arXiv subjects

Zhongqi Wu

Publications and source records attributed to Zhongqi Wu.

2 recordsLinked to original sources

arXiv2026

Disjunctive Submodular Functions: Envelopes and Applications to Inventory and 0-1 Quadratic Optimization

This paper considers convex envelopes of disjunctive submodular functions---functions that are lattice family submodular over faces of a hypercube---and constructs the first strongly polynomial algorithm for their separation when there are two facial disjunctions. Submodular functions, whose convex envelopes are characterized by the Lovász extension, have occupied a fundamental role in constructing relaxations for combinatorial and nonlinear optimization problems. However, disjunctive submodular function envelopes have not been explored besides the use of ellipsoid algorithm, which remains practically intractable. Our algorithm is derived in three steps by expressing the disjunctive function as a minimum of two extended submodular functions, introducing a variable lifting technique, and constructing the sublinear envelope in the lifted space. The paper also makes several other contributions. First, we provide a disjunctive formulation for the case where each submodular function admits a linear programming formulation. Second, we derive the closed-form sublinear envelope characterization for intersecting submodular functions, yielding new structural insights into a multi-product inventory sales maximization problem. Third, we fully characterize the convex envelope of a bilinear function defined over a cycle graph in the original variable space. Finally, we show computationally that the cycle inequalities close approximately 60\% of the gap for complete and Hadamard graphs, over 30\% of the gap for complete bipartite graphs, and over 80\% of the gap for sparse graphs such as cactus and Halin graphs. The resulting relaxations are also more efficient to solve than previous extended space formulations.

math.OC↗

arXiv2026

Assortment Optimization under Logit-Based Multi-Purchase Choice Models

Problem definition: We study assortment optimization under logit-based multi-purchase choice models. Because customers frequently purchase multiple distinct products together in a single transaction, incorporating such behavior into assortment decisions can unlock additional revenue opportunities. However, doing so is computationally challenging because the offered assortment induces a combinatorial set of feasible bundles that compete nonlinearly for customer demand. Methodology/results: We develop an exact assortment optimization framework for general multi-purchase patterns. We first introduce a hypergraph representation of Logit-MP models that unifies several models in the literature, including BundleMVL-K and Multivariate-MNL, and can be estimated from transaction data using a sparsity-based procedure. Building on this representation, we develop strong mixed-integer programming formulations that leverage recent advances from multilinear optimization and a perspective reformulation. The resulting formulations provide tighter linear programming relaxations than the prevalent Big-M approach. We further characterize classes of multi-purchase patterns, represented by different hypergraph structures, under which the formulations reduce to linear programs. Computational experiments demonstrate substantial improvements in both solution quality and scalability. Managerial implications: Using real transaction data, we show that a sparse hypergraph captures the dominant multi-purchase patterns while generating higher revenues. We also show that our formulations extend to heterogeneous customer populations and remain computationally viable, making the framework practical for large-scale retail assortment planning.

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