I would like to find an analytical solution to a linear optimization problem optimizinig over multiple time steps. Following a reduced version of the LP with variables denoted in capital letters and parameters in small letters:
minimize $f = c_{i,C}X_C + c_{i,R}X_R + c_{i,SE}X_{SE} + c_{i,SP}X_{SP} + \sum_t c_{v,C}Y_C(t) + c_{v,S} (Y_{SI}(t) + Y_{SO}(t))$
s.t. equality constraints for each time step:
$g_1(t): Y_R(t) + Y_C(t) + Y_{SO} - Y_{SI} = d(h)$
$g_2(t): Y_R(t) = \phi_R(t)X_R + Y_{CU}(t)$
s.t. an intertemporal equality constraint for each time step:
$g_3(t): Y_{SL}(t) = Y_{SL}(t-1) + \eta_IY_{SI}(t) - \frac{Y_{SO}(t)}{\eta_O}$
s.t. an equality constraint across the entire optimization horizon:
$g_4: Y_{SL}(t_1) = Y_{SL}(t_{end}) $
s.t. inequality constraints for each time step:
$h_1(t): Y_C(t) \leq X_C$
$h_2(t): Y_{CU}(t) \leq \omega_R(t)*X_R $
$h_3(t): Y_L(t) \leq X_{SE}$
$h_4(t): Y_{SI}(t) \leq X_{SP}$
$h_5(t): Y_{SO}(t) \leq X_{SP}$
s.t. an inequality constraint holding for the entire optimization horizon:
$h_6: \sum_t Y_R(t) \geq \phi^R_M*\sum_t (Y_R(t) + Y_C(t)) + (1-\phi^R_M)*\sum_t (Y_{SI}(t) - Y_{SO}(t) )$
and s.t. non-negativity constraints on all variables, i.e.:
$h_7(t): Y_C(t) \geq 0$
$h_8: X_{C} \geq 0$
Applying numerical methods of optimization programs with concrete figures for all parameters this can be solved. I am wondering if there is a way to apply the KKT-Conditions (or some other smart idea) to analytically derive a solution.
I guess, however, that there is no closed-form solution as there are too many possible cases, which alter all complementary slackness conditions.