First-Order Characterizations of Strong Quasiconvexity and a Quasimonotonicity Gap
We study first-order conditions for Gâteaux differentiable strongly quasiconvex functions on open convex subsets of normed spaces. First, we prove that the value-order gradient condition characterizes strong quasiconvexity with the same modulus $σ$, improving the modulus $σ/2$ of Vladimirov, Nesterov, and Chekanov and of Hadjisavvas and Lara. Second, we answer negatively a question of Lara, Marcavillaca, and Vuong: for every $σ>0$ there is a $C^\infty$ function on the real line that satisfies their generalized quasimonotonicity condition with modulus $σ$, even in its nonstrict form, yet is strongly quasiconvex with no positive modulus. Third, we show that adding an interior test along segments to this condition yields an exact characterization of strong quasiconvexity. Finally, a critical-point condition on the same compensated profile gives a second, elementary characterization.