Due to the autonomous characteristic and heterogeneity of the individual agents in active distribution network (ADN) with multi-microgrids (MMG), this paper proposes a fully decentralized adjustable robust operation framework achieving the coordinated operation between ADN and MMG. The improved linear decision rules (LDRs) based microgrid adjustable robust operation model is proposed to reduce the solution conservatism in dealing with renewable energy uncertainty. The LDRs model is then reformulated as a computationally tractable solution such that the proposed adjustable robust extension of decentralized operation can handle renewable energy uncertainty while reducing the computation burden of decentralized optimization. Then, a tailored fast alternating direction method of multipliers algorithm with a predictor-corrector type acceleration step is developed to improve the convergence rate of decentralized optimization. The effectiveness of the proposed model is validated on a modified IEEE 69-bus distribution system with four microgrids.& COPY; 2023 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).