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Optimizing over the circle:

Now we provide our first thought experiment: Consider the problem of optimizing a linear function over the feasible region defined by the constraints

<span>$</span>\sin(2\pi\frac{i}{10^6}) x + \cos(2\pi\frac{i}{10^6}) y \leq
1,\,\forall i\in\{1,\ldots,10^6\},<span>$</span>
i.e. the feasible region is essentially a unit circle in <span>$</span>\mathbb{R}^2<span>$</span>. Note that for all objective functions, the corresponding optimal point will be defined by two linear constraints that are very close to be parallel. What will happen to the numerical solution to the problem? Can you guess? The situation is depicted in the figure below:

To perform the experiment, we execute the code, where we randomly select an objective vector, find the optimal solution to the resulting optimization problem, and compute several relevant quantities:

  • The worst distance between the reported primal solution, and the theoretical solution to the problem of actually optimizing over a perfect circle, over all previous runs.
  • The worst bound violation reported by Gurobi over all previous runs.
  • The worst constraint violation reported by Gurobi over all previous runs.
  • The worst dual violation reported by Gurobi over all previous runs.
  • The number of previous experiments.
  • Accumulated number of simplex iterations.
  • The <span>$</span>\kappa<span>$</span> (KappaExact attribute) value for the current optimal basis.

Sample output is shown below:

Added 2 Vars and 1048576 constraints in 19.19 seconds
Errors: 8.65535e-08 0 2.94137e-07 2.77556e-17 Iter 0 10 Kappa 3150.06
Errors: 4.81978e-07 0 3.22359e-07 2.77556e-17 Iter 1 21 Kappa 3009.12
Errors: 4.81978e-07 0 3.4936e-07 1.11022e-16 Iter 2 33 Kappa 2890.58
Errors: 1.53201e-06 0 9.78818e-07 1.11022e-16 Iter 6 79 Kappa 1727.89
Errors: 1.61065e-06 0 8.26005e-07 1.11022e-16 Iter 46 536 Kappa 1880.73
Errors: 1.61065e-06 0 8.84782e-07 1.11022e-16 Iter 52 602 Kappa 1817.27
Errors: 1.61065e-06 0 9.4557e-07 1.11022e-16 Iter 54 625 Kappa 1757.96
Errors: 1.69167e-06 0 9.78818e-07 1.11022e-16 Iter 64 742 Kappa 1727.89
Errors: 1.69167e-06 0 3.8268e-07 1.66533e-16 Iter 88 1022 Kappa 2761.99
Errors: 1.69167e-06 0 9.04817e-07 1.66533e-16 Iter 92 1067 Kappa 1797.06
Errors: 1.69167e-06 0 2.94137e-07 2.22045e-16 Iter 94 1089 Kappa 3150.06
Errors: 1.69167e-06 0 3.29612e-07 2.22045e-16 Iter 95 1101 Kappa 2975.84
Errors: 1.69167e-06 0 3.4936e-07 2.22045e-16 Iter 98 1137 Kappa 2890.58
Errors: 1.69167e-06 0 9.25086e-07 2.22045e-16 Iter 99 1147 Kappa 1777.3
Errors: 1.69167e-06 0 9.78818e-07 2.22045e-16 Iter 107 1237 Kappa 1727.89
Errors: 1.69167e-06 0 9.99895e-07 2.22045e-16 Iter 112 1293 Kappa 1709.61
Errors: 1.84851e-06 0 9.78818e-07 2.22045e-16 Iter 132 1523 Kappa 1727.89
Errors: 1.96603e-06 0 9.99895e-07 2.22045e-16 Iter 134 1545 Kappa 1709.61

Surprisingly the reported errors are rather small. Why is this? There are at least two contributing factors: the model has a bounded feasible region (in this case the range of both variables is <span>$</span>[-1,1]<span>$</span>). In addition, the distance from one extreme point (a point at the intersection of two neighboring constraints) to its neighbor is also relatively small, so all <span>$</span>\varepsilon<span>$</span>-optimal solutions are close to each other.

We encourage you to play with this code, perturb some of the input data, and analyze the results. You will see the discrepancies between the theoretical and the numerical optimal solution will be comparable to the sizes of the perturbations.

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