At present, there are a few methods of multimodal optimization technique, which are used in order to find a solution case by case. The global search techniques which are used in all of these methods are usually nonsequential ones. In this paper, we make an attempt at using only a sequential technique in this global search.
As is commonly known, a necessary condition for the differentiable scalar function to become optimal is that its first derivative is equal to zero. When the function is defined over on n-dimensional space, the first derivatives generate hypersurfaces. A point which is any intersection of n hypersurfaces of derivatives is defined as a pole. The intersection of n-1 hypersurfaces, taking off any one from these n hypersurfaces, forms a curve, which is defined as a pole line. A curve evidently links a pole with another. The principle of the method proposed in this paper is to search sequentially these poles using the network of these pole lines. The existence and uniqueness of the pole line, the relation between the pole line and the pole, and the selecting of a resolution from poles are proved.
This method needs that the performance measure is of the class C2 because of the existence of continuous hypersurfaces, and furthermore that some unequality constraints are of the class C1 and close the feasible domain because of closing the network of pole lines. Otherwise, it may occur that an optimal solution is not found in this method, for all of extremums found in this method are connected with a certain network of pole lines. In particular, it is impossible to find the solution which exists on the boundary of unequality constraints, but in this case the method finding it using Lagrange multiplier is briefly described. Furthermore, the convergence of solutions in this method under certain conditions and two simple experimentations are reported.