In [1] a small non-convex MINLP is stated. The elements are:
The set group was populated using the GAMS statement:
The solution can look like:
This is a small model, so we can try a few different global and local solvers.
Interesting:
This formulation has more linear constraints, but in return, we get a much simpler non-linear objective. Note that we also got rid of the minus sign in the objective. This was probably put in there to convert the problem to a minimization problem.
Some of the solvers perform better on this model:
Couenne and especially Antigone do much better on this formulation.
Gurobi can solve non-convex quadratic models, so it is interesting to reformulate the problem into a form that Gurobi can digest.
Note that this requires the Gurobi option nonConvex=2.
Gurobi solves this model very quickly, although the node count is not that low.
- We have binary variables \(x_i \in \{0,1\}\) for \(i=1,\dots,52\)
- The data is comprised of three arrays: \(a_i\), \(b_i\) and \(c_i\)
- We want to minimize the following objective: \[\begin{align} - &\left(\frac{1166}{2000} \sum_i a_i \cdot x_i\right) \times \\ & \left(\frac{1}{2100} \sum_i b_i \cdot x_i + 0.05\right) \times \\ & \left(\frac{1}{1500} \sum_i c_i \cdot x_i + 1.5\right)\end{align}\]
- Of the first four elements \(x_i\), only one has the value 1, and the other three are 0. Same thing for the second, third, etc. groups of four.
Data
---- 66 SET iitems
i1 , i2 , i3 , i4 , i5 , i6 , i7 , i8 , i9 , i10, i11, i12, i13, i14, i15
i16, i17, i18, i19, i20, i21, i22, i23, i24, i25, i26, i27, i28, i29, i30
i31, i32, i33, i34, i35, i36, i37, i38, i39, i40, i41, i42, i43, i44, i45
i46, i47, i48, i49, i50, i51, i52
---- 66 SET ggroups
group1 , group2 , group3 , group4 , group5 , group6 , group7 , group8 , group9 , group10
group11, group12, group13
---- 66 SET groupgrouping of items
i1 i2 i3 i4 i5 i6 i7 i8 i9
group1 YES YES YES YES
group2 YES YES YES YES
group3 YES
+ i10 i11 i12 i13 i14 i15 i16 i17 i18
group3 YES YES YES
group4 YES YES YES YES
group5 YES YES
+ i19 i20 i21 i22 i23 i24 i25 i26 i27
group5 YES YES
group6 YES YES YES YES
group7 YES YES YES
+ i28 i29 i30 i31 i32 i33 i34 i35 i36
group7 YES
group8 YES YES YES YES
group9 YES YES YES YES
+ i37 i38 i39 i40 i41 i42 i43 i44 i45
group10 YES YES YES YES
group11 YES YES YES YES
group12 YES
+ i46 i47 i48 i49 i50 i51 i52
group12 YES YES YES
group13 YES YES YES YES
---- 66 PARAMETER data
a b c
i1 251.000179.000179.000
i2 179.000251.000179.000
i3 215.000215.000118.000
i4 251.000179.000
i5 63.00045.00045.000
i6 45.00063.00045.000
i7 54.00054.00030.000
i8 63.00045.000
i9 47.00034.00034.000
i10 34.00047.00034.000
i11 40.00040.00022.000
i12 47.00034.000
i13 141.000101.000101.000
i14 101.000141.000101.000
i15 121.000121.00067.000
i16 141.000101.000
i17 47.00034.00034.000
i18 34.00047.00034.000
i19 40.00040.00022.000
i20 47.00034.000
i21 94.00067.00067.000
i22 67.00094.00067.000
i23 81.00081.00044.000
i24 94.00067.000
i25 47.00034.00034.000
i26 34.00047.00034.000
i27 40.00040.00022.000
i28 47.00034.000
i29 157.000108.000108.000
i30 108.000157.000108.000
i31 133.000133.00071.000
i32 157.000108.000
i33 126.00085.00085.000
i34 85.000126.00085.000
i35 106.000106.00056.000
i36 126.00085.000
i37 126.00085.00085.000
i38 85.000126.00085.000
i39 106.000106.00056.000
i40 126.00085.000
i41 110.00074.00074.000
i42 74.000110.00074.000
i43 92.00092.00049.000
i44 110.00074.000
i45 110.00074.00074.000
i46 74.000110.00074.000
i47 92.00092.00049.000
i48 110.00074.000
i49 63.00040.00040.000
i50 40.00063.00040.000
i51 52.00052.00027.000
i52 63.00040.000
The set group was populated using the GAMS statement:
set group(g,i) 'grouping of items'; |
The term 0.5 is to make sure we don't take the floor of integer values (the floor function can then deliver different results depending on the very last bit - possibly dangerous when we deal with floating-point computations).
Direct MINLP formulation
A direct MINLP model formulation can look like:
| MINLP model A |
|---|
| \[\begin{align}\min\> - &\left(\frac{1166}{2000} \sum_i \color{darkblue}a_i \cdot \color{darkred}x_i\right) \times \\ & \left(\frac{1}{2100} \sum_i \color{darkblue}b_i \cdot \color{darkred}x_i + 0.05\right) \times \\ & \left(\frac{1}{1500} \sum_i \color{darkblue}c_i \cdot \color{darkred}x_i + 1.5\right)\\ & \sum_{i|\color{darkblue}{\mathit{group}}(g,i)} \color{darkred}x_i = 1 &&\forall g\\ & \color{darkred}x_i \in \{0,1\}\end{align}\] |
The solution can look like:
---- 89 VARIABLE x.L
i1 1.000, i5 1.000, i9 1.000, i14 1.000, i17 1.000, i21 1.000, i25 1.000, i29 1.000
i34 1.000, i38 1.000, i41 1.000, i46 1.000, i49 1.000
---- 89 VARIABLE z.L = -889.346obj
Of course, we see 13 items selected (one for each group of 4).
This is a small model, so we can try a few different global and local solvers.
| Solver | Type | Obj | Time | Nodes |
|---|---|---|---|---|
| Couenne | global | -889.3463 | 18.25 | 9972 |
| Baron | global | -889.3463 | 7.18 | 389 |
| Antigone | global | Stuck in preprocessing | ||
| Bonmin | local | -889.3463 | 89 | 2280 |
Interesting:
- Antigone has problems during preprocessing
- The local solver Bonmin finds the global optimum but is somewhat slow
Alternative formulation
There is an alternative formulation. We can substitute out the linear parts of the objective function. This will make the model larger (extra variables and equations), but less non-linear (e.g. in terms of non-linear variables).
| MINLP model B |
|---|
| \[\begin{align}\max & \prod_k\color{darkred}v_k \\ & \color{darkred}v_1 = \frac{1166}{2000} \sum_i \color{darkblue}a_i \cdot \color{darkred}x_i \\ & \color{darkred}v_2 = \frac{1}{2100} \sum_i \color{darkblue}b_i \cdot \color{darkred}x_i + 0.05 \\ & \color{darkred}v_3 = \frac{1}{1500} \sum_i \color{darkblue}c_i \cdot \color{darkred}x_i + 1.5\\ & \sum_{i|\color{darkblue}{\mathit{group}}(g,i)} \color{darkred}x_i = 1 &&\forall g\\ & \color{darkred}x_i \in \{0,1\}\end{align}\] |
This formulation has more linear constraints, but in return, we get a much simpler non-linear objective. Note that we also got rid of the minus sign in the objective. This was probably put in there to convert the problem to a minimization problem.
The solution is obviously the same:
---- 122 VARIABLE x.L
i1 1.000, i6 1.000, i9 1.000, i13 1.000, i17 1.000, i22 1.000, i25 1.000, i29 1.000
i33 1.000, i38 1.000, i42 1.000, i46 1.000, i49 1.000
---- 122 VARIABLE v.L obj factors
obj1 713.592, obj2 0.582, obj3 2.140
---- 122 VARIABLE z.L = 889.346obj
Some of the solvers perform better on this model:
| Solver | Type | Obj | Time | Nodes |
|---|---|---|---|---|
| Couenne | global | 889.3463 | 5.1 | 4398 |
| Baron | global | 889.3463 | 9.5 | 309 |
| Antigone | global | 889.3463 | 2 | 5 |
| Bonmin | local | 889.3463 | 105 | 3336 |
Non-convex quadratic formulation
Gurobi can solve non-convex quadratic models, so it is interesting to reformulate the problem into a form that Gurobi can digest.
| MIQCP model |
|---|
| \[\begin{align}\max\> & \color{darkred}v_1 \cdot \color{darkred}v_{23} \\& \color{darkred}v_{23} = \color{darkred}v_{2} \cdot \color{darkred}v_{3}\\ & \color{darkred}v_1 = \frac{1166}{2000} \sum_i \color{darkblue}a_i \cdot \color{darkred}x_i \\ & \color{darkred}v_2 = \frac{1}{2100} \sum_i \color{darkblue}b_i \cdot \color{darkred}x_i + 0.05 \\ & \color{darkred}v_3 = \frac{1}{1500} \sum_i \color{darkblue}c_i \cdot \color{darkred}x_i + 1.5\\ & \sum_{i|\color{darkblue}{\mathit{group}}(g,i)} \color{darkred}x_i = 1 &&\forall g\\ & \color{darkred}x_i \in \{0,1\}\end{align}\] |
Note that this requires the Gurobi option nonConvex=2.
| Solver | Type | Obj | Time | Nodes |
|---|---|---|---|---|
| Gurobi | global | 889.3463 | 0.26 | 6985 |
Gurobi solves this model very quickly, although the node count is not that low.
Conclusion
Even this small MINLP model has some interesting wrinkles and reformulations.
References
- R Binary Integer Optimization with Groups, https://stackoverflow.com/questions/63584707/r-binary-integer-optimization-with-groups