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• Basics of finite-dimensional convex analysis
  
• Convex analysis applies to various kinds of optimization problems.
  
• Lagrange multipliers and duality, nonlinear optimization
  
• Basic concepts in convex analysis and its use in convex optimization
  
• Convex sets : Convexity-preserving operations, derived sets (closure, relative interior, extreme points, faces)
  projection onto convex sets, separation ,conic approximations
  
• Convex functions : epigraphical view , infimal convolution, perspective transform, smoothing , differentiable and non-differentiable functions
  support functions: correspondence with convex sets; norms and duals; gauges and polars ,conjugate functions
  
• Convex optimization : problem types: linear and quadratic optimization, conic programming (second-order and semi-definite optimization)
  applications of sparse optimization, compressed sending, matrix completion , Lagrange and conjugate duality ,first-order methods for unconstrained and nonsmooth problems, including proximal gradient, ADMM, Frank-Wolfe ,interior methods for conic problems