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Weakly supervised object detection (WSOD) is a challenging task that requires simultaneously learning object detectors and estimating object locations under the supervision of image category labels. Many WSOD methods that adopt multiple instance learning (MIL) have nonconvex objective functions and, therefore, are prone to get stuck in local minima (falsely localize object parts) while missing full object extent during training. In this article, we introduce classical continuation optimization into MIL, thereby creating continuation MIL (C-MIL) with the aim to alleviate the nonconvexity problem in a systematic way. To fulfill this purpose, we partition instances into class-related and spatially related subsets and approximate MIL’s objective function with a series of smoothed objective functions defined within the subsets. We further propose a parametric strategy to implement continuation smooth functions, which enables C-MIL to be applied to instance selection tasks in a uniform manner. Optimizing smoothed loss functions prevents the training procedure from falling prematurely into local minima and facilities learning full object extent. Extensive experiments demonstrate the superiority of CMIL over conventional MIL methods. As a general instance selection method, C-MIL is also applied to supervised object detection to optimize anchors/features, improving the detection performance with a significant margin. [ABSTRACT FROM AUTHOR]