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...
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 ), . ...
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) ,...
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 maximum pointwise supremum composition with scalar functions vector composition minimization perspective the conjug...
(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 ...
2nd-order conditions: for twice differentiable f with convex domain • f is convex if and only if ∇ 2 f(x) 0 for all x ∈ domf • if ∇ 2 f(x) ≻ 0 for all x ∈ domf, then f is strictly convex Convex functions 3–8 Examples quadratic function: f(x) = (1/...
(y x)first-order approximation of f is global underestimatorConvex functions 3–7Second-order conditionsf is twice differentiable if domf is open and the Hessian 2f(x) ∈ Sn, 2f(x)ij = 2f(x) xixj , i, j = 1,...,n, exists at each x ∈ domf 2nd-order conditions: for twice ...
In this paper we discuss log-convex solutions of the second order f:R+→R+ to the functional equation with initial condition given by(E)f(x+1)=g(x)f(x)for all x∈R+,f(1)=1. We prove that if g satisfies an appropriate asymptotic condition, then (E) admits at most one solution...