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...
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...
objective / cost function inequality constraints equality constraints feasible optimal value optimal point active inactive 【Why Convex Optimization?】 Contains various types of problems, e.g., many ML and OR tasks. Repeatability: different runs give the same results. ...
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. . . . . . . . . . ....
composition with affine function: f(Ax +b) is convex if f is convex examples • log barrier for linear inequalities f(x) = − m i=1 log(b i −a T i x), domf = {x | a T i x < b i , i = 1, . . . , m} • (any) norm of affine function: f(x) = ...
example on RnRn and Rm×nRm×n restriction of a convex function to a line extended-value extension first-order condition second-order condition examples epigraph and sublevel set Jense's inequality operations that preserve convexity positive weighted sum & composition with affine function pointwise max...
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 ...