They are built on the k -th elementary symmetric function of the eigenvalues, k =1,2,…, n . Our motivation came from a paper by verák [S]. The proof of our result relies on the theory of the so-called k -Hessian equations, which have been intensively studied recently; see [CNS...
Convex functions Page 3–3 Examples on Rn and Rm×n a?ne functions are convex and concave; all norms are convex examples on Rn ? ? examples on Rm×n (m × n matrices) ? a?ne function f (x ) = aT x + b ? p 1/p for p ≥ 1; ?x ? norms: ?x ?p = ( n ∞ = maxk...
Rnand Rm×naffine functions are convex and concave; all norms are convex examples on Rn affine function f(x) = aTx + b norms: xp = (∑n i=1 |xi| p)1/pfor p ≥ 1; x∞ = maxk |xk| examples on Rm×n(m × n matrices) affine function f(X) = tr(ATX) + b =m∑i=1 ...
More generally, a continuous, twice differentiable function of several variables isconvexon a convex setif and only if its Hessian matrix is positive semidefiniteon the interior of the convex set. (Wikipedia) If we want to check if a function is convex, one easy way is to use our old frien...
Reduction to a scalar function ? Showing that f is obtained through operations preserving convexity Convex Optimization 7 Lecture 3 Second-Order Conditions Let f be twice di?erentiable and let dom(f ) = Rn [in general, it is required that dom(f ) is open] The Hessian ?2f (x) is a ...
Let r.v. XX in convex set C⊆RNC⊆RN, and convex function ff defined over CC. Then, E[X]∈C,E[f(X)]E[X]∈C,E[f(X)] is finite, andf(E[X])≤E[f(X)]f(E[X])≤E[f(X)]Sketch of proof: extending f(∑αx)≤∑αf(x)f(∑αx)≤∑αf(x) and ∑α=1∑α=...
2.1 for the definition) and {\operatorname {D}}^{2}h_{K} the Hessian matrix of hK. Here, we write [A]j for the jth elementary symmetric function of the eigenvalues of a symmetric matrix A and use the convention that [A]0=1. We write \mathcal {H}^{n-1} for (n−1)-...
norm balls and cones polyhedra positive semidefinite cone operations that preserve convexity intersection affine function perspective and linear-fractional function generalized inequalities minimum and minimal elements separating hyperplane thoerem supporting hyperplane theorem dual cones and generalized inequalities ...
• pointwise maximum and supremum • composition • minimization • perspective Convex functions 3–13 Positive weighted sum & composition with affine function nonnegative multiple: αf is convex if f is convex, α≥ 0 sum: f 1 +f 2 convex if f 1 , f 2 convex (extends to in...
In addition, let gradf and Hessf denote the gradient and Hessian of a function f on M (defined with respect to the Riemannian metric and Levi-Civita connection of M). Proposition B.2 Let A be a strongly convex subset of a complete Riemannian manifold M, and f:A→R. (i) Assume f ...