Zi Yuan Eric Shao | Searcharxiv

arXiv2026

A Certifying MCKP Framework for $Γ$-Robust Discrete Pricing

We study finite-menu portfolio pricing under a ratio margin requirement, price-admissibility constraints, and integer-budget demand uncertainty. The pricing model reduces exactly to a multiple-choice knapsack problem (MCKP), while the coupled robust constraint is equivalent to a finite family of fixed-threshold MCKPs indexed by zero and every original option deviation. For each threshold, classical upper-hull geometry yields an LP bound and a feasible one-item rounding rule whose loss is bounded by a single adjacent hull-value jump; under bounded jumps and linear objective growth, the corresponding relative bound is $O(1/n)$. An exact-arithmetic branch-and-bound construction provides either a global optimum or a valid anytime gap over the complete threshold family, and direct evaluation of the original sorted-$Γ$ protection term checks every returned pricing decision. The contribution is an end-to-end, pricing-specific certifying framework that connects these ingredients, states their boundary conditions, and implements the full original breakpoint family without tolerance clustering.

math.OC↗

arXiv2026

Simultaneous Group-Envelope Bounds for $Γ$-Robust Multiple-Choice Knapsack Problems

Many robust planning problems are solved by checking a family of ordinary optimization problems, one for each uncertainty threshold. Repeatedly building and solving those relaxations can dominate runtime. We show that, when a decision chooses exactly one option from each group, the entire threshold family of multiple-choice knapsack relaxations can instead be bounded together. A cancellation removes threshold-specific baselines and reduces each group's contribution to two simple envelopes. After sorting, one multiplier is evaluated across all thresholds in time linear in their number and nearly linear in the number of options. The resulting interval bound is valid, matches the ordinary relaxation on a single feasible threshold, and is at least as strong as the matched group-clique comparator. A certified one-dimensional search controls multiplier error, while exact checks resolve numerically ambiguous feasibility and comparison cases. In an otherwise identical adaptive search, the method wins all 60 instance-level median comparisons and achieves a 2.37-fold geometric-mean speedup. Additional ablations, independent linear-program checks, and nine instances transformed from a published archive test the mechanism beyond the internal generator. A separate integer audit shows the boundary: faster relaxation bounds need not help when integer search dominates.

math.OC↗