Hardness

The question nobody asks precisely

"Is this solution robust?" is usually answered with a yes or a no, which is a strange way to talk about a continuous property. Two plans can both be feasible under nominal data and both survive the worst case, and still be nothing alike in how they behave in between.

Hardness gives that middle ground a number. For an optimization problem whose coefficients live in intervals rather than at points, it aggregates per-constraint worst-case violation into a single scalar η ∈ [0, 1], so feasible solutions can be ranked rather than merely accepted.

Two engines that have to agree

The measure is computed two ways, and the fact that they agree is the point.

Closed form evaluates a generalized Irwin–Hall CDF exactly. It is the right tool for small problems and it is fast — microseconds per evaluation.

Monte Carlo estimates the same quantity from a violations matrix, with bootstrap confidence intervals. It scales where the closed form does not.

When an exact method and a sampling method built from different mathematics land on the same curve, you have evidence the definition is sound rather than an artifact of one implementation.

Where it gets interesting

Take a fixed budget — say a renewable generation portfolio with a hard $50M cap — and sweep one decision variable. Cost is constant by construction, so it cannot rank anything. Three different robustness measures then proceed to disagree about which plan to buy.

That disagreement is the whole argument for caring about the definition. One measure slopes monotonically the wrong way. Another is pinned flat at zero across half the range, unable to express an opinion at all. η picks a plan whose realized shortfall, under twenty thousand simulated futures, is a small fraction of what the naive choice delivers — at identical spend.

The part I am most attached to

The core theory is formalized in Agda and machine-checked. Not because a referee asked, but because a robustness measure that is itself unproven is a peculiar thing to publish.

That formalization is the audit trail: it says the properties claimed for η are not claimed on the strength of my careful reading of my own proof.

Composition

Hardness speaks the same problem schema as SIROM — lb_A, ub_A, lb_b, ub_b, integer_variables — deliberately. SIROM produces robust solutions; this scores them. Two peer-reviewed methods that compose without an adapter.