This course prepares senior undergraduates and beginning graduate students for research in networks, particularly energy networks. For 2025-26 academic year, it covers convex optimization theory, both smooth and nonsmooth optimization. It turns out that convexity is fundamental but smoothness (continuity, differentiability) is not, and this course explains how to extend directly the existence and characterization of optimal solutions to a nonsmooth setting. It then introduces a basic theory of stochastic optimization when problem parameters are uncertain, covering robust optimization, chance constrained optimization, scenario optimization, and two-stage optimization with recourse. These theories are applied to the optimal power flow problem that underlies numerous power system applications. Familiarity with linear algebra and real analysis is required. Previous exposure to smooth convex optimization will be helpful but not necessary.