The impelling function method applied to global optimization (Q1827019)

The author considers a global constrained oprimization problem such as the following \[ \text{Glob.Min} \,f(X),\quad\text{subject to }g(X)\geq 0. \] He first uses the penalty function method (PFM) for transforming this problem into an unconstrained optimization problem where he sets \[ F(X,r)= f(X)+ r \sum^m_{i=1} \{\min(0, g_i(X))\}^2, \] \(r\geq 0\) is a penalty factor. Then the PFM concept is extended to global optimization. An auxiliary function \(M(X)\) is proposed such that it decreases in the region \(G(X)\underset{\text{def}}{}= \{X/f(X)\geq f(X_1)\}\) where \(X_1\) is the known best local minimizer and has a stationary point in \(\{X/f(X)< f(X_1)\}\). For instance the paper suggests the impelling function \[ M(X)= a\{\min[0, f(X)- f(X_1)]\}^{2m}-\| X- X_1\|^p, \] where \(m\) and \(p\) are positive natural numbers (partically \(m= 1\), \(p= 2\)). The following of this interesting paper gives theoretical results for convergence towards a global minimum.