# Linearize a highly non-linear objective function

[EDIT] : The formula below is updated to remove the radical, 0.5 in the term $$(I_{i,v} \cdot \Delta t)$$ and constant temperature $$T$$ replces temperature as function of current.

[EDIT] :The values of the constants are added

[EDIT] : Plot of $$C_{loss,cyc} \ Versus \ I_{i,v} \ \ \ \ for \ \ SOC_{init} \in [0,1]$$ is shown below. Is it possible to approximate these family of functions using a simpler function like $$(1 - e^{-x})$$ ? Could you please suggest some functions?

[EDIT] : I have added the constraints to the problem below. Constants in the constraints are $$I_{max}, I_{c,max}, T^{dep}_{v},t_s, SOC^{dep}_{v}, SOC_{-1,v}, SOC_{xtra}$$

I would like to minimize the following objective function.

$$$$\min_{I_{i,v}} \ \sum^{N_v}_{v}\sum^{TT_v}_{i} \ C_{loss,cyc}$$$$

where the expression for $$C_{loss,cyc}$$ is described below.

The following are constants: $$a = -4.092 \cdot 10^{-4} , b = -2.167 , c = 1.408 \cdot 10^{-5}, d = 6.130, E_a = 7806, R = 8.3145 , T_{ref} = 28, \Delta t = 0.1, SOC_{init} \in [0,1], C_{bat} = 270, T = 50$$

The only continuous decision variable is $$I_{i,v}$$ It is highlighted below

[EDIT] : Explaination of the optimization problem.

In a charging station, that has $$N_v$$ vehicles that needs to be charged to some state of charge $$SOC^{dep}_v$$ within some departure time $$T^{dep}_v$$. The charging schedule for each vehicle,v is thus split into timeslots, i of width $$\Delta t$$ each for $$TT_{v}$$ number of slots. Within a timeslot, i the current $$I_{i,v}$$ is constant. $$t_s$$ is just starting time of optimization implementation. The objective is trying to minimize the Electric vehicle's battery capacity loss.

I want to perform linearlization of the objective function in order to achieve near real-time performance for one hundred decision variables in an optimization problem.

I know about first order taylor approximation but what are some of the other methods I can use especially with regards to this type of objective function with exponential and power terms to linearlize it.

You can also suggest any similifications to this model. If there is another way to solve this quickly without having to linearize it. Please suggest as well.

Assume that our solver is NOT the best so we need to simplify the model mathematically to compensate for the solvers shortcomings.

• You could do a few SLP (sequential linear programming) iterations. Of course, I just would try out a few NLP solvers first. This is a fairly small problem. If you have a good starting point a good NLP solver may be very fast. Dec 1, 2021 at 4:52
• @Erwin Kalvelagen , thanks. Using solvers is not a problem. I took ln on both sides and did a implementation gurobi. We want to publish papers so we want to have some analytical simplification. So could you suggest any analytical approaches specifically for this type of objectives. Could you recommend any book or online resource where I can do this? Thx. Dec 1, 2021 at 5:54
• I will do some self advertisement here. Your objective, although nonconvex, is separable. You could try what we propose in halshs.archives-ouvertes.fr/LAAS-ROC/hal-03336022v2 or some of the references that we cite in this class of problems Dec 1, 2021 at 10:02
• Do you know whether the $C$ function is convex in $I_{i,v}$ for each pair $i, v$?
– prubin
Dec 1, 2021 at 16:54
• Do any of $a, b, c, d, E_a, R, T_{ref}, \Delta t, SOC_{init}, C_{bat}, R_{th}, V$ depend on $i,v$? If not i would all $I_{i,v}$ would assume the same value meaning you could just minimize one term and multiple that by $|N_v|*|TT_v|$? Dec 3, 2021 at 16:52