Locally convex-regions approximation using an incremental quadratic-based fuzzy clustering

Abstract

Choosing an appropriate local optimal region in order to satisfy the location priorities and to guarantee enough robustness against measurement biases is desired in many optimization problems. To fulfill such aim, all locally convex regions which potentially contain optimal points must be approximated. In this paper, using a quadratic-based fuzzy clustering, approximation of locally convex regions of Multiple-Convex Functions (MCFs) is intended. At first, using an incremental fuzzy clustering approach, the input space is partitioned as hyper-rectangle regions in which Locally Quadratic Models (LQMs) are identified. Based on the Hessian matrices of LQMs, some clusters, that potentially contain convex regions, are chosen. Around a certain patch of each chosen cluster, a high-order model is fitted, through which a Gradient-based Ordinal Differential Equation (GODE) is defined. Estimating the domain of attraction of each defined GODE, a locally convex region is approximated. Then, robustness of the approximated convex regions, against unknown bounded biases of input variables, is discussed. A theorem is stated which conservatively determines the sub-regions remaining convex even in presence of the uncertainty. To explain the methodology of the proposed method, an illustrative example is given. Then, the suggested method is applied to the power economic dispatch (PED) problem. The achieved results demonstrate the capability of the proposed method.