Application of a Derivative-Free Method with Projection Skill to Solve an Optimization Problem

Improving numerical forecasting skill in the atmospheric and oceanic sciences by solving optimization problems is an important issue. One such method is to compute the conditional nonlinear optimal perturbation(CNOP), which has been applied widely in predictability studies. In this study, the Differential Evolution(DE) algorithm, which is a derivative-free algorithm and has been applied to obtain CNOPs for exploring the uncertainty of terrestrial ecosystem processes, was employed to obtain the CNOPs for finite-dimensional optimization problems with ball constraint conditions using Burgers' equation. The aim was first to test if the CNOP calculated by the DE algorithm is similar to that computed by traditional optimization algorithms, such as the Spectral Projected Gradient(SPG2) algorithm. The second motive was to supply a possible route through which the CNOP approach can be applied in predictability studies in the atmospheric and oceanic sciences without obtaining a model adjoint system, or for optimization problems with non-differentiable cost functions. A projection skill was first explanted to the DE algorithm to calculate the CNOPs. To validate the algorithm, the SPG2 algorithm was also applied to obtain the CNOPs for the same optimization problems. The results showed that the CNOPs obtained by the DE algorithm were nearly the same as those obtained by the SPG2 algorithm in terms of their spatial distributions and nonlinear evolutions. The implication is that the DE algorithm could be employed to calculate the optimal values of optimization problems, especially for non-differentiable and nonlinear optimization problems associated with the atmospheric and oceanic sciences.

A multi-objective model for cordon-based congestion pricing schemes with nonlinear distance tolls

Congestion pricing is an important component of urban intelligent transport system.The efficiency,equity and the environmental impacts associated with road pricing schemes are key issues that should be considered before such schemes are implemented.This paper focuses on the cordon-based pricing with distance tolls,where the tolls are determined by a nonlinear function of a vehicles' travel distance within a cordon,termed as toll charge function.The optimal tolls can give rise to:1) higher total social benefits,2) better levels of equity,and 3) reduced environmental impacts(e.g.,less emission).Firstly,a deterministic equilibrium(DUE) model with elastic demand is presented to evaluate any given toll charge function.The distance tolls are non-additive,thus a modified path-based gradient projection algorithm is developed to solve the DUE model.Then,to quantitatively measure the equity level of each toll charge function,the Gini coefficient is adopted to measure the equity level of the flows in the entire transport network based on equilibrium flows.The total emission level is used to reflect the impacts of distance tolls on the environment.With these two indexes/measurements for the efficiency,equity and environmental issues as well as the DUE model,a multi-objective bi-level programming model is then developed to determine optimal distance tolls.The multi-objective model is converted to a single level model using the goal programming.A genetic algorithm(GA) is adopted to determine solutions.Finally,a numerical example is presented to verify the methodology.

NEW ALGORITHM FOR FIR FILTER DESIGN WITH DISCRETE COEFFICIENTS

This paper develops a new algorithm based on the Projected Gradient Algorithm (PGA) for the design of FIR digital filters with "sum of power of two" coefficients. It is shown that the integer programming involved in the FIR filter design can be solved by this algorithm. It is compared with the reported method for a SemiDefinite Programming (SDP) relaxation- based design. The simulations demonstrate that the new algorithm often yields the similar error performances of the FIR filter design, but the average CPU time of this approach is significantly reduced.

An active set algorithm for nonlinear optimization with polyhedral constraints

A polyhedral active set algorithm PASA is developed for solving a nonlinear optimization problem whose feasible set is a polyhedron. Phase one of the algorithm is the gradient projection method, while phase two is any algorithm for solving a linearly constrained optimization problem. Rules are provided for branching between the two phases. Global convergence to a stationary point is established, while asymptotically PASA performs only phase two when either a nondegeneracy assumption holds, or the active constraints are linearly independent and a strong second-order sufficient optimality condition holds.