Polymatroid Inequalities Research Articles

Joint chance-constrained programs and the intersection of mixing sets through a submodularity lens

A particularly important substructure in modeling joint linear chance-constrained programs with random right-hand sides and finite sample space is the intersection of mixing sets with common binary variables (and possibly a knapsack constraint). In this paper, we first revisit basic mixing sets by establishing a strong and previously unrecognized connection to submodularity. In particular, we show that mixing inequalities with binary variables are nothing but the polymatroid inequalities associated with a specific submodular function. This submodularity viewpoint enables us to unify and extend existing results on valid inequalities and convex hulls of the intersection of multiple mixing sets with common binary variables. Then, we study such intersections under an additional linking constraint lower bounding a linear function of the continuous variables. This is motivated from the desire to exploit the information encoded in the knapsack constraint arising in joint linear CCPs via the quantile cuts. We propose a new class of valid inequalities and characterize when this new class along with the mixing inequalities are sufficient to describe the convex hull.

Lifted polymatroid inequalities for mean-risk optimization with indicator variables

We investigate a mixed 0-1 conic quadratic optimization problem with indicator variables arising in mean-risk optimization. The indicator variables are often used to model non-convexities such as fixed charges or cardinality constraints. Observing that the problem reduces to a submodular function minimization for its binary restriction, we derive three classes of strong convex valid inequalities by lifting the polymatroid inequalities on the binary variables. Computational experiments demonstrate the effectiveness of the inequalities in strengthening the convex relaxations and, thereby, improving the solution times for mean-risk problems with fixed charges and cardinality constraints significantly.

Capacitated closed-loop supply chain network design under uncertainty

This study optimizes the design of a closed-loop supply chain network, which contains forward and reverse directions and is subject to uncertainty in demands for new & returned products. To address uncertainty in decision-making, we formulate a two-stage stochastic mixed-integer non-linear programming model to determine the distribution center locations and their corresponding capacity, and new & returned product flows in the supply chain network to minimize total design and expected operating costs. We convert our model to a conic quadratic programming model given the complexity of our problem. Then, the conic model is added with certain valid inequalities, such as polymatroid inequalities, and extended with respect to its cover cuts so as to improve computational efficiency. Furthermore, a tabu search algorithm is developed for large-scale problem instances. We also study the impact of inventory weight, transportation weight, and marginal value of time of returned products by the sensitivity analysis. Several computational experiments are conducted to validate the effectiveness of the proposed model and valid inequalities.

Ambiguous Chance-Constrained Binary Programs under Mean-Covariance Information

We consider chance-constrained binary programs, where each row of the inequalities that involve uncertainty needs to be satisfied probabilistically. Only the information of the mean and covariance matrix is available, and we solve distributionally robust chance-constrained binary programs (DCBP). Using two different ambiguity sets, we equivalently reformulate the DCBPs as 0-1 second-order cone (SOC) programs. We further exploit the submodularity of 0-1 SOC constraints under special and general covariance matrices, and utilize the submodularity as well as lifting to derive extended polymatroid inequalities to strengthen the 0-1 SOC formulations. We incorporate the valid inequalities in a branch-and-cut algorithm for efficiently solving DCBPs. We demonstrate the computational efficacy and solution performance using diverse instances of a chance-constrained bin packing problem.

Warehouse location and two-echelon inventory management with concave operating cost

In this paper, we study a location–inventory network design problem which jointly optimises the warehouse location, the warehouse–retailer assignments, the warehouse–retailer echelon inventory replenishment and the safety stock-level decisions over an infinite planning horizon. The consideration of the facility operating cost, the safety stock cost and the two-echelon inventory cost results in an MIP model with several nonlinear terms. Due to the complex trade-offs among the various costs and multiple nonlinear terms in the model, traditional solution approaches no longer work for this problem. We outline a polymatroid cutting-plane approach based on the submodular property of the cost terms to address this problem. Computational results demonstrate that the cutting-plane method based on polymatroid inequalities can efficiently solve randomly generated instances with moderate sizes.

A computational study for common network design in multi-commodity supply chains

In this paper, we study a supply chain network design problem which consists of one external supplier, a set of potential distribution centers, and a set of retailers, each of which is faced with uncertain demands for multiple commodities. The demand of each retailer is fulfilled by a single distribution center for all commodities. The goal is to minimize the system-wide cost including location, transportation, and inventory costs. We propose a general nonlinear integer programming model for the problem and present a cutting plane approach based on polymatroid inequalities to solve the model. Randomly generated instances for two special cases of our model, i.e., the single-sourcing UPL&TAP and the single-sourcing multi-commodity location-inventory model, are provided to test our algorithm. Computational results show that the proposed algorithm can solve moderate-sized problem instances efficiently.

On the Complexity of Some Enumeration Problems for Matroids

Let $M$ be a matroid defined by an independence oracle on ground set $S$, and let $A\subseteq S$. We present an incremental polynomial-time algorithm for enumerating all minimal (maximal) subsets of $S$ which span (do not span) $A$. Special cases of these problems include the generation of bases, circuits, hyperplanes, flats of given rank, circuits through a given element, generalized Steiner trees, and multiway cuts in graphs, as well as some other applications. We also consider some tractable and NP-hard generation problems related to systems of polymatroid inequalities and (generalized) packing and spanning in matroids.

An inequality for polymatroid functions and its applications

An integral-valued set function f : 2 V↦ Z is called polymatroid if it is submodular, non-decreasing, and f(∅)=0. Given a polymatroid function f and an integer threshold t⩾1, let α= α( f, t) denote the number of maximal sets X⊆ V satisfying f( X)< t, let β= β( f, t) be the number of minimal sets X⊆ V for which f( X)⩾ t, and let n=| V|. We show that if β⩾2 then α⩽β ( log t)/c , where c= c( n, β) is the unique positive root of the equation 1=2 c(n c/ log β −1) . In particular, our bound implies that α⩽(nβ) log t for all β⩾1. We also give examples of polymatroid functions with arbitrarily large t, n, α and β for which α⩾β (0.551 log t)/c . More generally, given a polymatroid function f:2 V↦ Z and an integral threshold t⩾1, consider an arbitrary hypergraph H such that | H|⩾2 and f( H)⩾ t for all H∈ H . Let S be the family of all maximal independent sets X of H for which f( X)< t. Then | S|⩽| H| ( log t)/c(n,| H|) . As an application, we show that given a system of polymatroid inequalities f 1( X)⩾ t 1,…, f m ( X)⩾ t m with quasi-polynomially bounded right-hand sides t 1,…, t m , all minimal feasible solutions to this system can be generated in incremental quasi-polynomial time. In contrast to this result, the generation of all maximal infeasible sets is an NP-hard problem for many polymatroid inequalities of small range.