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 it exists, the Hessian is positive semi-definite. Level sets are convex. a·f(x) + b·g(x) is convex for convex f,g and a,b > 0. max(f(x), g(x)) is convex for convex f(x) and g(x). 【Convex Optimization Terminology】 optimization variable objective / cost function i...
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...
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...
Norms are convex Examples on Rn ? A?ne function f (x) = a x + b with a ∈ Rn and b ∈ R ? Euclidean, l1, and l∞ norms ? General lp norms n 1/p x p = i=1 | xi | p for p ≥ 1 Convex Optimization 5 Lecture 3 Examples on Rm×n The space Rm×n is the space of...
3–3Examples on 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) ...
1.2 Special Convex Functions: Affinity and Closedness . 1.3 First Examples . . . . . . . . . . . . . 2 Functional Operations Preserving Convexity 2.1 Operations Preserving Closedness . . . 2.2 Dilations and Perspectives of a Function 2.3 Infimal Convolution. . . . . . . . . . ....
3–3Examples on 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) ...
In the following, we introduce an approach to determine the quasi-convexity and quasi-concavity based on the bordered Hessian matrix of a function. This approach could be utilized to determine the quasi-concavity and quasi-convexity of a twice-differentiable function without plotting the sublevel or...
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 ...