本系列blog所有内容大多都来自于凸优化一书，请多多指教
函数的定义：函数f:Rn → R 若dom f 是凸集（即对任意的x,y属于dom f,θ1x+θ2y也属于domf,θ1+θ2=1 and θ1,θ2>=0）,且 for all x, y ∈ domf, and θ with 0 ≤ θ ≤ 1
f(θx + (1 − θ)y) ≤ θf(x) + (1 − θ)f(y).
则该函数为凸函数，凸函数有一个十分有用的性质，对接下来证明二阶充要条件十分关键。
A function is convex if and only if it is convex when restricted to any line that
intersects its domain. In other words f is convex if and only if for all x ∈ domf and
all v, the function g(t) = f(x+tv) is convex (on its domain, {t | x+tv ∈ domf}).
这个性质将多维空间上函数的凸性证明约束到了一条直线上（这条直线是任意的，只要它仍在dom f内即可，且其定义域依旧是凸集，有兴趣的同学可以自行验证）
接下来一阶，二阶条件的证明均采用了这个性质的思路（即先证明f:R → R，再证明一般情况）
先给出凸函数的一阶充要条件（这里我们假设其一阶导存在）
f(y) ≥ f(x) + ∇f(x)T (y − x)
函数的一阶充要条件在f:R → R上是显而易见的，这里不给出详细的证明过程，具体内容参见原文凸优化一书，这里给出几何解释
The affine function of y given by f(x)+∇f(x)T (y−x) is, of course, the first-order
Taylor approximation of f near x. The inequality (3.2) states that for a convex
function, the first-order Taylor approximation is in fact a global underestimator of
the function. Conversely, if the first-order Taylor approximation of a function is
always a global underestimator of the function, then the function is convex.
也就是说任意一点的泰勒一阶逼近函数是永远都处于整个原函数的下方。
再扩展到一般情况(运用先去提到过的凸函数的性质)
with f : Rn → R. Let x, y ∈ Rn and
consider f restricted to the line passing through them, i.e., the function defined by
g(t) = f(ty + (1 − t)x), so g′(t) = ∇f(ty + (1 − t)x)T (y − x).
First assume f is convex, which implies g is convex, so by the argument above
we have g(1) ≥ g(0) + g′(0), which means
f(y) ≥ f(x) + ∇f(x)T (y − x).（证明必要性）
Now assume that this inequality holds for any x and y, so if ty +(1−t)x ∈ domf
and ˜ty + (1 − ˜t)x ∈ domf, we have
f(ty + (1 − t)x) ≥ f(˜ty + (1 − ˜t)x) + ∇f(˜ty + (1 − ˜t)x)T (y − x)(t − ˜t),
i.e., g(t) ≥ g(˜t) + g′(˜t)(t − ˜t).（证明充分性） We have seen that this implies that g is convex.
接下来再看二阶充要条件
We now assume that f is twice differentiable, that is, its Hessian or second derivative
∇2f exists at each point in domf, which is open. Then f is convex if and
only if domf is convex and its Hessian is positive semidefinite: for all x ∈ domf,
∇2f(x)-0∈S+(海森矩阵是半正定，即对任意的向量v，v’ * ∇2f(x) * v >=0)
该条件在函数f:R→ R 上就表示为
f′′(x) ≥ 0 (and domfconvex, i.e., an interval),
该条件的证明首先要通过一阶条件，具体过程懒得打了直接放书上习题答案的图
本篇所有内容均是对凸优化第三章第一节的小部分整理，归纳。