for all α∈Rα∈R, are convex. A function is quasiconcave if −f−f is quasiconvex, i.e., every superlevel set {x | f(x)≥α}{x | f(x)≥α} is convex. A function that is both quasiconvex and quasiconcave
In addition, one of the main result of this section (Theorem 9.5) gives, under suitable assumptions, a characterization of convexity of a function in terms of its associated Hessian matrix. This boils down to check positive (semi)definiteness of matrices and thus, there will be separate ...
If we want to check if a function is convex, one easy way is to use our old friend the Hessian matrix. However, instead of checking if it is positivedefiniteas we did in the previous article, this time, we need to check if it is positivesemidefinite. What is the difference? Theorem:...
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 ...
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 ...
因此, h_{t+1}\le h_t(1-\frac{\alpha}{4(\beta-\alpha)})\le h_t(1-\frac{\alpha}{4\beta})\le h_te^{-\frac{\gamma}{4}},Theorem 2.4 就得证了。 3、Reductions of function quality 第2部分的梯度下降算法都是针对 \gamma\text{-well-conditioned} 的目标函数。显然并不是所有的优化...
the exterior dirichlet problem for homogeneous complex k-hessian equation. arxiv:2208.03794 guan, b.: the dirichlet problem for complex monge–ampère equations and regularity of the pluri-complex green function. comm. anal. geom. 6 (4), 687–703 (1998) mathscinet google scholar ...
Since a nonzero linear function takes infinitely large values, the boundedness of 〈x,x1*+⋯+xm*〉 implies that (1.17)x1*+⋯+xm*=0. Now taking into account Eqs. (1.16) and (1.17), we have the desired result.■ Theorem 1.11 Let K1, K2,…, Km be convex cones. If K=K1∩K2...
Conjugate gradient method是另一种无约束优化方法(Unconstrained Optimization Method)。能用于优化二次函数(Quadratic Function)。基本理论建立在Q-orthogonality和Conjugate direction theorem上。收敛很快,但是用起来比较麻烦,也没见什么人用过。 Q为正定矩阵(Positive-definite Matrix) ...
The function tg\left(x/t\right) is the perspective of g\left( x \right) . Since the perspective of a function preserves convexity and g\left( x \right) is convex, tg\left(x/t\right) is convex on \left\{x:x\in\mathbb{R},\,\,t>0\right\} . Besides, if h is convex, -h...