Equivalence of objective functions of optimization problems

Question

If the objective function is $\min\limits_{x} \sum\limits_{i=1}^n e^{-a_ix_i}$, can I transform the objective into $\max\limits_{x} \sum\limits_{i=1}^n a_ix_i$?

Answer 1

Yes. What you're doing is basically making use of two general facts:

1. Any monotonic transformation preserves the max/min (note that $\ln$ is a monotonic transformation). This should be pretty intuitive to see.

2. Maximizing $f(x)$ is equivalent to minimizing $-f(x)$. Again, this is fairly intuitive.

Applying these two to your problem, in order, will result in the desired transformation.

Edit: There is a slight mistake in the second expression you have. It should be $\max \sum a_ix_i$, not $-a_ix_i$, assuming of course that $a_i>0$

Edit 2: As pointed out in the comment and second answer, this is incorrect. My answer applies to a product, not a sum. See the second answer.

Answer 2

If one of the $a_i$, say $a_1$, is $0$ and all others are positive, then maximizing $ \sum\limits_{i=1}^n -a_ix_i$ is the same as piling everything onto $x_1$.

On the other hand, in this situation minimizing $\sum\limits_{i=1}^n e^{-a_ix_i}$ is indifferent to the value of $x_1$ and spending your budget on other $x_i$ makes the sum smaller.

Answer 3

The two optimization problems $$\min _x \prod_i e^{-a_i x_i} \quad \text{and} \quad \min_x \sum_i -a_i x_i $$ are equivalent by taking the log of the first.

The two optimization problems $$\min _x \sum_i e^{-a_i x_i} \quad \text{and} \quad \min_x \sum_i -a_i x_i $$ are NOT necessarily equivalent. If your constraints are just $0 < x < c$ then in either one the minimum is going to be achieved when $x = c$ (assuming $a_i >0$). If you have linear inequality constraints $Ax \leq b$ where $A$ is not invertible (consider for example $\sum x_i \leq c$), or if you had nonlinear constraints, it's not obvious to me that the two objectives are always going to have the same minimizer. A negative exponent is going to zero very quickly, so if you had a limited budget ($\sum x_i \leq c$), you wouldn't want to spend too much trying to get any particular $- a_ix_i$ very small. On the other hand, if there is no exponent, you'd just want to put everything on the largest $a_i$.