A Certifying MCKP Framework for $\Gamma$-Robust Discrete Pricing

Abstract

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-$\Gamma$ 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.