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\mathbb{R}^n 中的子集S 上的凹函数。
f : Rn → R is convex if and only if the function g : R → R, g (t ) = f (x + tv ), dom g = {t | x + tv ∈ dom f } is convex (in t ) for any x ∈ dom f , v ∈ Rn So, can check convexity of f by checking convexity of functions of one variable example....
f is convex if and only if ? 2 f (x) ? f is strictly convex if ? 2 f (x) Convex Optimization 0 for all x ∈ dom(f ) 0 for all x ∈ dom(f ) 8 Lecture 3 Examples Quadratic function: f (x) = (1/2)x P x + q x + r with a symmetric n × n matrix P ?f (x)...
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...
On Discrete Hessian Matrix and Convex ExtensibilityTheoretical or Mathematical/ Hessian matricesmathematical programming/ discrete Hessian matrixconvex extensibilityinteger lattice pointsL-convex functionM-convex functiondiscrete functionsdiscrete convex analysis...
Hessian equationsSupport functionMinkowski additionWe study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every u ∈ Q ∩ C ( R n ) a function H : R n × R → R ∪ { ±∞ } , such that H ( X , t ) is nondecreasing in t...
Convex Hessian Hessian matrix Matrix Positive Replies: 24 Forum: Topology and Analysis Proof that the exponential function is convex I try to proof it but i got stuck right here, i want your opinions Can I get a solution if i continue by this way? or Do I have to take another? and...
example (n=2n=2) x1,x2,x3x1,x2,x3 are efficient; x4,x5x4,x5 are not Convex functions basic properties and examples oerations that preserve convexity the conjugation function quasiconvex functions log-concave and log-convex functions convexity with respect to generalized inequality definition f ...
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)-...
In this paper, a discrete Hessian matrix and a local quadratic expansion are defined for L-convex functions. We characterize L-convex functions in terms of the discrete Hessian matrix and the local quadratic expansion.关键词: discrete optimization discrete convex function hessian matrix local ...