Characterizations of Pseudolinear and Semi-strictly Quasilinear Functions
Vsevolod I. Ivanov
TL;DR
The paper develops a comprehensive framework for characterizing pseudolinear and semistrictly quasilinear functions in Banach spaces using Clarke-Rockafellar derivatives. It delivers a derivative-free complete characterization via a positive function $p(x,y,x^*)$ satisfying $f(y)-f(x)=p(x,y,x^*)\langle x^*,y-x\rangle$, and a Lipschitz-based derivative-free form with $b(x,y,\lambda)$ that yields $0<\lambda b(x,y,\lambda)<1$ and independence from subgradients. It further links semistrictly quasilinear to pseudolinear by showing the necessary conditions become sufficient under semistrictly quasilinear, and provides both Fréchet differentiable and subdifferential-based criteria. The results unify and extend prior work on pseudoconvexity and quasiconvex/quasiconcave structures, offering practical, verifiable criteria for pseudolinearity in nonsmooth settings with potential applications in optimization theory.
Abstract
In this paper we obtain several new complete characterizations of pseudolinear functions. Two of the results are of first-order and one is derivative free. All results are derived in terms of the Clarke-Rockafellar subdifferential. Additionally, we prove a characterization of the semi-strictly quasilinear functions. It is similar to the derivative free characterization of the pseudolinear functions. We also find the conditions such that a semi-strictly quasilinear function become pseudolinear.