For a function on RR, quasiconvexity requires that each sublevel set to be an interval. All convex functions are quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generaliza
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。
Hessian equationsSupport functionMinkowski additionWe study the class Q of quasiconvex functions (i.e. functions with convex sublevel sets), by associating to every u is an element of Q boolean AND C(R-n) a function H:R-n x R -> R boolean OR {+/-infinity}, such that H(X, t) ...
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....
Example: is the banana function convex? We saw that the Hessian of our banana function was: ∇2f(x,y)=(1200x2−400y+2−400x−400x200)∇2f(x,y)=(1200x2−400y+2−400x−400x200) Its principal minors of rang 1 are: ...
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...
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)...
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 Hausdorff measure and ωk for the (k−1)-...
Example 21.1 1. The function f(x)=x2 is strongly convex (and, hence, convex and strictly convex) with l = 2. 2. The functions f(x)=x4 and f(x)=ex are strictly convex (and, hence, convex, but not strongly convex). The next two lemmas will be used hereinafter. View chapter ...
The full Hessian matrix is checked against a possibly very small negative eigenvalue due to rounding errors. The calling sequence is similar to quadprog except that there is no starting point x0 argument. Option is limited to a tolerance tol and a maximum of iterations. The exitflag convention...