Notes on Convex Optimization
1.1 Sub-level sets. - Definition: The $\alpha$-sublevel set of a function $f$ contains all the feasible input $x$ of $f$ having the value $f(x)<=\alpha$. - Properties: Sublevel sets of a convex function are convex for all $\alpha$ ($f(x) \leq \alpha$, $f(y) \leq \alpha$, $f(\theta x + (1-\theta) y) \leq \alpha $). If $f$ is concave, then its $\alpha$-superlevel set is a convex. - Usage: To establish convexity of a convex set by associating it as a sublevel set of a convex function or a superlevel set of a concave function. 1.2 Epi-graph. - Definition: epi f = $\{ (x, t) | x \in dom f, f(x) \leq t \}$. Epi means 'above'. - Properties: A function is convex iff its epigraph is a convex set. A function is concave iff its hypograph is a convex set. - Usage: Many results for convex functions can be proved (or interpreted) geometrically using epigraphs, and applying results for convex sets. 1.3 Closed set. - Definition: A set is closed if it contains its bo...