We tackle the nested and overlapping event detection task and propose a novel search-based neural network (SBNN) structured prediction model that treats the task as a search problem on a relation graph of trigger-argument structures. Unlike existing structured prediction tasks such as dependency parsing, the task targets to detect DAG structures, which constitute events, from the relation graph. We define actions to construct events and use all the beams in a beam search to detect all event structures that may be overlapping and nested. The search process constructs events in a bottom-up manner while modelling the global properties for nested and overlapping structures simultaneously using neural networks. We show that the model achieves performance comparable to the state-of-the-art model Turku Event Extraction System (TEES) on the BioNLP Cancer Genetics (CG) Shared Task 2013 without the use of any syntactic and hand-engineered features. Further analyses on the development set show that our model is more computationally efficient while yielding higher F1-score performance.