RENS

[nguidotti]

This PR implements RENS (Relaxation Enforced Neighborhood Search, see [1, 2]) heuristic. RENS construct and then solve the following sub-MIP:

$$\begin{align} & z^* = \min \, \vec{c}^T \vec{x} \\ & \text{subject to} & \mathbf{A} \vec{x} = \vec{b} \\ & & l_i \leq x_i \leq u_i, \forall i \in \mathcal{N} \setminus \mathcal{I} \\ & & \lfloor \check{x}_j \rfloor \leq x_i \leq \lceil \check{x}_j \rceil, \forall i \in \mathcal{I} \\ & & \vec{x} \in \mathbb{R}^n \\ & & x_j \in \mathbb{Z}, \forall j \in \mathcal{I} \end{align}$$

where $\mathcal{N}$ is the set of all variables and $\mathcal{I}$ is the set of integer values. In essence, all integer variables with integral values are fixed, while the ones with fractional values are restricted to be rounded up or down.

This re-use the same infrastructure as RINS, i.e., it passes through presolve and can recurse. It requires some refactoring of the sub-MIP code to support these additional features.

Closes#1584.

Results

MIPLIB2017, GH200, 10min

================================================================================
root-heuristics-1 (1) vs rens-1 (2)
================================================================================
------------------------------------------------------------------------------------------------------------------------------
| | Run 1 | Run 2 | Abs. Diff. | Rel. Diff. (%) |
------------------------------------------------------------------------------------------------------------------------------
| Imported 240 240 +0 --- |
| Feasible 227 227 +0 --- |
| Optimal 87 92 +5 --- |
| Solutions with <0.1% primal gap 145 144 -1 --- |
| Nodes explored (mean) 1.366e+07 1.309e+07 -5.731e+05 -4.19 |
| Nodes explored (shifted geomean) 8947 5351 -3596 -40.2 |
| Relative MIP gap (mean) 0.2709 0.248 -0.02291 -8.45 |
| Relative MIP gap (shifted geomean) 0.08762 0.087 -0.0006245 -0.713 |
| Solve time (mean) 419 405.6 -13.35 -3.19 |
| Solve time (shifted geomean) 189.4 176 -13.38 -7.06 |
| Primal gap (mean) 9.237 9.634 +0.3963 +4.29 |
| Primal gap (shifted geomean) 0.3752 0.3633 -0.01198 -3.19 |
| Primal integral (mean) 20.14 20.08 -0.06726 -0.334 |
| Primal integral (shifted geomean) 2.071 1.978 -0.09335 -4.51 |
------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------
| Name | status 1 | status 2 |
----------------------------------------------------------------------
| app1-2 feasible optimal |
| cbs-cta feasible optimal |
| glass-sc feasible optimal |
| mzzv11 feasible optimal |
| n2seq36q feasible optimal |
| neos-5093327-huahum optimal feasible |
| triptim1 feasible optimal |
----------------------------------------------------------------------
================================================================================
root-heuristics-2 (1) vs rens-2 (2)
================================================================================
------------------------------------------------------------------------------------------------------------------------------
| | Run 1 | Run 2 | Abs. Diff. | Rel. Diff. (%) |
------------------------------------------------------------------------------------------------------------------------------
| Imported 240 240 +0 --- |
| Feasible 225 228 +3 --- |
| Optimal 87 91 +4 --- |
| Solutions with <0.1% primal gap 143 145 +2 --- |
| Nodes explored (mean) 1.322e+07 1.306e+07 -1.595e+05 -1.21 |
| Nodes explored (shifted geomean) 8201 6479 -1722 -21 |
| Relative MIP gap (mean) 0.2695 0.2752 +0.005638 +2.09 |
| Relative MIP gap (shifted geomean) 0.0867 0.08533 -0.001362 -1.57 |
| Solve time (mean) 415.6 407.4 -8.174 -1.97 |
| Solve time (shifted geomean) 179.6 175.6 -4.007 -2.23 |
| Primal gap (mean) 10.28 9.136 -1.141 -11.1 |
| Primal gap (shifted geomean) 0.4062 0.3614 -0.04481 -11 |
| Primal integral (mean) 22.11 19.93 -2.177 -9.85 |
| Primal integral (shifted geomean) 2.074 1.961 -0.1133 -5.46 |
------------------------------------------------------------------------------------------------------------------------------
----------------------------------------------------------------------
| Name | status 1 | status 2 |
----------------------------------------------------------------------
| app1-2 feasible optimal |
| cmflsp50-24-8-8 feasible optimal |
| glass-sc feasible optimal |
| neos-3216931-puriri timeout feasible |
| neos-5104907-jarama timeout feasible |
| peg-solitaire-a3 timeout optimal |
----------------------------------------------------------------------

References

[1] T. Berthold, “RENS: The optimal rounding,” Math. Prog. Comp., vol. 6, no. 1, pp. 33–54, Mar. 2014, doi: 10.1007/s12532-013-0060-9.
[2] T. Achterberg, “Constraint Integer Programming,” PhD, Technischen Universität Berlin, Berlin, 2007. doi: 10.14279/depositonce-1634.

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[nguidotti]

/ok to test 7a9e5f4

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