Efficient smooth minorants for global optimization of univariate functions with the first derivative satisfying the interval Lipschitz condition

Abstract

AbstractIn 1998, the paper Sergeyev (Math Program 81(1):127–146, 1998) has been published where a smooth piece-wise quadratic minorant has been proposed for multiextremal functions f(x) with the first derivative $$f'(x)$$ f ′ ( x ) satisfying the Lipschitz condition with a constant L, i.e., $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than L and decrease with the slope smaller than $$-L$$ - L . This minorant has been successfully applied in several efficient global optimization algorithms and used in engineering applications. In the present paper, it is supposed that the first derivative $$f'(x)$$ f ′ ( x ) cannot increase with the slope higher than a constant $$\beta $$ β and decrease with the slope smaller than $$\alpha $$ α . The interval $$[\alpha ,\beta ]$$ [ α , β ] is called the Lipschitz interval (clearly, in this case the Lipschitz constant $$L = \max \{|\alpha |, |\beta | \}$$ L = max { | α | , | β | } ). For this class of functions, smooth piece-wise estimators (minorants and majorants) have been proposed and applied in global optimization. Both theoretically and experimentally (on 200 randomly generated test problems) it has been shown that in cases where $$ |\alpha | \ne |\beta |$$ | α | ≠ | β | the new estimators can give a significant improvement w.r.t. those proposed in Sergeyev (Math Program 81(1):127–146, 1998), for example, in the framework of branch-and-bound global optimization methods.