一阶条件(first order condition) 二阶条件(second order condition) 凸函数的例子 下水平集(sublevel sets) Epigraph 琴生不等式(Jesen inequality) 保持函数凸性的操作 若干凸函数非负的加权和 与仿射函数的复合 逐点取最大值(上确界) 函数复合 特殊情况下的下确界函数 函数的透视 共轭函数(Conjugate Function) ...
Examples onRnRnandRm×nRm×n Restriction of a convex function to a line Extended-value extension First-order condition Second-order conditions Examples Epigraph and sublevel set Jensen’s inequality Operations that preserve convexity Positive weighted sum & composition with affine function Pointwise maximum...
这样的定义在函数表示上有一定的意义。 一阶条件(first order condition) 可微的凸函数满足一阶条件: 这个不等式揭示了凸函数的局部特性,那就是在一点的切平面是整个函数的global underestimate。 如果上面的不等号严格成立,那么这个函数是严格凸的。这里的条件是充分必要的。 二阶条件(second order condition) 如果...
Second-order conditions Suppose f is twice differentiable, its Hessian \nabla^2 f exists at each point of \bold{dom}\;f . Then $f$ is convex iff. \bold{dom}\; f is convex and: \nabla^2 f \succeq 0 The condition requires the derivative is nondecreasing....
Convex functions First-order condition Page 3–5 f is dicondition ?erentiable if dom f is open and the gradient First-order f is di?erentiable if dom f is open and the gradient ? f (x) ? f (x) ? ? f (x) ? ? f ( x ) = ? f (x ) ? f (x ) , ? f (x ), . ...
凸方程 Convex Function 定义:定义域D(f)为凸集,且对于任意两个属于D(f)的两个数x,y ; θ∈ R, 0 ≤θ≤ 1,满足 f(θx + (1 − θ)y) ≤θf(x) + (1 − θ)f(y). first-order approximation: first order condition for convexity ...
Second-order conditions ? First-order conditions ? Reduction to a scalar function ? Showing that f is obtained through operations preserving convexity Convex Optimization 10 Lecture 3 First-Order Condition f is di?erentiable if dom(f ) is open and the gradient ?f (x) ?f (x) ?f (x) ,...
(x − x0) second order condition: for f twice differentiable, f is convex ⇐⇒ for all x ∈ dom f, ∇2f(x) ≽ 06Convex Optimization Shuguang CuiSimple examples• linear and affine functions are convex and concave • quadratic function f(x) = xTPx + 2qTx + r convex ...
59 The value distribution of the Hurwitz zeta function with an irrational shift 51:58 Sums of Fibonacci numbers close to a power of 2 33:04 Quadratic Twists of Modular L-functions 50:26 On the Quality of the ABC-Solutions 39:35 Negative moments of the Riemann zeta-function 49:44 Least ...
An approach concerning second-order optimality conditions is proposed. 关键词: optimal control hyperbolic differential inclusion second-order necessary condition second variation DOI: 10.1006/jmaa.2000.7170 被引量: 9 年份: 2001 收藏 引用 批量引用 报错 分享 ...