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Recently deep learning-based image compression has shown the potential to outperform traditional codecs. However, most existing methods train multiple networks for multiple bit rates, which increase the implementation complexity. In this paper, we propose a new variable-rate image compression framework, which employs generalized octave convolutions (GoConv) and generalized octave transposed-convolutions (GoTConv) with built-in generalized divisive normalization (GDN) and inverse GDN (IGDN) layers. Novel GoConv- and GoTConv-based residual blocks are also developed in the encoder and decoder networks. Our scheme also uses a stochastic rounding-based scalar quantization. To further improve the performance, we encode the residual between the input and the reconstructed image from the decoder network as an enhancement layer. To enable a single model to operate with different bit rates and to learn multi-rate image features, a new objective function is introduced. Experimental results show that the proposed framework trained with variable-rate objective function outperforms the standard codecs such as H.265/HEVC-based BPG and state-of-the-art learning-based variable-rate methods.