Convert Non-linear objective function from Maximization to Minimization


I have the following non linear objective function. How can I convert this maximization objective function to a minimization problem?

I have read conflicting material online, some say to multiple the problem by -1 others say to multiply by the reciprocal.
Overall, I like the decision variable not to be in the denominator as well.

Screenshot 2022-07-19 114929
Represented in Julia as

@NLobjective(model, Max, sum(b_dict[r] * (p_dict[r] * l_star[r] / l_[r]) for r in od))

where b_dict[r] , p_dict[r] and l_star[r] are lists of integers (representing demand, priority, and known arc length)

and l_[r] is a decision variable defined as @variable(model,l_[od] >= 0)

The most usual practice is to multiply by -1. That has the advantage of not introducing complications that my lead to unexpected gradients/Hessian.

Got it,

It is possible to multiply the reciprocal as well though, correct?

Consider maximizing -x^2. The reciprocal is -1/x^2. This has no minimum, and the derivative at x=0 is undefined. For a problem like this, a gradient-based optimizer will encounter problems. On the other hand, if you just multiply by -1, you won’t have any trouble.