It is well-known that if a convex function has a minimum, then that minimum is global. The minimizers, however, may not be unique. There are certain subclasses, such as strictly convex functions, that do have unique minimizers when the minimum exists, but other subclasses, such as ...
minimum and minimal elements via dual inequality Convex functions definition examples on RR example on RnRn and Rm×nRm×n restriction of a convex function to a line extended-value extension first-order condition second-order condition examples epigraph and sublevel set Jense's inequality operations ...
The most important property of strongly convex function is that it attains a unique minimum on any closed convex set Q. Therefore, if \(x^* = \arg \min \nolimits _{x \in Q} f(x)\), then for any \(x \in Q\) and \(\alpha \in (0,1)\) we have ...
is positive, and there exists c∈C,d∈Dc∈C,d∈D that achieve the minimum distance. Definea=d−c,b=∥d∥22−∥c∥222a=d−c,b=‖d‖22−‖c‖222Now we need to prove that the affine functionf(x)=aTx−b=(d−c)T(x−(1/2)(d+c))f(x)=aTx−b=(d−c)T...
Then for each point x∈ X there is a unique point π(x ) ∈ Q that is closest to x— i.e., that satisfies ||x − π(x )|| = dist(x , Q ). Furthermore, this function π : X→ Q is continuous. It is called the closest point projection onto Q . Proof . Uniqueness fol...
We start with some foundational definitions and a few preliminary remarks. Letbe a convex function with pertaining set (1.1) of all minimizing points. Thisminimum setcan equivalently be described in terms of the right and left derivativeandoff. Indeed, it follows from Theorem 23.2 of Rockafellar...
FunctionResult NetworkElement QueryAssociationsParameters QueryAssociationsResult SynthesizeAssociationGeometriesParameters TraceLocation TraceParameters TraceResult ValidateNetworkTopologyParameters ValidateNetworkTopologyResult query/support AttachmentInfo support AddressCandidate AlgorithmicColorRamp ArealUnit AreasAndLengths...
If Q is closed convex then πQ{x} is unique, since ϕ(y):=‖x−y‖2 is a strictly convex function and, hence, has a unique minimum point Lemma 21.9 If Q is closed convex then 1. for all x∈ℝn and all y∈Q (21.63)(x−πQ{x},y−πQ{x})≤0 2. for all x,y...
A local minimum x∗ is said to be strict if there is no other local mini- mum within some open sphere centered at x∗. Local maxima are defined similarly. Proposition 3.1.1: If X is a convex subset of ℜn and f : ℜn → (−∞, ∞] is a convex function, then a local...
Theorem2.34.Let f be a real-valued concave function defined on a convex set S⊆R n. Then f is continuous on the interior of S.This result can not be strengthened to include the boundary points of S.For differentiable functions,we can conclude the following.Theorem2.35.Let f ...