Adomian’s decomposition method (ADM) is a nonnumerical method which can be adapted for solving nonlinear ordinary differential equations. In this paper, the principle of the decomposition method is described, and its advantages as well as drawbacks are discussed. Then an aftertreatment technique (AT) is proposed, which yields the analytic approximate solution with fast convergence rate and high accurscy through the application of Padé approximation to the series solution derived from ADM. Some concrete examples are also studied to show with numerical results how the AT works efficiently.

The decomposition method is a nonnumerical method of solving strongly nonlinear differential equations. In this paper, the method is adapted for the solution of the one-dimensional nonlinear Poisson’s equations governing the linearly graded p-n junctions in semiconductor devices, and the error analysis for the approximate analytic solutions obtained by the decomposition method is carried out. The simulation results show that the solutions obtained by the method are accurate and reliable, and that the quantitative analysis of the linearly graded p-n junctions can be conducted. This work indicates that the decomposition method has some advantages, which opens up a new way for the numerical analysis of semiconductor devices.

In this two-part series of papers, a new generalized minimax optimization model, termed variable programming (VP), is developed to solve dynamically a class of multi-objective optimization problems with non-decomposable structure. It is demonstrated that such type of problems is more general than existing optimization models. In this part, the VP model is proposed first, and the relationship between variable programming and the general constrained nonlinear programming is established. To illustrate its practicality, problems on investment and the low-side-lobe conformal antenna array pattern synthesis to which VP can be appropriately applied are discussed for substantiation. Then, theoretical underpinnings of the VP problems are established. Difficulties in dealing with the VP problems are discussed. With some mild assumptions, the necessary conditions for the unconstrained VP problems with arbitrary and specific activated feasible sets are derived respectively. The necessary conditions for the corresponding constrained VP problems with the mild hypotheses are also examined. Whilst discussion in this part is concentrated on the formulation of the VP model and its theoretical underpinnings, construction of solution algorithms is discussed in Part II.

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.

The application of the new nonlinear optimization algorithms to arrays design is demonstrated by the low-side-lobe pattern synthesis of conformal arrays. By adopting the information of array element realized gain patterns (RGP’s), we formulate synthesis problems as nonlinearly constrained optimization problems, and propose a new direct method to solve them. The technique allows one to find a new direct method to solve them. The technique allows one to find a set of array coefficients that yield a pattern meeting a specified side-lobe level and achieving the maximum directivity, if such a set of array coefficients exists. If the side-lobe specifications cannot be met with the given array, the technique will result in a set of coefficients that yield a pattern meeting the best attainable side-lobe level and having directivity as high as possible. Also, simplified synthesis problems for an axial dipole array in an infinite, perfectly conducting cylinder are discussed, and numerical synthesis results are given. Our synthesis technique works for general array patterns.