The question asks me to "show that if the optimal value of the objective function of a linear programming problem is attained at several extreme points, then it is also attained at any convex combination of these extreme points."
I try to write a convex combination of all extreme points $$f=c_1x_1+…+c_nx_n.$$
However, a theorem of extreme point is that "not a convex combination of other points". This makes me confused. If a convex combination can not be an extreme point then how can I get the optimal value of the objective function？
Best Answer
Note: I changed your notations to more sensible ones.
Suppose $x^{(j)}, j \in \{1,N\}$ are such extreme points at which optimal value of the objective function $f$ is attained. Denote such optimal value $M = f(x^{(j)})$. Let $c = (c_1,\dots,c_n)^T$. Then $f(x) = \sum_{i=1}^n c_ix_i = c^T x$. Now let $x = \sum_{j = 1}^N \lambda_j x^{(j)}$ be a convex combination of $\{x^{(1)},\dots,x^{(N)}\}$ with $\lambda_j \ge 0$ and $\sum_{j=1}^N \lambda_j = 1$.
\begin{align} f(x) &= f\left( \sum_{j=1}^N \lambda_j x^{(j)} \right) \\ &= c^T \left( \sum_{j=1}^N \lambda_j x^{(j)} \right) \\ &= \sum_{j=1}^N \lambda_j c^T x^{(j)} \tag{$v \mapsto c^Tv$ is linear} \\ &= \sum_{j=1}^N \lambda_j f\left( x^{(j)} \right) \\ &= \sum_{j=1}^N \lambda_j M \\ &= M \tag{$\sum_{j=1}^N \lambda_j = 1$} \end{align} This confirms the first statement in the question body: any convex combination of "optimizers" is also an "optimizer".
Next, to address OP's doubt.
Optimal points and extreme points are two concepts. The former can be either finitely are infinitely many, but the later is finitely many (as long as the dimension of the feasible region is finite). From the Fundamental Theorem of Linear Programming, an LPP has optimal feasible solution if and only if it has optimal solution at the extreme points in the feasible region. Since we only have finitely many such extreme points, we can use an algorithm (notably the simplex algorithm) to find the optimal value of the objective function.