Marcos Goycoolea

Harvest Scheduling Subject to Maximum Area Restrictions: Exploring Exact Approaches

We consider a spatial problem arising in forest harvesting. For regulatory reasons, blocks harvested should not exceed a certain total area, typically 49 hectares. Traditionally, this problem, called the adjacency problem, has been approached by forming a priori blocks from basic cells of 5 to 25 hectares and solving the resulting mixed-integer program. Superior solutions can be obtained by including the construction of blocks in the decision process. The resulting problem is far more complex combinatorially. We present an exact algorithmic approach that has yielded good results in computational tests. This solution approach is based on determining a strong formulation of the linear programming problem through a clique representation of a projected problem.

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.

Imposing Connectivity Constraints in Forest Planning Models

Connectivity requirements are a common component of forest planning models, with important examples arising in wildlife habitat protection. In harvest scheduling models, one way of addressing preservation concerns consists of requiring that large contiguous patches of mature forest are maintained. In the context of nature reserve design, it is common practice to select a connected region of forest, as a reserve, in such a way as to maximize the number of species and habitats protected. Although a number of integer programming formulations have been proposed for these forest planning problems, most are impractical in that they fail to solve reasonably sized scheduling instances. We present a new integer programming methodology and test an implementation of it on five medium-sized forest instances publicly available in the Forest Management Optimization Site repository. Our approach allows us to obtain near-optimal solutions for multiple time-period instances in fewer than four hours.

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.

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.

A heuristic to generate rank-1 GMI cuts

Gomory mixed-integer (GMI) cuts are among the most effective cutting planes for general mixed-integer programs (MIP). They are traditionally generated from an optimal basis of a linear programming (LP) relaxation of a MIP. In this paper we propose a heuristic to generate useful GMI cuts from additional bases of the initial LP relaxation. The cuts we generate have rank one, i.e., they do not use previously generated GMI cuts. We demonstrate that for problems in MIPLIB 3.0 and MIPLIB 2003, the cuts we generate form an important subclass of all rank-1 mixed-integer rounding cuts. Further, we use our heuristic to generate globally valid rank-1 GMI cuts at nodes of a branch-and-cut tree and use these cuts to solve a difficult problem from MIPLIB 2003, namely timtab2, without using problem-specific cuts.

Two-Step MIR Inequalities for Mixed Integer Programs

Two-step mixed integer rounding (MIR) inequalities are valid inequalities derived from a facet of a simple mixed integer set with three variables and one constraint. In this paper we investigate how to effectively use these inequalities as cutting planes for general mixed integer problems. We study the separation problem for single-constraint sets and show that it can be solved in polynomial time when the resulting inequality is required to be sufficiently different from the associated MIR inequalities. We discuss computational issues and present numerical results based on a number of data sets.

Numerically Safe Gomory Mixed-Integer Cuts

We describe a simple process for generating numerically safe cutting planes using floating-point arithmetic and the mixed-integer rounding procedure. Applying this method to the rows of the simplex tableau permits the generation of Gomory mixed-integer cuts that are guaranteed to be satisfied by all feasible solutions to a mixed-integer programming problem (MIP). We report on tests with the MIPLIB 3.0 and MIPLIB 2003 test collections as well as with MIP instances derived from the TSPLIB traveling salesman library.

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.

On the Exact Separation of Mixed Integer Knapsack Cuts

During the last decades, much research has been conducted deriving classes of valid inequalities for single-row mixed integer programming polyhedrons. However, no such class has had as much practical success as the MIR inequality when used in cutting plane algorithms for general mixed integer programming problems. In this work we analyze this empirical observation by developing an algorithm which takes as input a point and a single-row mixed integer polyhedron, and either proves the point is in the convex hull of said polyhedron, or finds a separating hyperplane. The main feature of this algorithm is a specialized subroutine for solving the Mixed Integer Knapsack Problem which exploits cost and lexicographic dominance. Separating over the entire closure of single-row systems allows us to establish natural benchmarks by which to evaluate specific classes of knapsack cuts. Using these benchmarks on Miplib 3.0 instances we analyze the performance of MIR inequalities. Computations are performed in exact arithmetic.