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。
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...
Proof for theses examples are written on P73.Sublevel sets The \alpha -sublevel set of a convex function f: \mathbb{R}^n \rightarrow \mathbb{R} is: C_{\alpha} = \{x \in \bold{dom}\; f| f(x) \le \alpha \} Sublevel sets are convex sets, easy to be proved: Suppose f(x...
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...
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. . . . . . . . . . ....
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 ...
Examples Calling Sequence Convex(f,t,x(t)) Parameters f - expression int,x(t), andx'(t) t - independent variable x(t) - unknown function (or list of functions) Description • TheConvex(f, t, x(t))command determines if the integrand is convex. ...
the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete ...