Branching via Cutting Plane Selection: Improving Hybrid Branching

Cutting planes and branching are two of the most important algorithms for solving mixed-integer linear programs. For both algorithms, disjunctions play an important role, being used both as branching candidates and as the foundation for some cutting planes. We relate branching decisions and cutting planes to each other through the underlying disjunctions that they are based on, with a focus on Gomory mixed-integer cuts and their corresponding split disjunctions. We show that selecting branching decisions based on quality measures of Gomory mixed-integer cuts leads to relatively small branch-and-bound trees, and that the result improves when using cuts that more accurately represent the branching decisions. Finally, we show how the history of previously computed Gomory mixed-integer cuts can be used to improve the performance of the state-of-the-art hybrid branching rule of SCIP. Our results show a 4\% decrease in solve time, and an 8\% decrease in number of nodes over affected instances of MIPLIB 2017

A computational study of cuts derived from the Chvatal-Gomory cut for interger programming problems

Orientador: Vinicius Amaral ArmentanoDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Em 1958, Gomory propôs uma desigualdade válida ou corte a partir do tableau do método simplex para programação linear, que foi utilizado no primeiro método genérico para resolução de problemas de programação inteira. Em 1960, o corte foi estendido para problemas de programação inteira mista. Em 1973, Chvátal sugeriu um corte derivado da formulação original do problema de programação inteira, e devido à equivalência com o corte de Gomory, este passou a ser chamado de corte de Chvátal-Gomory. A importância do corte de Gomory só foi reconhecida em 1996 dentro do contexto do método branch-and-cut para resolução de problemas de programação inteira e programação inteira mista. Desde então, este corte é utilizado em resolvedores comerciais de otimização. Recentemente, diversos cortes novos derivados do corte de Chvátal-Gomory foram propostos na literatura para programação inteira. Este trabalho trata do desenvolvimento de algoritmos para alguns destes cortes, e implementação computacional em um contexto de branch-and-cut, no ambiente do resolvedor CPLEX. A eficácia dos cortes é testada em instâncias dos problemas da mochila multidimensional, designação generalizada e da biblioteca MIPLIB.Abstract: In 1958, Gomory proposed a valid inequality or cut from the tableau of the simplex method for linear programming, which was used in the first generic method for solving integer programming problems. In 1960, the cut was extended to handle mixed integer programming problems. In 1973, Chvátal suggested a cut that is generated from the original formulation of an integer programming problem, and due to the equivalence with the Gomory cut, it was named Chvátal-Gomory cut. The importance of the Gomory cut was recognized only in 1996 in the context of the branch-and-cut method for solving (mixed) integer programming problems. Today, such a cut is utilized in optimization commercial solvers. Recently, several new cuts derived from the Chvátal-Gomory cut have been proposed in the literature for integer programming. This work deals with the development of algorithms and computational implementations for some of the new proposed cuts, in a context of the branch-and-cut method, by using the CPLEX solver. The efficiency of the cuts is tested on instances of the multi-dimensional knapsack, generalized assignment problems, and instances from the MIPLIB library.MestradoAutomaçãoMestre em Engenharia Elétric

Computational Methods for Discrete Conic Optimization Problems

This thesis addresses computational aspects of discrete conic optimization. Westudy two well-known classes of optimization problems closely related to mixedinteger linear optimization problems. The case of mixed integer second-ordercone optimization problems (MISOCP) is a generalization in which therequirement that solutions be in the non-negative orthant is replaced by arequirement that they be in a second-order cone. Inverse MILP, on the otherhand, is the problem of determining the objective function that makes a givensolution to a given MILP optimal.Although these classes seem unrelated on the surface, the proposedsolution methodology for both classes involves outer approximation of a conicfeasible region by linear inequalities. In both cases, an iterative algorithmin which a separation problem is solved to generate the approximation isemployed. From a complexity standpoint, both MISOCP and inverse MILP areNP--hard. As in the case of MILPs, the usual decision version ofMISOCP is NP-complete, whereas in contrast to MILP, we provide the firstproof that a certain decision version of inverse MILP is rathercoNP-complete.With respect to MISOCP, we first introduce a basic outer approximationalgorithm to solve SOCPs based on a cutting-plane approach. As expected, theperformance of our implementation of such an algorithm is shown to lag behindthe well-known interior point method. Despite this, such a cutting-planeapproach does have promise as a method of producing bounds when embedded withina state-of-the-art branch-and-cut implementation due to its superior ability towarm-start the bound computation after imposing branching constraints. Ourouter-approximation-based branch-and-cut algorithm relaxes both integrality andconic constraints to obtain a linear relaxation. This linear relaxation isstrengthened by the addition of valid inequalities obtained by separatinginfeasible points. Valid inequalities may be obtained by separation from theconvex hull of integer solution lying within the relaxed feasible region or byseparation from the feasible region described by the (relaxed) conicconstraints. Solutions are stored when both integer and conic feasibility isachieved. We review the literature on cutting-plane procedures for MISOCP andmixed integer convex optimization problems.With respect to inverse MILP, we formulate this problem as a conicproblem and derive a cutting-plane algorithm for it. The separation problem inthis algorithm is a modified version of the original MILP. We show that thereis a close relationship between this algorithm and a similar iterativealgorithm for separating infeasible points from the convex hull of solutions tothe original MILP that forms part of the basis for the well-known result ofGrotschel-Lovasz-Schrijver that demonstrates the complexity-wiseequivalence of separation and optimization.In order to test our ideas, we implement a number of software librariesthat together constitute DisCO, a full-featured solver for MISOCP. Thefirst of the supporting libraries is OsiConic, an abstract base classin C++ for interfacing to SOCP solvers. We provide interfaces using thislibrary for widely used commercial and open source SOCP/nonlinear problemsolvers. We also introduce CglConic, a library that implements cuttingprocedures for MISOCP feasible set. We perform extensive computationalexperiments with DisCO comparing a wide range of variants of our proposedalgorithm, as well as other approaches. As DisCO is built on top of a libraryfor distributed parallel tree search algorithms, we also perform experimentsshowing that our algorithm is effective and scalable when parallelized

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201

A hybrid method for capacitated vehicle routing problem

The vehicle routing problem (VRP) is to service a number of customers with a fleet of vehicles. The VRP is an important problem in the fields of transportation, distribution and logistics. Typically the VRP deals with the delivery of some commodities from a depot to a number of customer locations with given demands. The problem frequently arises in many diverse physical distribution situations. For example bus routing, preventive maintenance inspection tours, salesmen routing and the delivery of any commodity such as mail, food or newspapers.We focus on the Symmetric Capacitated Vehicle Routing Problem (CVRP) with a single commodity and one depot. The restrictions are capacity and cost or distance. For large instances, exact computational algorithms for solving the CVRP require considerable CPU time. Indeed, there are no guarantees that the optimal tours will be found within a reasonable CPU time. Hence, using heuristics and meta-heuristics algorithms may be the only approach. For a large CVRP one may have to balance computational time to solve the problem and the accuracy of the obtained solution when choosing the solving technique.This thesis proposes an effective hybrid approach that combines domain reduction with: a greedy search algorithm; the Clarke and Wright algorithm; a simulating annealing algorithm; and a branch and cut method to solve the capacitated vehicle routing problem. The hybrid approach is applied to solve 14 benchmark CVRP instances. The results show that domain reduction can improve the classical Clarke and Wright algorithm by 8% and cut the computational time taken by approximately 50% when combined with branch and cut.Our work in this thesis is organized into 6 chapters. Chapter 1 provides an introduction and general concepts, notation and terminology and a summary of our work. In Chapter 2 we detail a literature review on the CVRP. Some heuristics and exact methods used to solve the problem are discussed. Also, this Chapter describes the constraint programming (CP) technique, some examples of domain reduction, advantages and disadvantage of using CP alone, and the importance of combining CP with MILP exact methods. Chapter 3 provides a simple greedy search algorithm and the results obtained by applying the algorithm to solve ten VRP instances. In Chapter 4 we incorporate domain reduction with the developed heuristic. The greedy algorithm with a restriction on each route combined with domain reduction is applied to solve the ten VRP instances. The obtained results show that the domain reduction improves the solution by an average of 24%. Also, the Chapter shows that the classical Clarke and Wright algorithm could be improve by 8% when combined with domain reduction. Chapter 4 combines domain reduction with a simulating annealing algorithm. In Chapter 4 we use the combination of domain reduction with the greedy algorithm, Clarke and Wright algorithm, and simulating annealing algorithm to solve 4 large CVRP instances.Chapter 5 incorporates the Branch and Cut method with domain reduction. The hybrid approach is applied to solve the 10 CVRP instances that we used in Chapter 4. This Chapter shows that the hybrid approach reduces the CPU time taken to solve the 10 benchmark instances by approximately 50%. Chapter 6 concludes the thesis and provides some ideas for future work. An appendix of the 10 literature problems and generated instances will be provided followed by bibliography

Mathematical Multi-Objective Optimization of the Tactical Allocation of Machining Resources in Functional Workshops

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers and to maintain control of the tied-up working capital. We introduce new multi-item, multi-level capacitated resource allocation models with a medium--to--long--term planning horizon. The model refers to functional workshops where costly and/or time- and resource-demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimize the maximum excess resource loading above a given loading threshold while incurring as low qualification costs as possible and minimizing the inventory.In Paper I, we propose a new bi-objective mixed-integer (linear) optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. In Paper II, we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. In Paper III, we extend the TRAP with an inventory of semi-finished as well as finished parts, resulting in a tri-objective mixed-integer (linear) programming model. We create a criterion space partitioning approach that enables solving sub-problems simultaneously. In Paper IV, using our knowledge from our previous work we embarked upon a task to generalize our findings to develop an approach for any discrete tri-objective optimization problem. The focus is on identifying a representative set of non-dominated points with a pre-defined desired coverage gap

Mathematical Optimization of the Tactical Allocation of Machining Resources in Aerospace Industry

In the aerospace industry, efficient management of machining capacity is crucial to meet the required service levels to customers (which includes, measures of quality and production lead-times) and to maintain control of the tied-up working capital. We introduce a new multi-item, multi-level capacitated planning model with a medium-to-long term planning horizon. The model can be used by most companies having functional workshops where costly and/or time- and resource demanding preparations (or qualifications) are required each time a product needs to be (re)allocated to a machining resource. Our goal is to identify possible product routings through the factory which minimizes the maximum excess resource loading above a given loading threshold, while incurring as low qualification costs as possible. In Paper I (Bi-objective optimization of the tactical allocation of jobtypes to machines), we propose a new bi-objective mathematical optimization model for the Tactical Resource Allocation Problem (TRAP). We highlight some of the mathematical properties of the TRAP which are utilized to enhance the solution process. Another contribution is a modified version of the bi-directional $\epsilon$ -constraint method especially tailored for our problem. We perform numerical tests on industrial test cases generated for our class of problem which indicates computational superiority of our method over conventional solution approaches. In Paper II (Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs), we address the uncertainty in the coefficients of one of the objective functions considered in the bi-objective TRAP. We propose a new bi-objective robust efficiency concept and highlight its benefits over existing robust efficiency concepts. We also suggest a solution approach for identifying all the relevant robust efficient (RE) solutions. Our proposed approach is significantly faster than an existing approach for robust bi-objective optimization problems

China\u27s Industrial Policy and its Impact on U.S. Companies, Workers and the American Economy

[Excerpt] China’s industrial policies have had a profound effect on the U.S. economy. The trade deficit with China in goods reached $266 billion in 2008, resulting in slower U.S. economic growth and fewer jobs here than if the trade relationship were more balanced between imports and exports. Witnesses differed as to the degree that the overall U.S. trade deficit would decline if the trading relationship between the two countries were brought into balance. But it is significant that the U.S. deficit with China represented 33 percent of the total U.S. trade deficit with the world and 42.6 percent of the deficit with non-oil exporting countries. In addition, it is not just the size of the deficit that policymakers should examine, but the changing nature of its composition. The United States in 2008 ran a record$72.7 billion trade deficit with China in advanced technology products