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Open question · OPEN-SOFT-HARD-ZEROTEMP

Does soft optimization approach the hard problem?

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Meaning

Understand the zero-temperature limit and what survives hardening.

Scientific context

When, if ever, the stationary points or optima of a softened family converge to hard stationary points or optima as the temperature goes to zero is open. Until it is answered, hardening a soft fit is an operation whose result must be verified on the hard objective rather than inferred from the soft one.

Statement

Determine when stationary points or optima of softened randomized affine/Voronoi families converge to hard stationary/optimal partitions as temperature tends to zero.

Relationships

Related settled results

Where it comes from

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Local graph
Local map of Soft-to-hard zero-temperature limit2 prerequisites to the left, 0 results it enables or questions it raises to the right, 0 boundary cases below.Fixed-temperature soft optimization has standard nonconvex stationary guaranteesFixed-temperature sof…Hard finite geometric objective is piecewise constant in boundary parametersHard finite geometric…Soft-to-hard zero-temperature limitSoft-to-hard zero-tem…
Prerequisites to the left, what it enables or raises to the right, boundary cases below. Open in the map.

The question in full

Soft-to-hard zero-temperature limit

When do stationary points or optima of a temperature-softened affine/Voronoi family converge to hard stationary points or optima as τ→0\tau\to0? Separate objective convergence along a parameter path, convergence of global optima, and convergence of local branches.

From OPEN_PROBLEMS.md