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

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