- non-negative scaling
- sum: \(f_{1}+ f_{2}\) is convex if \(f_1, f_2\) is convex
- infinite sums: \(\sum_{i=1}^{\infty} f_{i}\) is convex
- integral: if \(f\qty(x,a)\) is convex in \(x\), \(\int_{a} f\qty(x,a) \dd{a}\) is convex
- pre-composition with affine function: \(f\qty(Ax + b)\) is convex if \(f\) is convex
- pointwise maximum: \(f_{1}, …, f_{m}\) is convex, then \(f\qty(x) = \max \qty(f_{1} \qty(x)\dots f_n \qty(x))\) is convex
- supremum: if \(f\qty(x,y)\) is convex in \(x\) far each \(\text{sup}_{y \in Y} f\qty(x,y)\)
- partial minimization: \(f\qty(x) = \text{inf}_{y \in C} f\qty(x,y)\) (find the smallest value of \(f\) over \(y \in C\), or the point at which its approached)
composition with scalar functions
\(g : \mathbb{R}^{n} \to \mathbb{R}\), \(h: \mathbb{R} \to \mathbb{R}\), and let \(f = h\qty(g\qty(x)) = h \odot g\)
composition \(f\) is convex if:
- \(g\) convex, \(h\) convex, extended-value extension \(\tilde{h}\) non decreasing
- \(g\) concave, \(h\) convex, extended-value extension \(\tilde{h}\) non increasing
general composition rule that preserve convexity
composition of \(g : \mathbb{R}^{n} \to \mathbb{R}^{k}\), and \(h: \mathbb{R}^{k} \to \mathbb{R}\) is \(f\qty(x) = h\qty(g\qty(x)) = h\qty(g_{1}\qty(x), \dots, g_{k}\qty(x))\)
\(f\) is convex if \(h\) is convex and for each \(i\), one of the the following holds:
- \(g_{i}\) convex, \(\tilde h\) nondecreasing in its \(i\) th element
- \(g_{i}\) concave, \(\tilde h\) nonincreasing in its \(i\) th element
- \(g_{i}\) affine
examples
sum of the \(r\) largest elements of a set is convex since we can multiply them with many-hot selectors which gives you combinations an and then max them together