In this paper, we develop a new algorithm, Annealed Skewed SGD - AskewSGD - for training deep neural networks (DNNs) with quantized weights. First, we formulate the training of quantized neural networks (QNNs) as a smoothed sequence of interval-constrained optimization problems. Then, we propose a new first-order stochastic method, AskewSGD, to solve each constrained optimization subproblem. Unlike algorithms with active sets and feasible directions, AskewSGD avoids projections or optimization under the entire feasible set and allows iterates that are infeasible. The numerical complexity of AskewSGD is comparable to existing approaches for training QNNs, such as the straight-through gradient estimator used in BinaryConnect, or other state of the art methods (ProxQuant, LUQ). We establish convergence guarantees for AskewSGD (under general assumptions for the objective function). Experimental results show that the AskewSGD algorithm performs better than or on par with state of the art methods in classical benchmarks.