Let f: R → R be a convex function. A number u ∈ R is called a subderivative of the function f at ˉx if u ⋅ (x − ˉx) ≤ f(x) − f(ˉx) for all x ∈ R. The set of all subderivatives of f at ˉx is called the …

Lemma 1.2 (Nonnegative weighted sums). The weighted sum of mconvex functions f 1;:::;f m f:= Xm i=1 if i (3) is convex as long as the weights 1;:::; 2R are nonnegative. Proof. By convexity …

Sep 26, 2014 · It is clear that a convex function can be non-differentiable at countably many points, for example f(x) = ∫ ⌊x⌋dx.

Apr 18, 2023 · Convex functions have a unique global minimum, making optimization easier and more reliable. Non-convex functions, on the other hand, can have multiple local minima, …

How and when does non-differentiability happen [at argument \(x\)]? Here are some ways: 1. The function jumps at \(x\), (is not continuous) like what happens at a step on a flight of stairs. 2. …

There are similar characterizations for strongly convex functions. For example, f is strongly convex if and only if there exists m>0 such that f(y) f(x) + rTf(x)(y x) + mjjy xjj2; 8x;y2dom(f); or …

Example 1: Show analytically that function f defined below is non differentiable at x = 0 x = 0. One way to answer the above question, is to calculate the derivative at x = 0 x = 0. We start by …

For a differentiable convex function f, we know from previous lectures that x is a minimizer if and only if rf(x) =0. The following theorem provides a generalization to potentially non-differentiable …

A function is convex iff its epigraph is a convex set. Example: =max | =1… , . Since epi is the intersect of epi , epi is convex. Thus, function is convex.

Dec 15, 2024 · A simple example of non-differentiable optimization is approximation of a kink origination from an absolute value function. The simple function f = | x | {\displaystyle f=|x|} is …