Recall the definition of convexity:

• Given any two points in a shape, all points in the line segment between them are ALSO inside of the shape.

Formally, for a 2-dimensional set of constraints (i.e. two variables), this means that the set $\{λx_1 + (1 - λ)x_2: λ∈ℝ\}$ is fully contained in the shape for any $x$.

Epigraph

$epi(f) = \left\{{y \choose x}:y≥f(x), x∈ℛ^n\right\}$ for a function $f: ℛ^n→ℛ$ is the set of infinite points above the function.

• Then, $f$ is convex iff its epigraph is.

Local Optimum

For a point $\overline{p}$, we say it is a local optimum if, for some $λ∈ℝ$

• We draw a circle with radius $λ$ around it, and all feasible points in the circle are less optimal (e.g. for a minimization problem, all feasible points are above $\overline{p}$

NLP

• May assume objective function is linear WLOG
• Local optimum is the overall optimal solution if the feasible region is convex.