15. Given that $A$ is a set of 100 different positive integers, and has the property: if $a, b, c \in A$, then $a, b, c$ are the side lengths of a non-degenerate triangle, let $S(A)$ denote the sum of the perimeters of all such triangles, find the minimum value of $S(A)$.
15. Let 100 different positive integers be $a_{1}339^{2}$ satisfying the given conditions. If $B=\left\{b_{1}\right.$, $\left.b_{2}, \cdots, b_{100}\right\}$, then $b_{1} \geqslant 240, \cdots, b_{i} \geqslant a_{i}, S(B) \geqslant S(A), S(A)$ has the minimum value.

To find $S(A): S(A)=S_{1}+S_{2}+S_{3}$, where $S_{1}=\sum_{a \in A} 3 a, S_{2}=\sum_{\substack{a, b \in A \\ a \neq b}}(2 a+b)$, $S_{3}=\sum_{\substack{a, b, c \in A \\ a \neq b \neq c}}(a+b+c)$, let $S=\sum_{a \in A} a=28950, S_{1}=3 S$; calculate $S_{2}$, if $x \in A$, $x$ as the two equal sides of an isosceles triangle, then the other side is chosen from $A-\{x\}$, there are $2 \times 99$ such cases, $x$ as one side, the other two sides are equal, chosen from $A-\{x\}$, there are 99 such cases, so $S_{2}=3 \times 99 S$ $=297 S$; calculate $S_{3}$, if $x \in A$, the three sides are all different, the other two sides are chosen from $A-\{x\}$, there are $\mathrm{C}_{99}^{2}$ ways, so $S_{3}=\mathbf{C}_{99}^{2} S=4851 S, S(A)=3 S+297 S+4851 S=149121450$.