This paper presents a hybrid heuristic-triangle evolution (TE) for global optimization. It is a real coded evolutionary algorithm. As in differential evolution (DE), TE targets each individual in current population and attempts to replace it by a new better individual. However, the way of generating new individuals is different. TE generates new individuals in a Nelder-Mead way, while the simplices used in TE is 1 or 2 dimensional. The proposed algorithm is very easy to use and efficient for global optimization problems with continuous variables. Moreover, it requires only one (explicit) control parameter. Numerical results show that the new algorithm is comparable with DE for low dimensional problems but it outperforms DE for high dimensional problems.
In this paper, the rotated cone fitting problem is considered. In case the measured data are generally accurate and it is needed to fit the surface within expected error bound, it is more appropriate to use l∞ norm than l2 norm. l∞ fitting rotated cones need to minimize, under some bound constraints, the maximum function of some nonsmooth functions involving both absolute value and square root functions. Although this is a low dimensional problem, in some practical application, it is needed to fitting large amount of cones repeatedly, moreover, when large amount of measured data are to be fitted to one rotated cone, the number of components in the maximum function is large. So it is necessary to develop efficient solution methods. To solve such optimization problems efficiently, a truncated smoothing Newton method is presented. At first, combining aggregate smoothing technique to the maximum function as well as the absolute value function and a smoothing function to the square root function, a monotonic and uniform smooth approximation to the objective function is constructed. Using the smooth approximation, a smoothing Newton method can be used to solve the problem. Then, to reduce the computation cost, a truncated aggregate smoothing technique is applied to give the truncated smoothing Newton method, such that only a small subset of component functions are aggregated in each iteration point and hence the computation cost is considerably reduced.
