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I have a rather simple scenario based optimization problem: Maximize $$ Q_1{_s}(A_1{_s}-Q_1{_s}-bQ_2{_s})+ Q_2{_s}(A_2{_s}-Q_2{_s}-bQ_1{_s})-[(Q_1{_s}-K_1)^+ + (Q_2{_s}-K_2)^+]c $$ subject to $Q_1{_s}+Q_2{_s} \leq K_1+K_2$, $Q_1{_s},Q_2{_s} \geq 0$. Here $A_1$, $A_2$, $b$, $K_1$, $K_2$ and $c$ are parameters, where $s$ is scenario. ($K_1$, $K_2$ are actually variables that are decided in previous stage which then become inputs to the second stage model given above). The question is about the concavity of the objective function with respect to $Q_1$, $Q_2$ because of the $(Q_i-K_i)^+$ terms. Are there simple Linear Programming tricks to re-cast this problem (without making it an IP) and then test the concavity of the objective function (possibly using Hessian matrix)?

When I try to solve this using CPLEX it states that the Hessian is not negative definite. If we consider the terms of the Hessian $H=[\{a, b\}, \{b, c\}]$ then $c<0$ in my case whereas $a, b<0$. I was wondering what is the best way to approach - reformulate or add additional restrictions so that the Hessian becomes negative definite (numerically as well as theoretically).

Thank you in advance.

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