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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