Direct imaging of exoplanets requires overcoming the enormous contrast between the exoplanet and its host star, to distinguish the reflected light from the exoplanet, from the diffracted light of the star. Direct optimization of contrast, using nonlinear optimization techniques of the Adaptive Optics (AO) system coupled with coronagraphy, shows significant promise in achieving high contrast, beyond the limits of what can be achieved with traditional AO systems. Using a coronagraph optic as a static'' phase modifying element, and a deformable mirror as a dynamic’’ element, we create an adaptive coronagraph, capable of engineering the point spread function (PSF) of the imaging system, to create a deep, dark hole in the focal plane, within which the exoplanet can be imaged. We present the results of simulations of a system, consisting of a vector Apodizing Phase Plate (vAPP) coronagraph, a deformable mirror (DM), and an imaging camera. The vAPP coronagraph reroutes starlight within the pupil plane, to create a designated dark hole region, which in the ideal case would be devoid of starlight in the focal plane. Off-axis exoplanet light is transmitted through to the dark hole and hence can be imaged. Atmospheric turbulence is simulated to generate a distorted wavefront, and a nonlinear, gradient climbing based optimization algorithm is implemented to drive the DM to optimize a merit function. This merit function is chosen with a dual objective to maximize average raw contrast in the dark hole, while maintaining a sufficiently high Strehl ratio. Preliminary results show that in a setup with a coronagraph designed to create a 6×6 (ensuremathłambda/D)$^2$ rectangular dark hole with a raw contrast of 10$^-5$, the optimization procedure results in a raw contrast of 10$^-7$ at the dark hole while maintaining a Strehl ratio above 40%. It is observed that by tweaking the merit function, this non-linear optimization procedure can be adjusted to result in either higher Strehl or higher contrast. We discuss potential strategies to extend the non-linear optimization techniques to real-time, non-linear control for the AO system, thereby achieving a real-time, dynamic, adaptive coronagraph. Toward this end, we investigate the results of using the fast wavefront sensor data to reconstruct the wavefront phase, virtually propagate this through the science optical path, and optimize contrast on this virtual science image, as opposed to using the slower science camera to optimize contrast on the true science image. One potential approach to implement true real-time control would be to use deep neural networks, trained using deep deterministic policy gradients, to identify and remove speckles of diffracted starlight in the dark hole region in real-time.