A New Strategy for Solving a Class of Constrained Nonlinear Optimization Problems Related to Weather and Climate Predictability

There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.

Convergence analysis of a nonlinear Lagrange algorithm for general nonlinear constrained optimization problems

The convergence analysis of a nonlinear Lagrange algorithm for solving nonlinear constrained optimization problems with both inequality and equality constraints is explored in detail. The estimates for the derivatives of the multiplier mapping and the solution mapping of the proposed algorithm are discussed via the technique of the singular value decomposition of matrix. Based on the estimates, the local convergence results and the rate of convergence of the algorithm are presented when the penalty parameter is less than a threshold under a set of suitable conditions on problem functions. Furthermore, the condition number of the Hessian of the nonlinear Lagrange function with respect to the decision variables is analyzed, which is closely related to efficiency of the algorithm. Finally, the preliminary numericM results for several typical test problems are reported.

A Numerical Embedding Method for Solving the Nonlinear Optimization Problem

A numerical embedding method was proposed for solving the nonlinear optimization problem. By using the nonsmooth theory, the existence and the continuation of the following path for the corresponding homotopy equations were proved. Therefore the basic theory for the algorithm of the numerical embedding method for solving the non-linear optimization problem was established. Based on the theoretical results, a numerical embedding algorithm was designed for solving the nonlinear optimization problem, and prove its convergence carefully. Numerical experiments show that the algorithm is effective.

AN SQP ALGORITHM WITH NONMONOTONE LINE SEARCHFOR GENERAL NONLINEAR CONSTRAINED OPTIMIZATION PROBLEM

In this paper, an SQP type algorithm with a new nonmonotone line search technique for general constrained optimization problems is presented. The new algorithm does not have to solve the second order correction subproblems for each iterations, but still can circumvent the so-called Maratos effect. The algorithm's global convergence and superlinear convergent rate have been proved. In addition, we can prove that, after a few iterations, correction subproblems need not be solved, so computation amount of the algorithm will be decreased much more. Numerical experiments show that the new algorithm is effective.

The Behaviors of Optimal Precursors during Wintertime Eurasian Blocking Onset

In this paper the optimal precursors for wintertime Eurasian blocking onset are acquired by solving a nonlinear optimization problem whose objective function is constructed based on a blocking index with a triangular T21, three-level, quasi-geostrophic global spectral model. The winter climatological state is chosen as the reference basic state. Numerical results show that the optimal precursors are characterized by a baroclinic pattern with a westward tilt with height, which are mainly located upstream of the blocking region. For an optimization time of 5 days, these perturbations are mainly localized over the Northeast Atlantic Ocean and continental Europe. With the extension of the optimization time to 8 days, these perturbations are distributed more upstream and extensively in the zonal direction. Wave spectrum analysis reveals that the optimal precursors are composed of not only synoptic-scale （wave numbers 5-18） waves, but planetary-scale （wave numbers 0-4） waves as well. The synoptic-scale optimal precursors are mainly located in the mid-latitude area, while the planetary-scale optimal precursors focus primarily on the high- latitude region. The formation of a strong planetary-scale positive blocking anomaly is accompanied by the reinforcement of synoptic-scale perturbations and further fragmentation into two branches, in which the northern branch is generally stronger than the southern one. The eddy forcing arising from the selfinteraction of synoptic-scale disturbances is shown to be crucial in triggering the dipole blocking anomaly, and the planetary-scale optimal precursor provides the initial favorable background conditions for blocking onset.

Solving infinite horizon nonlinear optimal control problems using an extended modal series method

This paper presents a new approach for solving a class of infinite horizon nonlinear optimal control problems (OCPs).In this approach,a nonlinear two-point boundary value problem (TPBVP),derived from Pontryagin's maximum principle,is transformed into a sequence of linear time-invariant TPBVPs.Solving the latter problems in a recursive manner provides the optimal control law and the optimal trajectory in the form of uniformly convergent series.Hence,to obtain the optimal solution,only the techniques for solving linear ordinary differential equations are employed.An efficient algorithm is also presented,which has low computational complexity and a fast convergence rate.Just a few iterations are required to find an accurate enough suboptimal trajectory-control pair for the nonlinear OCP.The results not only demonstrate the efficiency,simplicity,and high accuracy of the suggested approach,but also indicate its effectiveness in practical use.