Piecewise Linear Valued CSPs Solvable by Linear Programming Relaxation

Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. The computational complexity of VCSPs depends on the set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of the complexity of infinite-domain VCSPs with piecewise linear homogeneous cost functions. Such VCSPs can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by reducing the problem to a finite-domain VCSP which can be solved using the basic linear program relaxation. It follows that VCSPs for submodular PLH cost functions can be solved in polynomial time; in fact, we show that submodular PLH functions form a maximally tractable class of PLH cost functions.

Consistency Cuts for Dantzig-Wolfe Reformulations

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

The Gap Function: Evaluating Integer Programming Models over Multiple Right-Hand Sides

For an integer programming model with fixed data, the linear programming relaxation gap is considered one of the most important measures of model quality. There is no consensus, however, on appropriate measures of model quality that account for data variation. In particular, when the right-hand side is not known exactly, one must assess a model based on its behavior over many right-hand sides. Gap functions are the linear programming relaxation gaps parametrized by the right-hand side. Despite drawing research interest in the early days of integer programming, the properties and applications of these functions have been little studied. In this paper, we construct measures of integer programming model quality over sets of right-hand sides based on the absolute and relative gap functions. In particular, we formulate optimization problems to compute the expectation and extrema of gap functions over finite discrete sets and bounded hyperrectangles. These optimization problems are linear programs (albeit of an exponentially large size) that contain at most one special ordered-set constraint. These measures for integer programming models, along with their associated formulations, provide a framework for determining a model’s quality over a range of right-hand sides.

Rhythmic Control of Automated Traffic—Part II: Grid Network Rhythm and Online Routing

Connected and automated vehicle (CAV) technology is providing urban transportation managers tremendous opportunities for better operation of urban mobility systems. However, there are significant challenges in real-time implementation as the computational time of the corresponding operations optimization model increases exponentially with increasing vehicle numbers. Following the companion paper (Chen et al. 2021), which proposes a novel automated traffic control scheme for isolated intersections, this study proposes a network-level, real-time traffic control framework for CAVs on grid networks. The proposed framework integrates a rhythmic control method with an online routing algorithm to realize collision-free control of all CAVs on a network and achieve superior performance in average vehicle delay, network traffic throughput, and computational scalability. Specifically, we construct a preset network rhythm that all CAVs can follow to move on the network and avoid collisions at all intersections. Based on the network rhythm, we then formulate online routing for the CAVs as a mixed integer linear program, which optimizes the entry times of CAVs at all entrances of the network and their time–space routings in real time. We provide a sufficient condition that the linear programming relaxation of the online routing model yields an optimal integer solution. Extensive numerical tests are conducted to show the performance of the proposed operations management framework under various scenarios. It is illustrated that the framework is capable of achieving negligible delays and increased network throughput. Furthermore, the computational time results are also promising. The CPU time for solving a collision-free control optimization problem with 2,000 vehicles is only 0.3 second on an ordinary personal computer.

Discrete Optimization: The Case of Generalized BCC Lattice

Recently, operations research, especially linear integer-programming, is used in various grids to find optimal paths and, based on that, digital distance. The 4 and higher-dimensional body-centered-cubic grids is the nD (n≥4) equivalent of the 3D body-centered cubic grid, a well-known grid from solid state physics. These grids consist of integer points such that the parity of all coordinates are the same: either all coordinates are odd or even. A popular type digital distance, the chamfer distance, is used which is based on chamfer paths. There are two types of neighbors (closest same parity and closest different parity point-pairs), and the two weights for the steps between the neighbors are fixed. Finding the minimal path between two points is equivalent to an integer-programming problem. First, we solve its linear programming relaxation. The optimal path is found if this solution is integer-valued. Otherwise, the Gomory-cut is applied to obtain the integer-programming optimum. Using the special properties of the optimization problem, an optimal solution is determined for all cases of positive weights. The geometry of the paths are described by the Hilbert basis of the non-negative part of the kernel space of matrix of steps.

A Linear Programming Relaxation DEA Model for Selecting a Single Efficient Unit with Variable RTS Technology

The selection-based problem is a type of decision-making issue which involves opting for a single option among a set of available alternatives. In order to address the selection-based problem in data envelopment analysis (DEA), various integrated mixed binary linear programming (MBLP) models have been developed. Recently, an MBLP model has been proposed to select a unit in DEA with variable returns-to-scale technology. This paper suggests utilizing the linear programming relaxation model rather than the MBLP model. The MBLP model is proved here to be equivalent to its linear programming relaxation problem. To the best of the authors’ knowledge, this is the first linear programming model suggested for selecting a single efficient unit in DEA under the VRS (Variable Returns to Scale) assumption. Two theorems and a numerical example are provided to validate the proposed LP model from both theoretical and practical perspectives.

A Deterministic Method for Solving the Sum of Linear Ratios Problem

Since the sum of linear ratios problem (SLRP) has many applications in real life, for globally solving it, an efficient branch and bound algorithm is presented in this paper. By utilizing the characteristic of the problem (SLRP), we propose a convex separation technique and a two-part linearization technique, which can be used to generate a sequence of linear programming relaxation of the initial nonconvex programming problem. For improving the convergence speed of this algorithm, a deleting rule is presented. The convergence of this algorithm is established, and some experiments are reported to show the feasibility and efficiency of the proposed algorithm.

Dynamic Pricing (and Assortment) Under a Static Calendar

This work is motivated by our collaboration with a large consumer packaged goods (CPG) company. We have found that whereas the company appreciates the advantages of dynamic pricing, they deem it operationally much easier to plan out a static price calendar in advance. We investigate the efficacy of static control policies for revenue management problems whose optimal solution is inherently dynamic. In these problems, a firm has limited inventory to sell over a finite time horizon, over which heterogeneous customers stochastically arrive. We consider both pricing and assortment controls, and derive simple static policies in the form of a price calendar or a planned sequence of assortments, respectively. In the assortment planning problem, we also differentiate between the static vs. dynamic substitution models of customer demand. We show that our policies are within 1-1/e (approximately 0.63) of the optimum under stationary demand, and 1/2 of the optimum under nonstationary demand, with both guarantees approaching 1 if the starting inventories are large. We adapt the technique of prophet inequalities from optimal stopping theory to pricing and assortment problems, and our results are relative to the linear programming relaxation. Under the special case of stationary demand single-item pricing, our results improve the understanding of irregular and discrete demand curves, by showing that a static calendar can be (1-1/e)-approximate if the prices are sorted high-to-low. Finally, we demonstrate on both data from the CPG company and synthetic data from the literature that our simple price and assortment calendars are effective. This paper was accepted by Hamid Nazerzadeh, big data analytics.

A Stochastic Integer Programming Approach to Air Traffic Scheduling and Operations

Air traffic management measures comprise tactical operating procedures to minimize delay costs and strategic scheduling interventions to control overcapacity scheduling. Although interdependent, these problems have been treated in isolation. This paper proposes an integrated model of scheduling and operations in airport networks that jointly optimizes scheduling interventions and ground-holding operations across airports networks under operating uncertainty. It is formulated as a two-stage stochastic program with integer recourse. To solve it, we develop an original decomposition algorithm with provable solution quality guarantees. The algorithm relies on new optimality cuts—dual integer cuts—that leverage the reduced costs of the dual linear programming relaxation of the second-stage problem. The algorithm also incorporates neighborhood constraints, which shift from exploration to exploitation at later stages. We also use a scenario generation approach to construct representative scenarios from historical records of operations—using integer programming. Computational experiments show that our algorithm yields near-optimal solutions for the entire U.S. National Airspace System network. Ultimately, the proposed approach enhances airport demand management models through scale integration (by capturing network-wide interdependencies) and scope integration (by capturing interdependencies between scheduling and operations).