Convex Optimization Problems

After an introduction to the basics of convex sets and functions, we prove the \(C^{1}\) -characterization of convexity and thereby express the set of global minimal points of unconstrained optimization problems as the solution set of an equation. With the \(C^{2}\) -characterization of convexity, we also derive a handy way to check the convexity of functions. Optimality conditions and estimates for the optimal value of constrained convex optimization problems are based on duality statements, which we subsequently prove. Afterwards, we discuss a series of algorithmic approaches for convex optimization problems, the most efficient of which also rely on the explicit exploitation of duality.