I have an MINLP and its linearized formulation problem given below where the objective (nonconvex) and constraint C4 are nonlinear. We linearized them by applying some known techniques.
However, when I solve both Original MINLP and Linearized MILP, I find MINLP and MILP not giving same objective. Also, when we increase $P_{\rm{Max}}$, the gap between MINLP and MILP increases (MINLP giving higher objective than the linearized MILP). Why is that? I used Knitro solver. Or is it the consequence of the solver?
$\text{Maximize}\prod\limits_{t=1}^{K_1 L}(1+p_{\mathcal{M}_t}h_{\mathcal{M}_t})^{c_{\mathcal{M}_t}}$
subject to
$\text{}\text{C1:} \hspace{1mm}{c}_{\mathcal{M}_t}\in\{0,1\}\hspace{25mm}\forall t,\hspace{1mm}t=1,\cdots,{K}{L}$
$\text{}\text{C2:}\hspace{1mm}\sum_{\begin{matrix} t\in\{l,l+k{L}\},\hspace{1mm}k=1,\cdots, K\end{matrix}}c_{\mathcal{M}_t}=1, \hspace{3.8mm}\forall l,\hspace{1mm} l=1,\cdots,{L}$
$\text{}\text{C3: } {p}_{\mathcal{M}_t}\ge 0, \hspace{25mm}\hspace{4mm}\forall t,\hspace{2mm}t=1,\cdots, {K}{L}$
$\text{C4: } \sum\limits_{t=1}^{KL}c_{\mathcal{M}_t}p_{\mathcal{M}_t}\le P_{\text{Max}} $
$\text{C5:} \sum\limits_{t=K_1L+1+(k-1)L}^{K_1L+L+(k-1)L}c_{\mathcal{M}_t}\rm{log_2}(1+p_{\mathcal{M}_t}h_{\mathcal{M}_t})\ge R_k^{\rm{target}}L, \hspace{1mm} k=K_1+1,\cdots, K $
where $ c_{\mathcal{M}_t}$ (binary integer) and $ p_{\mathcal{M}_t}$ (continuous) are optimization variables.
I linearized the objective function using the fact
$(1+p_{\mathcal{M}_t}h_{\mathcal{M}_t})^{c_{\mathcal{M}_t}}=1+c_{\mathcal{M}_t}p_{\mathcal{M}_t}h_{\mathcal{M}_t}$ which is true for binary $c_{\mathcal{M}_t}$.
then I introduced a slack variable $\xi_{\mathcal{M}_t}$ such that
${\xi}_{\mathcal{M}_t}\le 1+\rm{minimum}\{p_{\mathcal{M}_t},P_{\rm{Max}}c_{\mathcal{M}_t}\}h_{\mathcal{M}_t}$
Then the objective function becomes
$\prod\limits_{t=1}^{K_1 L}\xi_{\mathcal{M}_t}$, which we can transform into geometric mean optimization problem, thus concave.
Note that constraint C4 is still nonlinear.
I introduced a new slack variable and formulated the following additional (linear) constraints to make it linear
$\text{N1:} \hspace{2mm}\lambda_{\mathcal{M}_t}\le P_{\rm{Max}}c_{\mathcal{M}_t},\hspace{20mm}\forall t,\hspace{1mm}t=1,\cdots, KL$
$\text{N2:}\hspace{2mm}\lambda_{\mathcal{M}_t}\ge 0, \hspace{30.4mm}\forall t,\hspace{1mm}t=1,\cdots,KL$
$\text{N3:}\hspace{2mm} \lambda_{\mathcal{M}_t}\le {p}_{\mathcal{M}_t}, \hspace{24.5mm}\hspace{2mm}\forall t,\hspace{1mm}t=1,\cdots, KL $
$\text{N4:} \lambda_{\mathcal{M}_t}\ge p_{\mathcal{M}_t}-P_{\rm{Max}}(1-c_{\mathcal{M}_t}),\hspace{1.0mm}\hspace{1.2mm}\forall t,\hspace{2mm}t=1,\cdots, KL$
${\text{N5:}} \sum_{t=1}^{KL}\lambda_{\mathcal{M}_t}=P_{\rm{Max}}$