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...
Some properties of the Hessian Matrix of a Strictly Convex Function. : Journal für die reine und angewandte Mathematik (Crelles Journal)IndexesAdvertisement.doi:10.1109/TCC.2013.26Mark R. GreeneELSEVIERNdt International
如果作为Rn+1的子集,集合epif是凸的,则我们将函数f定义为S上的凸函数。 4.1.3 定义(凹函数)concave function 空间\mathbb{R}^n上的子集S上的函数f是凹函数,当它的相反函数-f是凸函数时,这个函数在子集S上就是凹函数。 也即是,在空间\mathbb{R}^{n+1}中的子集epi(-f)是一个凸集时,函数f就是空间\...
where hK is the support function of K (see Sect. 2.1 for the definition) and 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 for (n−1)-dimensional ...
mathematical programming/ discrete Hessian matrixconvex extensibilityinteger lattice pointsL-convex functionM-convex functiondiscrete functionsdiscrete convex analysiscombinatorial propertiessemidefinite programming/ A0210 Algebra, set theory, and graph theory B0210 Algebra B0260 Optimisation techniques C1110 Algebra...
Keywords Definitions of L- and M-Convexity L-Convex Sets M-Convex Sets Properties of L-Convex Functions Properties of M-Convex Functions L- and M-Convexity Duality Network Duality Subdifferentials Algorithms...DOI: 10.1007/978-0-387-74759-0_325 被...
? ? ? ? ? basic properties and examples operations that preserve convexity the conjugate function quasiconvex functions log-concave and log-convex functions convexity with respect to generalized inequalities IOE 611: Nonlinear Programming, Winter 2011 3. Convex functions Page 3–1 De?nition De?
凸函数(convex function):关于凸函数,有三种定义 A function f:\mathcal{K}\mapsto\mathbb{R} is convex if for any \mathbf{x},\mathbf{y}\in\mathcal{K}, \forall\alpha\in[0,1],f((1-\alpha)\mathbf{x}+\alpha\mathbf{y})\le(1-\alpha)f(\mathbf{x})+\alpha f(\mathbf{y}) 从图...
MHBF convex iff Hessian matrix positive semidefinite Hey! A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is convex if for all $x,y\in \mathbb{R}^n$ the inequality $$f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)$$ holds for all $t\in [0,1]$. Show that a twice continuously...
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 ...