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Add new quadratization methods to QCQP - #158
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Add `ToReformulatedPolynomialBridge` (constraint and objective), which substitutes high-degree subexpressions of a `MOI.ScalarNonlinearFunction` with bounded auxiliary variables before conversion to a polynomial, and `QCQP.optimizer(model; preprocessing)` stacking these bridges on top of `QCQP.Optimizer`. `ToPolynomialBridge` now only bridges `MOI.ScalarNonlinearFunction` and non-integer exponents now throw instead of being rounded.
Add `AbstractQuadratizationMethod` with `GreedyQuadratization`, `CompositeQuadratization` (smart decomposition followed by a greedy, balanced or sequential phase, with optional variable reuse) and `OptimalQuadratization` (binary program minimizing the number of auxiliary variables). The method is chosen with the `QuadratizationMethod` optimizer attribute and `EnforceConsistencyConstraints` optionally adds the redundant constraints `v1 * v2 == v3 * v4`. `QCQP.Optimizer` no longer claims support for `MOI.ScalarNonlinearFunction` so that it is always converted to a polynomial.
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This PR adds a bunch of polynomial to QCQP reformulation methods.
CompositeQuadratization: a multi-parameter quadratization method. It is discussed in detail in my master's thesis. I'm not 100% happy with the interface and would like to know your opinion about it.
OptimalQuadratization: in most cases not very practical, but it creates the quadratization with the mathematically minimal amount of auxiliary variables.
GreedyQuadratizion: the new name of the old default PolyJuMP.QCQP method.
Adding a quadratization method requires only creating a type that inherits from
AbstractQuadratizationMethod', create adecompose()` overload for this type, and provide the type as an attribute to the optimizer.It also adds consistency constraints of type
v1 * v2 == v3 * v4if this is true for the monomials represented by these (auxiliary) variables. It is an optional parameter that's deactivated by default.None of these parameters have an "optimal" setting, it is recommended to identify what works best for your specific problem by trial and error.