# Help needed in understanding the following trick in solving the optimization problem

Suppose in an optimization problem we have the following constraints $$\sum_{i=1}^K\frac{1}{x_{i}}\leq T,$$ $$0 where $$X,T$$ are some real constants. In this case, is it ok to use the following relation (due to Jensen inequality) $$K\frac{K}{\sum_{i=1}^Kx_i}\leq \sum_{i=1}^K\frac{1}{x_i}$$ and use the fact that the above Jenson inequality becomes equal when all $$x_i$$'s are equal. Then, replace the original constraint $$\sum_{i=1}^K\frac{1}{x_{i}}\leq T$$ in the optimization problem with the new constraint $$K\frac{K}{\sum_{i=1}^Kx_i}\leq T$$ with obviously all $$x_i$$'s in this constraint being equal. How we can justify this trick? Any help in this regard will be much appreciated. Thanks in advance.

There is no trick, if $$a\leq b$$ and $$b\leq c$$ then $$a\leq c$$:
$$K\frac{K}{\sum_{i=1}^Kx_i}\leq \sum_{i=1}^K\frac{1}{x_i}\;\;\;\;{\rm and }\;\;\;\;\;\sum_{i=1}^K\frac{1}{x_{i}}\leq T,$$ so $$K\frac{K}{\sum_{i=1}^Kx_i}\leq T$$
• But is there a condition in which the two problems will be equivalent. (For example if we prefer to have smaller values of $x_i$'s etc)? – Frank Moses Jan 29 '19 at 10:12