Is this formulation linear or non-linear?

Question

Can you help me figure out if this formulation constitutes a non-linear problem? I believe It is a linear problem but my solver (GAMS) is unable to produce a acceptable solution.

$x,y$ and $\text{state}$ are variables and the rest are parameters.

$\sum\limits_{n=1}^{N}\left[\sum\limits_{i=1}^{T}x_{n,i}\cdot p_{i}-y_{n,i}\cdot p_{i}\right]$

$ \forall_{n} \forall_{i} x_{n,i},y_{n,i}\geq 0 \wedge x_{n,i},y_{n,i}\leq M_{n} $

$\forall_{n} \forall_{i} \text{state}_{n,i}=\text{state}_{n,i-1}+y_{n,i}-x_{n,i}$

$ \forall_{n} \forall_{i} \text{statemin}_{n} \leq \text{state}_{n,i} \leq \text{statemax}_{n}$

$ \text{flow}_{l,i} = A \cdot K_{i}$

$ \text{flow}_{l,i} \leq \text{fmax}_{l}$

$ K_{i} = L_{i} - (G_{i} + y_{i} - x_{i})$

Where $A$ is an $N \times N$ matrix. Any feedback is appreciated,

The GAMS code is the following:

** Define the structure to connect with the matlab code
*$onempty
$include matglobs.gms

set      t /1*%timeSteps%/,
         b /1*%bus%/,
         l /1*%lines%/
         ;

Positive Variable x(b,t),
                  y(b,t)
                  state(b,t)
                  ;

Free Variable    res, unit(b), revenue, flow(l,t), K(b,t);

parameters       size(b), rate(b), fmax(l), P(b,t), A(l,b), price(t);

$if exist matdata.gms $include matdata.gms

Equations

stateCalc1(b,t)
stateCalc2(b,t)
Initial_y(b,t)
Initial_x(b,t)
stateMax(b,t)
stateMin(b,t)

max_x(b,t)
max_y(b,t)

K_Calc(b,t)
flow_Calc(l,t)
lim_K(l,t)

Con10(b)
Con11
Obj
;

stateCalc1(b,t)$(ord(t)=1)..      state(b,t) =e= size(b)/2;
stateCalc2(b,t)$(ord(t)>1)..      state(b,t) =e= state(b,t-1) + y(b,t) - x(b,t);

Initial_y(b,t)$(ord(t)=1)..           y(b,t) =e= 0;
Initial_x(b,t)$(ord(t)=1)..           x(b,t) =e= 0;

stateMax(b,t)..                   state(b,t) =l= size(b);
stateMin(b,t)..                   state(b,t) =g= 0;

max_x(b,t)..        x(b,t) =l= rate(b)*size(b); 
max_y(b,t)..        y(b,t) =l= rate(b)*size(b);


K_Calc(b,t)..       K(b,t) =e= P(b,t)+y(b,t)-x(b,t);
flow_Calc(l,t)..    flow(l,t) =e= sum(b, A(l,b)*K(b,t));
lim_K(l,t)..        flow(l,t) =l= fmax(l);

Con10(b)..               sum(t, x(b,t)*price(t) - y(b,t)*price(t)) =e= unit(b);
Con11..                  sum(b, unit(b)) =e= revenue;

Obj..                    revenue =e= res;

Model Opt_Bat /all/;

Solve Opt_Bat using LP maximazing res;

Display state.l, size;

$libinclude matout res.l

To be noted that $M_{n} = size_{n} * rate_{n}$.

Answer 1

The model you describe is linear. There are a couple of reasons why GAMS wouldn't like it though: (I) did you define the right solver for your problem? and (ii) GAMS initialises any uninitiated variables to 0 - it then proceeds to evaluate all constraints at the initial values before sending the problem to a solver. If the initial values are infeasible (including the implicit 0 values), then GAMS will refuse to solve your model.

Another possibility for what you report is that your problem is integer, in which case there can be multiple solutions unless you specify that you desire a very small MIP gap for convergence.