Design of integrated photonic devices continues to drive innovation in electro-optical systems for many applications ranging from communications to sensing and computing. Traditional design methods for integrated photonics involve using fundamental physical principles of guided-wave behavior to engineer optical functionalities for specific application requirements. While these traditional approaches may be sufficient for basic functionalities, the set of physically realizable optical capabilities these methods remains limited. Instead, photonic design can be formulated as an inverse problem where the target device functionality is specified, and a numerical optimizer creates the device with appropriate geometrical features within specified constraints. However, even with inverse design methods, achieving arbitrarily-specified phase offsets on-chip remains an important problem to solve for the reliability of interferometry-based nanophotonic applications. In order to address difficulties in achieving simultaneous phase and power optimization in inverse nanophotonic design, in this paper, we develop a set of optimization approaches that can enable user-specified phase differences in single-wavelength and multi-wavelength nanophotonic devices. By specifying phase offset targets for each output, we prevent convergence failures resulting from the changes in the figure of merit and gradient throughout the iterative optimization process. Additionally, by introducing phase-dependent figure of merit terms through an adaptive scheduling approach during the optimization, we accelerate device convergence up to a factor of 4.4 times. Our results outline a clear path towards the optimization of nanophotonic components with arbitrary phase-handling capabilities, with potential applications in a wide variety of integrated photonic systems and platforms.