bull and bear markets. Firstly, under the criterion of maximizing the expected HARA utility of the insurer's terminal wealth, the HJB equation for the value function is obtained by applying dynamic programming principle. Secondly, due to the complexity of the structure of HARA utility, we use Legendre transform to change the original HJB equation into its dual one, whose solution is easy to conjecture. Closed-form solution of optimal investment-reinsurance strategy is obtained by constructing the solution form of the dual equation and the variable change technique. Finally, some numerical simulations are presented to illustrate the impacts of model parameters on the optimal reinsurance-investment strategy.

Different from RMB deposit, structured RMB deposit is a financial product that enables investors to obtain higher returns than RMB deposit based on taking certain risks. Under the framework of uncertainty theory, this paper deals with the pricing of break-even stock index-linked structured RMB deposit products without a yield ceiling and stock index-linked structured RMB deposit products, and gives corresponding numerical examples. Finally, Matlab software is used to analyze the risks of these two financial products and provide reasonable suggestions for investors of these products.

The single-machine scheduling problems of due window assignment with position-dependent weights are investigated with the aim of minimizing the total weighted sum of the starting time, the size of due window and total tardiness of  due window in a just-in-time environment. The goal is to find the optimal job processing sequence and due window starting time $d_k^1$ (finishing time $d_k^2$). Under common, slack and different due window assignments, the corresponding optimal properties are obtained from the theoretical analysis, it is proved that the problems can be solved in polynomial-time. For the common and slack due window assignments, the time complexity is $O(n^2 \log n)$, while the different due window scheduling problem can be solved in $O(n \log n)$ time, where $n$ is the number of given jobs.

Multi-band wavelets have applied in many areas such as signal processing and numerical analysis due to their richer parameter space to have a more flexible time-frequency tiling, to give better energy compaction than 2-band wavelets. The sub-band operators are studied, an optimization model is built, and the wavelets with the smallest norm of the sub-band operator can be selected from the symmetric multi-band biorthogonal wavelets with free parameters, which can be applied in digital image processing based on wavelet theory and its fast algorithms. Firstly, the sub-band operator which is an infinite-dimensional matrix, is introduced, circular matrix theory is developed, and the norm of a sub-band operator associated to four-band wavelets with fast calculation structure is obtained. It is easy to obtain the norm of the sub-band operator with some structure by construction a function and computing the maximum. Secondly, a model to minimize the norm is built and four-band biorthogonal wavelets filter bands with fast calculation structure are designed. Lastly, an example is provided to illustrate the proposed results.

By means of critical point theory, we investigate homoclinic solution problems for the Kirchhoff-type difference equations with periodic coefficients. First, we verify that the graph of the energy functional satisfies the mountain pass geometrical properties. Such mountain pass geometry produces a Palais-Smale sequence. Second, we exploit one global property condition to guarantee that this Palais-Smale sequence is bounded. Further, by using the subset of $l^{2}$ consisting of functions with compact support and periodicity of coefficients, we obtain the existence of one nontrivial homoclinic solution for the Kirchhoff-type difference equations with periodic coefficients. Finally, two examples are given to illustrate our main results.

Both positivity-preserving and iteration acceleration techniques are key issues when employing numerical methods to solve neutron transport equation. An algorithm combining $S_2$ synthetic acceleration method and positivity-preserving techniques to solve neutron transport equation that defined in one-dimensional planar geometry. The linear discontinuous Galerkin methods are applied to solve both the transport equation and $S_2$ equations. Two kind of simple limiters are involved. The numerical results for solving the classical Reed problem testify the efficiency of our methods.