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I am working on an optimization problem where I aim to maximize revenue. My current model has the following objective function: $$ Sales(P_i) * log(P_i - const_i))$$ where $P_i$ represents the price of product $i$. The Sales function is a mixed integer linear function due to the inclusion of some categorical variables. I should note also that Sales function, which is linear, is modeled to predict the log of sales (a common practice in econometrics) hence the need to log the unit margin part in order to obtain the true optimum.

Is there a way to transform this objective function into a quadratic form so I can use quadratic solvers? If not, is there any approximation that I can do?

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    $\begingroup$ If you restrict $P_i$ to a finite set of values, you can linearize the objective. Your nonlinear function of $P_i$ becomes a precomputed objective coefficient for a binary variable. See or.stackexchange.com/questions/6545/… $\endgroup$
    – RobPratt
    Commented Aug 9, 2023 at 11:32
  • $\begingroup$ Another possibility is to do a (concave if you are maximizing) piecewise-linear approximation of the objective function. Once you get a solution, you can try refining the approximation in the vicinity, solve again with the new objective, repeating ad nauseum. $\endgroup$
    – prubin ♦
    Commented Aug 9, 2023 at 15:36
  • $\begingroup$ @RobPratt method worked for me (much better computational time than Baron solver). Did not try the second method. $\endgroup$
    – MarcM
    Commented Aug 10, 2023 at 14:11

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