[SCIP] Solving a Convex MINLP

[Guillaume BS]

Dear SCIP team,

I am trying to solve a relatively simple convex MINLP of the shape:

min c^t x
s.t.:    g_i(x,y) \leq 0      i \in 1..m
           Ax + By \leq d
           x \in R^n, y \in B^n'

with all functions g_i convex (cf "Convex MINLP" in https://www.scipopt.org/doc/html/WHATPROBLEMS.php ).
I try to solve the MINLP (with variables y in {0,1}, and its convex relaxation (with variables y in [0,1]   ).

In the documentation, a parameter "constraints/nonlinear/assumeconvex" is described, which allows assuming that all constraint functions are convex.

The convex relaxed NLP with y \in [0,1]:

  *
is easily solved by SCIP when the  "constraints/nonlinear/assumeconvex" parameter equals TRUE
  *
is not easily solved by SCIP (several minutes for a toy instance with a few variables and constraints) when  "constraints/nonlinear/assumeconvex" equals FALSE ;  from the logs, it seems SCIP can't prove the complexity of the constraints and runs a spatial branch-and-bound (several nodes are generated, despite having only continuous variables).

The MINLP with y \in {0,1}:

  *
is not correctly solved when "constraints/nonlinear/assumeconvex" is true: a feasible solution, with values of the binary variable y fixed "randomly", is returned by SCIP.
  *
is very slow when "constraints/nonlinear/assumeconvex" is false

So it seems the  "constraints/nonlinear/assumeconvex" parameters not only controles the convexity of the constraints in the NLP relaxation, but also the convexity of the variables' domains - which would make it improper to solve an MINLP.

Is this understanding correct and, if so, is there a way to solve a MINLP when all constraints are known to be convex - but the variables domain isn't ? (for instance by stating, for each nonlinear constraint, that it should be considered as convex?)

Many thanks,
Guilaume
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