Anytime Solver Evaluation with a Normalized Signed Primal Integral and Explicit Reference Policies - Extended Version

Abstract

Anytime solvers return usable solutions before they terminate and improve them while time remains. Their progress is commonly summarized by a primal integral, a final gap, or convergence curves. Each summary leaves consequential choices open: how to score a run before its first feasible solution, whether poor incumbents are truncated, and whether the reference value is updated after the experiment or version-frozen beforehand. These choices become visible when runs produce no valid incumbent or improve a published best-known value. We study a normalized signed primal integral built from a bounded relative gap. It assigns an intrinsic worst value to an empty run, requires no acceptance threshold, and assigns negative instantaneous gaps to incumbents that beat a frozen reference. We compare the smooth difference-over-sum kernel with a signed version of Berthold's max-normalized gap and use the former as a working default. We also distinguish analysis-time from version-frozen reference policies and recommend reading the score with the raw final gap, mean convergence curve, and target-attainment curve. We evaluate these choices on a five-arm routing campaign and two model-fidelity ladders. The two signed kernels preserve every panel ordering in this study, whereas a common acceptance threshold compresses the distances between arms and reverses one panel ordering. Updating 54 of the 212 reference values changes score levels and removes all negative scores, while leaving the observed panel orderings unchanged. An exploratory screen also identifies 12 panel comparisons in which similar integral scores conceal materially different endpoints or attainment rates. The implementation, frozen inputs, and generators are openly released.