On an approach to construct continuation algorithms in global optimization (Q1348544)

The problem of global minimization \[ F(x) \to \min_{x\in X}, \] where \(F(\dots)\) is a twice continuously differentiable function on \(X\), \[ X= \prod^{(k)}=\{x=(x_1,\dots ,x_k)\in {R_k} \mid 0\leq x_i \leq 1, i=1,\dots ,k\}=\bigotimes_{n=1}^k [0. 1]. \] is considered. The following notations are introduced \[ X^{\text{opt}}=\arg\min_{x\in X} F(x), \] \[ X^{\text{lopt}}=\{x\in X \mid \exists \varepsilon>0:F(x)\leq F(x'), \forall x'\ni B_{\varepsilon}(x)\;X\}, \] \[ B_{\varepsilon}(x)= \{x'\in {R_k}\mid \quad \|x'- x\|<\varepsilon \}, \] where \(X^{\text{opt}}\), \(X^{\text{lopt}}\) are the set of optimal solutions and the set of locally-optimal solutions of the stated global optimization problem, correspondingly. To be precise the problem consists in searching a point \(x\in X^{\text{opt}}\) where \(X^{\text{lopt}}\neq X^{\text{opt}}.\) There are many methods of solving the stated problem and the continuation method is one of the important ones. Usually the continuation method is used for a wide class of minimization problems. As a result the efficiency of the proper algorithms is low. Contrary to this in the article a highly specialized class of global minimization problems is considered and the continuation algorithms are constructed that essentially use the specific features of the problem disscused.