Dynamic Threshold Optimization - A New Approach?

Dynamic Threshold Optimization (DTO) adaptively “compresses” the decision space (DS) in a global search and optimization problem by bounding the objective function from below. This approach is different from “shrinking” DS by reducing bounds on the decision variables. DTO is applied to Schwefel’s Problem 2.26 in 2 and 30 dimensions with good results. DTO is universally applicable, and the author believes it may be a novel approach to global search and optimization. Problem Statement In a bounded hyperspace = Ω : { max min | i i i x x x x ≤ ≤ r , d N i ,..., 1 = } (decision space), the i x being decision variables, determine the locations and values of the global maxima of the objective function ) ,..., , ( 2 1 d N x x x f , that is, max{ d N x x f R ⊂ Ω ∈ r r : ) ( , R → R ⊂ Ω d N f : }. The value of ) (x f r at each point x r is its “fitness.” Dynamic Threshold Optimization DTO is conceptually quite simple. Figure 1 is a schematic illustration of how it works in a one-dimensional (1-D) DS. Objective function ) (x f is multimodal with many local maxima and a single global maximum, and the problem is to locate that maximum (coordinates and value). DTO bounds ) (x f from below using a series of successively increasing “thresholds,” in effect compressing DS in the direction of the dependent variable (from “below”) instead of, as is sometimes done, shrinking DS by reducing the independent variable’s domain (from the “sides”). Locating the global maximum is easier in the compressed DS because unwanted local maxima are progressively filtered out as the “floor” (threshold) rises. Because DTO is a general geometric technique, it is algorithm-independent so that it can be used with any global search and optimization algorithm. Although DTO is described in the context of maximization, it can be applied to minimization with obvious modifications because ) ( min ) ( max x f x f r r − = . Procedure ] *, *, ), ( [ min q q x x q OPT r r is a global search and optimization routine that returns (i) the d N coordinates * x r of a maximum of function ) (x q r , (ii) its value * q , and (iii) a minimum value min q (no coordinates). ] [⋅ OPT may comprise any search and optimization algorithm (singly or in combination with others) regardless of its type, deterministic, stochastic, or hybrid; and different algorithms may be used on successive calls to ] [⋅ OPT . DTO is initialized by applying ] [⋅ OPT to ) (x f r without any threshold. Its return values then are used to define a starting threshold that subsequently is updated by applying ] [⋅ OPT to the auxiliary function