The filled function transformations for constrained global optimization
The purpose of this paper is to introduce the main idea of the filled function transformation. One shows that there exist several ways to make global optimization (1) global-min f(x) subject to \(c_ i(x)=0\), \(i=1,...,m'\); \(c_ i(x)\geq 0\), \(i=m'+1,...,m.\) The first is to use the exact penalty function \[ P_ e(x,\mu)=f(x)+\sum^{m'}_{i=1}\mu_ i| c_ i(x)| +\sum^{m}_{i=m'+1}\mu_ i\max [0,-c_ i(x)], \] to transform the problem (1) to an unconstrained global minimization problem. The second one constructs filled functions for problem (1) with inequality constraints only. For this, one divides \(R^ n\) into two parts: \(S_ 1=\{x/f(x)\geq f(x^*_ 1)\) or \(x\not\in \Omega_ f\}\), \(S_ 2=\{x/f(x)<f(x^*_ 1)\) and \(x\in \Omega_ f\}\), \((\Omega_ f\) is the feasible region of problem (1)) and one finds a filled function transformation \[ \bar U(x,A,h)=\eta (\| x-x_ 0\|)\phi [A(\max \{[f(x)-f(x^*_ 1)], \quad \max [-c_ i(x)]\}+h)], \] which has no stationary point in \(S_ 1\), but does have a minimizer in \(S_ 2.\) The third way is to use a square penalty function \[ F(x,\mu)=f(x)+\sum^{m'}_{i=1}\mu_ i[c_ i(x)]^ 2+\sum^{m}_{i=m'+1}\mu_ i[\max (0,c_ i(x))]^ 2 \] to transform the problem (1) to a sequence of unconstrained global optimization problems. The fourth way is to solve a sequence of new constrained local minimization problems with filled functions and a part of the original constraints.
unconstrained global minimization problem
scientific article; zbMATH DE number 4176370
scientific article