Convex Optimization - Jensen's Inequality

Let S be a non-empty convex set in $\mathbb{R}^n$ and $f:S \rightarrow \mathbb{R}^n$. Then f is convex if and only if for each integer $k>0$

$x_1,x_2,...x_k \in S, \displaystyle\sum\limits_{i=1}^k \lambda_i=1, \lambda_i\geq 0, \forall i=1,2,s,k$, we have $f\left ( \displaystyle\sum\limits_{i=1}^k \lambda_ix_i \right )\leq \displaystyle\sum\limits_{i=1}^k \lambda _if\left ( x \right )$

Proof

By induction on k.

$k=1:x_1 \in S$ Therefore $f\left ( \lambda_1 x_1\right ) \leq \lambda_i f\left (x_1\right )$ because $\lambda_i=1$.

$k=2:\lambda_1+\lambda_2=1$ and $x_1, x_2 \in S$

Therefore, $\lambda_1x_1+\lambda_2x_2 \in S$

Hence by definition, $f\left ( \lambda_1 x_1 +\lambda_2 x_2 \right )\leq \lambda _1f\left ( x_1 \right )+\lambda _2f\left ( x_2 \right )$

Let the statement is true for $n < k$

Therefore,

$f\left ( \lambda_1 x_1+ \lambda_2 x_2+....+\lambda_k x_k\right )\leq \lambda_1 f\left (x_1 \right )+\lambda_2 f\left (x_2 \right )+...+\lambda_k f\left (x_k \right )$

$k=n+1:$ Let $x_1, x_2,....x_n,x_{n+1} \in S$ and $\displaystyle\sum\limits_{i=1}^{n+1}=1$

Therefore $\mu_1x_1+\mu_2x_2+.......+\mu_nx_n+\mu_{n+1} x_{n+1} \in S$

thus,$f\left (\mu_1x_1+\mu_2x_2+...+\mu_nx_n+\mu_{n+1} x_{n+1} \right )$

$=f\left ( \left ( \mu_1+\mu_2+...+\mu_n \right)\frac{\mu_1x_1+\mu_2x_2+...+\mu_nx_n}{\mu_1+\mu_2+\mu_3}+\mu_{n+1}x_{n+1} \right)$

$=f\left ( \mu_y+\mu_{n+1}x_{n+1} \right )$ where $\mu=\mu_1+\mu_2+...+\mu_n$ and

$y=\frac{\mu_1x_1+\mu_2x_2+...+\mu_nx_n}{\mu_1+\mu_2+...+\mu_n}$ and also $\mu_1+\mu_{n+1}=1,y \in S$

$\Rightarrow f\left ( \mu_1x_1+\mu_2x_2+...+\mu_nx_n+\mu_{n+1}x_{n+1}\right ) \leq \mu f\left ( y \right )+\mu_{n+1} f\left ( x_{n+1} \right )$

$\Rightarrow f\left ( \mu_1x_1+\mu_2x_2+...+\mu_nx_n+\mu_{n+1}x_{n+1}\right ) \leq$

$\left ( \mu_1+\mu_2+...+\mu_n \right )f\left ( \frac{\mu_1x_1+\mu_2x_2+...+\mu_nx_n}{\mu_1+\mu_2+...+\mu_n} \right )+\mu_{n+1}f\left ( x_{n+1} \right )$

$\Rightarrow f\left ( \mu_1x_1+\mu_2x_2+...+\mu_nx_n +\mu_{n+1}x_{n+1}\right )\leq \left ( \mu_1+ \mu_2+ ...+\mu_n \right )$

$\left [ \frac{\mu_1}{\mu_1+ \mu_2+ ...+\mu_n}f\left ( x_1 \right )+...+\frac{\mu_n}{\mu_1+ \mu_2+ ...+\mu_n}f\left ( x_n \right ) \right ]+\mu_{n+1}f\left ( x_{n+1} \right )$

$\Rightarrow f\left ( \mu_1x_1+\mu_2x_2+...+\mu_nx_n+\mu_{n+1}x_{n+1}\right )\leq \mu_1f\left ( x_1 \right )+\mu_2f\left ( x_2 \right )+....$

Hence Proved.