# uf_subproblem

> Solve the sub-optimization problem in distributed algorithm for user friendly problem

---

## Description
uf_subproblem solves the sub-optimization problem assigned for each
charger during the distributed algorithm for user friendly problem.
This function is only for the function user_friendly.

## Syntax
```matlab
[xi,optval] = uf_subproblem(T,l,x,d,P_max,P_min,A,b,empty_ind,G,h,cs,pc1,pc2,pc3,bs,c,rho)
```

## Input Arguments
| Name | Type | Description |
|------|------|--------------|
| `T` | *positive integer* | Number of time slots comprising a scenario |
| `l` | *(2T)×1 vector* | Variable for building the cost function |
| `x` | *(3T)×1 vector* | Variable for building the cost function |
| `d` | *(2T)×1 vector* | Variable for building the cost function |
| `P_max` | *nonnegative scalar* | Maximum aggregate charging power |
| `P_min` | *nonpositive scalar* | Minimum aggregate charging power |
| `A` | *T×T matrix* | Local constraint matrix |
| `b` | *T×1 vector* | Local constraint matrix |
| `empty_ind` | *T×1 vector* | Index for empty time slot (for one charger) |
| `G` | *matrix* | Penalty matrix |
| `h` | *vector* | Penalty matrix |
| `cs` | *1×T vector* | Vector for calculating (cost saving) cost function |
| `pc1` | *1×T vector* | Vector for calculating (priority charging) cost function |
| `pc2` | *T×1 vector* | Vector for calculating (priority charging) cost function |
| `pc3` | *T×T matrix* | Matrix for calculating (priority charging) cost function |
| `bs` | *1×T vector* | vector for calculating (battery saving) cost function |
| `c` | *positive scalar* | Parameter for optimization algorithm |
| `rho` | *nonnegative scalar* | Penalty parameter |


## Output Arguments
| Name | Type | Description |
|------|------|--------------|
| `xi` | *(3T)×1 vector* | Optimal solution of the subproblem |
| `optval` | *scalar* | Optimal value |