Convex-concave extensions (Q1577720)

Interval arithmetic provides various techniques to construct upper and lower bounds of function values \(f(x)\), \(x\in X\) where \(X\) is a \(n\)-dimensional box. Well-known means for obtaining these bounds are interval extensions. The bounds, however, can be quite crude. The paper under review considers so-called convex-concave extensions of \(f\). They are obtained by splitting \(f\) into two summands, \(f= c+r\), where \(c\) is a convex function and \(r\) is a smooth function. (A more refined version is also discussed, which subdivides \(f\) into three summands.) Then appropriate interval extensions of the summands over \(X\) give lower convex bounds for \(f\). Analogously, upper concave bounds for \(f\) can be found. In general, these bounds are better than bounds that arise from standard interval extensions of \(f\) directly. Numerical examples, which include a large scale constraint optimization problem, demonstrate the practical use of convex-concave extensions.