Suppose I want to set the price $0 \le p_t \le p_{max} $ and based on the price, demand is determined $D_t(p_t)=a-bp_t$. Inventory level at each time is denoted by $I_t$ and it is determined by $I_t= I_{t-1}+x_t-D_t$ where $x_t$ is the quantity that is refilled. Now, the quantity that we can send to customers is determined based on the inventory level and demand, $z_t \le I_{t-1}+x_{t}$ or $z_t \le D_t(p_t)$. Now, consider a part of objective function $\sum_{t\in T}p_t.z_t$.
Gurobi can solve it regardless if it is convex or concave. Now, suppose we discretize the price with $L$ levels and price is re-written as $p_t=\sum_{l \in L}p_ly_{lt}$ where $\sum_{l \in L}y_{lt}=1 \quad \forall t \in T$.
Regardsless the gap between these two formulation, I am wondering which formulation is well-suited for solvers like Gurobi, because in the second formulation, the number of decision variables increases and there will be some binary variables. This will be more severe if it depends on scenarios, i.e., $p_{ts}$.
One more question, if I use the second formulation, should I linearized the objective function or Gurobi can do it by it-self? That is: $$ \max \sum_{t \in T}\sum_{l \in L}p_l W_{lt} \\ W_{lt} \le M.y_{lt} \\ W_{lt} \le z_t \\ \sum_{l \in L}y_{lt}=1 $$
