Daniel G. Espinoza

Certification of an optimal TSP tour through 85,900 cities

We describe a computer code and data that together certify the optimality of a solution to the 85,900-city traveling salesman problem pla85900, the largest instance in the TSPLIB collection of challenge problems.

Per-Seat, On-Demand Air Transportation Part I: Problem Description and an Integer Multicommodity Flow Model

The availability of relatively cheap small jet planes has led to the creation of on-demand air transportation services in which travelers call a few days in advance to schedule a flight. A successful on-demand air transportation service requires an effective scheduling system to construct minimum-cost pilot and jet itineraries for a set of accepted transportation requests. We present an integer multicommodity network flow model with side constraints for such dial-a-flight problems. We develop a variety of techniques to control the size of the network and to strengthen the quality of the linear programming relaxation, which allows the solution of small instances. In Part II, we describe how this core optimization technology is embedded in a parallel, large-neighborhood, local search scheme to produce high-quality solutions efficiently for large-scale real-life instances.

Computing with multi-row Gomory cuts

Recent advances on the understanding of valid inequalities from the infinite group relaxation has opened the possibility of finding a computationally effective extension to GMI cuts. In this paper, we investigate the computational impact of using a subclass of minimally valid inequalities from this relaxation on a wide set of instances.

Per-Seat, On-Demand Air Transportation Part II: Parallel Local Search

The availability of relatively cheap small jet aircrafts suggests a new air transportation business: dial-a-flight, an on-demand service in which travelers call a few days in advance to schedule transportation. A successful on-demand air transportation service requires an effective scheduling system to construct minimum-cost pilot and jet itineraries for a set of accepted transportation requests. In Part I, we introduced an integer multicommodity network flow model with side constraints for the dial-a-flight problem and showed that small instances can be solved effectively. Here, we demonstrate that high-quality solutions for large-scale real-life instances can be produced efficiently by embedding the core optimization technology in a local search scheme. To achieve the desired level of performance, metrics were devised to select neighborhoods intelligently, a variety of search diversification techniques were included, and an asynchronous parallel implementation was developed.

Computing with multi-row gomory cuts

Cutting planes for mixed integer problems (MIP) are nowadays an integral part of all general purpose software to solve MIP. The most prominent, and computationally significant, class of general cutting planes are Gomory mixed integer cuts (GMI). However finding other classes of general cuts for MIP that work well in practice has been elusive. Recent advances on the understanding of valid inequalities derived from the infinite relaxation introduced by Gomory and Johnson for mixed integer problems, has opened a new possibility of finding such an extension. In this paper, we investigate the computational impact of using a subclass of minimal valid inequalities from the infinite relaxation, using different number of tableau rows simultaneously, based on a simple separation procedure.We test these ideas on a set of MIPs, including MIPLIB 3.0 and MIPLIB 2003, and show that they can improve MIP performance even when compared against commercial software performance.

Local cuts for mixed-integer programming

A general framework for cutting-plane generation was proposed by Applegate et al. in the context of the traveling salesman problem. The process considers the image of a problem space under a linear mapping, chosen so that a relaxation of the mapped problem can be solved efficiently. Optimization in the mapped space can be used to find a separating hyperplane, if one exists, and via substitution this gives a cutting plane in the original space. We extend this procedure to general mixed-integer programming problems, obtaining a range of possibilities for new sources of cutting planes. Some of these possibilities are explored computationally, both in floating-point arithmetic and in rational arithmetic.

Lifting, tilting and fractional programming revisited

Lifting, tilting and fractional programming, though seemingly different, reduce to a common optimization problem. This connection allows us to revisit key properties of these three problems on mixed integer linear sets. We introduce a simple common framework for these problems, and extend known results from each to the other two.

Computing with Domino-Parity Inequalities for the Traveling Salesman Problem (TSP)

We describe methods for implementing separation algorithms for domino-parity inequalities for the symmetric traveling salesman problem. These inequalities were introduced by Letchford (2000), who showed that the separation problem can be solved in polynomial time when the support graph of the LP solution is planar. In our study we deal with the problem of how to use this algorithm in the general (nonplanar) case, continuing the work of Boyd et al. (2001). Our implementation includes pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of a 33,810-city instance from the TSPLIB.

A study of domino-parity and k -parity constraints for the TSP

Letchford (2000) introduced the domino-parity inequalities for the symmetric traveling salesman problem and showed that if the support graph of an LP solution is planar, then the separation problem can be solved in polynomial time. We generalize domino-parity inequalities to multi-handled configurations, introducing a superclass of bipartition and star inequalities. Also, we generalize Letchfords algorithm, proving that for a fixed integer k, one can separate a superclass of k-handled clique-tree inequalities satisfying certain connectivity characteristics with respect to the planar support graph. We describe an implementation of Letchfords algorithm including pruning methods to restrict the search for dominoes, a parallelization of the main domino-building step, heuristics to obtain planar-support graphs, a safe-shrinking routine, a random-walk heuristic to extract additional violated constraints, and a tightening procedure to allow us to modify existing inequalities as the LP solution changes. We report computational results showing the strength of the new routines, including the optimal solution of the TSPLIB instance pla33810.

A study of the Bienstock–Zuckerberg algorithm: applications in mining and resource constrained project scheduling

We study a Lagrangian decomposition algorithm recently proposed by Dan Bienstock and Mark Zuckerberg for solving the LP relaxation of a class of open pit mine project scheduling problems. In this study we show that the Bienstock–Zuckerberg (BZ) algorithm can be used to solve LP relaxations corresponding to a much broader class of scheduling problems, including the well-known Resource Constrained Project Scheduling Problem (RCPSP), and multi-modal variants of the RCPSP that consider batch processing of jobs. We present a new, intuitive proof of correctness for the BZ algorithm that works by casting the BZ algorithm as a column generation algorithm. This analysis allows us to draw parallels with the well-known Dantzig–Wolfe decomposition (DW) algorithm. We discuss practical computational techniques for speeding up the performance of the BZ and DW algorithms on project scheduling problems. Finally, we present computational experiments independently testing the effectiveness of the BZ and DW algorithms on different sets of publicly available test instances. Our computational experiments confirm that the BZ algorithm significantly outperforms the DW algorithm for the problems considered. Our computational experiments also show that the proposed speed-up techniques can have a significant impact on the solve time. We provide some insights on what might be explaining this significant difference in performance.