How does SCIP deal with power terms in the constraints?

[abb-omidi]

Dear support team,

I hope you are doing well. I have seen an interesting question regarding the power term in the optimization models, and I am willing to know how SCIP deals with that, specifically, how it linearizes the model internally. The terms in the model would be something like $(y/x)$ and $(a^{y})$ where $x$ and $y$ are unrestriced continuse variables, $\in \mathbb{R}$, and $a$ is a constant number. What I am trying, and also based on the SCIP doc. is:

model = Model()

x = model.addVar(name="x", vtype="C", lb = -100, ub = 100)
y = model.addVar(name="y", vtype="C", lb = -100, ub = 100)
s = model.addVar(name="s", vtype="C", lb = -100, ub = 100)

model.setObjective(s, "maximize")

model.addCons( (5**y) == 5*x, "c0")
model.addCons( (2**y) + (8**y) == 130, "c1")
model.addCons( s == (y/x), "c2")

model.redirectOutput()
model.printVersion()

model.setPresolve(SCIP_PARAMSETTING.OFF)
model.setSeparating(SCIP_PARAMSETTING.OFF)
model.setHeuristics(SCIP_PARAMSETTING.OFF)

model.optimize()

By solving this, SCIP throws an error:

TypeError                                 Traceback (most recent call last)
[/tmp/ipython-input-1872219861.py](https://localhost:8080/#) in <cell line: 0>()
     10 model.setObjective(s, "maximize")
     11 
---> 12 model.addCons( (5**y) == 5*x, "c0")
     13 model.addCons( (2**y) + (8**y) == 130, "c1")
     14 model.addCons( s == (y/x), "c2")

TypeError: unsupported operand type(s) for ** or pow(): 'int' and 'pyscipopt.scip.Variable'

• Is there any specific way to add power terms, also containing the variable(s) in the power, in SCIP?

Also, I have often used GAMS/SCIP to solve the problem. It surprisingly solves the problem optimally, and the output is:

presolving:
(round 1, fast)       0 del vars, 0 del conss, 0 add conss, 13 chg bounds, 0 chg sides, 0 chg coeffs, 0 upgd conss, 0 impls, 0 clqs
   (0.0s) symmetry computation started: requiring (bin +, int +, cont +), (fixed: bin -, int -, cont -)
   (0.0s) no symmetry present (symcode time: 0.00)
presolving (2 rounds: 2 fast, 1 medium, 1 exhaustive):
 0 deleted vars, 0 deleted constraints, 0 added constraints, 13 tightened bounds, 0 added holes, 0 changed sides, 0 changed coefficients
 0 implications, 0 cliques
presolved problem has 3 variables (0 bin, 0 int, 0 impl, 3 cont) and 3 constraints
      3 constraints of type <nonlinear>
Presolving Time: 0.00

 time | node  | left  |LP iter|LP it/n|mem/heur|mdpt |vars |cons |rows |cuts |sepa|confs|strbr|  dualbound   | primalbound  |  gap   | compl. 
  0.0s|     1 |     0 |     6 |     - |   694k |   0 |  11 |   3 |  16 |   0 |  0 |   0 |   0 | 7.028440e+00 |      --      |    Inf | unknown
L 0.0s|     1 |     0 |     6 |     - |  subnlp|   0 |  11 |   3 |  16 |   0 |  0 |   0 |   0 | 7.028440e+00 | 6.577991e+00 |   6.85%| unknown
  0.0s|     1 |     0 |     7 |     - |   703k |   0 |  11 |   3 |  17 |   1 |  1 |   0 |   0 | 7.028440e+00 | 6.577991e+00 |   6.85%| unknown
  0.0s|     1 |     0 |     7 |     - |   703k |   0 |  11 |   3 |  17 |   1 |  1 |   0 |   0 | 7.028440e+00 | 6.577991e+00 |   6.85%| unknown
  0.0s|     1 |     0 |     8 |     - |   703k |   0 |  11 |   3 |  18 |   2 |  2 |   0 |   0 | 7.028440e+00 | 6.577991e+00 |   6.85%| unknown

SCIP Status        : solving was interrupted [gap limit reached]
Solving Time (sec) : 0.00
Solving Nodes      : 1
Primal Bound       : +6.57799101062506e+00 (1 solutions)
Dual Bound         : +6.57800059142918e+00
Gap                : 0.00 %

------------------------------------------
object.L         =        6.578  
VARIABLE x.L     =        0.353  
VARIABLE y.L     =        2.322  

• How does GAMS really translate the problem in a way that SCIP can deal with that?

Best regards
Abbas

[Joao-Dionisio]

Hey @abb-omidi , long time no see!

I think the issue is not with the variable in the exponent, but rather with the basis. From what I understand, SCIP can only directly handle variables in exponents where the basis is e.
So if you want to do

2**y <= 5

You should be able to

scip.exp(scip.log(2)*y)

(I'm on mobile, so apologies for the formatting)

As for what GAMS does, I'll defer over to the one and only @svigerske

[svigerske]

Yes, like @Joao-Dionisio says. Many interfaces do this translation into exp-log automatically. Looks like PySCIPOpt doesn't yet.

[abb-omidi]

@Joao-Dionisio, @svigerske,

Thank you so much for your prompt response. Please, let me check what you suggested and I will get back to you asap.
However, as an alternative, I have tried to reformulate the problem based on only log function, and the result is corresponding to the optimal solution. Also, as a follow up question, I am interested to reformulate my alternative format as a conic problem and I was wondering if you could advise me how is log(x) = y reformulated as a conic form?

Best regards
Abbas

[svigerske]

The set {(x,y): log(x) = y} is nonconvex. So there is no cone formulation.
If you relax one side of the equality to become a convex inequality, then you can formulate it as exponential cone (5.3) with x2 fixed to 1.

(SCIP/PySCIPOpt doesn't do conic programming, though.)

[abb-omidi]

Dear @svigerske,

Thank you so much for your comments as well as provided link.

Regards