MR images are often reconstructed first and then used for medical image analysis tasks such as segmentation or classification. This sequential procedure can compromise the performance of the image analysis task. In this work, we propose a multi-task learning framework that jointly reconstructs underlying images and detects multiple sclerosis lesions. This framework outperforms the conventional sequential processing pipeline. We also introduce a multi-objective optimization as an effective and automated approach to balance the trade-off among multi-task losses. Experimental results suggest that taking into account subsequent detection tasks during image reconstruction may lead to enhanced detection performance.