In the book (Anders Klarbring and Peter W. Christensen, "An introduction to Structural Optimization"), the Pareto optimality has been briefly mentioned which is a method used in the case when we have an optimization problem that involves several objective functions to be minimized simultaneously:
$\text{minimize}~~ (f_1(x,y), f_2(x,y), ... , f_l(x,y))$
where $l$ is the number of objective functions. And so this method involves finding an optimal point called Patero optimal point $(x^*,y^*)$ that minimizes the following scalar objective function (instead of the multiple objective functions above):
$\sum_{i=1}^l w_i ~f_i(x,y)$
where $w_i \geq 0$, $i=1,...,l$, are so-called weight factors satisfying $\sum_{i=1}^l w_i =1$.
Thus, respecting certain constraints of the optimization problem, and by varying weights, different Pareto optima are obtained, and then the best one is chosen.
My questions about this method are: why do we impose the sum condition $\sum_{i=1}^l w_i =1$ on the weights? And to make sure I understand, if the weight of the first function, $w_1$, was the largest, does this mean that we care about minimizing the first function the most?
