In this paper, we propose an improved L-shaped method to solve large-scale two-stage convex 0-1 mixed-integer nonlinear stochastic programs with mixed-integer variables in both first and second stage decisions and with relatively complete recourse. To address the difficulties in solving large problems, we propose a Benders-like decomposition algorithm that includes both (strengthened) Benders cuts and Lagrangean cuts in the Benders master problem. The proposed algorithm is applied to solve a batch plant design problem under demand uncertainty, and a planning problem under demand and price uncertainty. It is shown that the proposed algorithm outperforms the commercial solvers, DICOPT, SBB, Alpha-ECP, and BARON, for the problems with a large number of scenarios. Also, although the proposed algorithm cannot close the duality gap, it is proved that it can yield a lower bound that is at least as tight as the one from Lagrangean decomposition. (C) 2018 Elsevier Ltd. All rights reserved.