Negative definiteness and concavity
From
phoenix1983@21:1/5 to
All on Mon Mar 28 17:54:37 2016
Hello experts, I have a rather simple scenario based problem: Maximize (Stage II): Q1(A1-Q1-bQ2)+ Q2(A2-Q2-bQ1)-[(Q1-K1)^+ + (Q2-K2)^+]*c. Subject to: Q1+Q2<=K1+K2 Q1,Q2>=0. Here A1, A2, b, K1, K2 and c are parameters. K1 and K2 are variables in Stage I
where I maximize the expected value of the Stage II problem.
The question is about the concavity of the Stage II objective function stated above, because of the (Qi-Ki)^+ terms. Are there simple LP 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 positive definite. If we consider the terms of the Hessian
$H=[\{a, b\}, \{b, c\}]$ then $c>0$ in my case whereas $a, b<0$. The determinant appears to be negative. I was wondering what is the best way to approach - reformulate or add additional restrictions so that the Hessian becomes positive definite (
numerically as well as theoretically).
Thank you in advance.
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