Combining variable neighborhood search with integer linear programming for the generalized minimum spanning tree problem

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. This work is supported by the RTN ADONET under grant 504438.     

An integer linear programming approach and a hybrid variable neighborhood search for the car sequencing problem

In this paper we present two major approaches to solve the car sequencing problem, in which the goal is to find an optimal arrangement of commissioned vehicles along a production line with respect to constraints of the form “no more than l_ccars are allowed to require a component c in any subsequence of m_cconsecutive cars”. The first method is an exact one based on integer linear programming (ILP). The second approach is hybrid: it uses ILP techniques within a general variable neighborhood search (VNS) framework for examining large neighborhoods. We tested the two methods on benchmark instances provided by CSP_LIB and the automobile manufacturer RENAULT for the ROADEF Challenge 2005. These tests reveal that our approaches are competitive to previous reported algorithms. For the CSP_LIB instances we were able to shorten the required computation time for reaching and proving optimality. Furthermore, we were able to obtain tight bounds on some of the ROADEF instances. For two of these instances the proposed ILP-method could provide new optimality proofs for already known solutions. For the VNS, the individual contributions of the used neighborhoods are also experimentally analyzed. Results highlight the significant impact of each structure. In particular the large ones examined using ILP techniques enhance the overall performance significantly, so that the hybrid approach clearly outperforms variants including only commonly defined neighborhoods.     

Iterative variable aggregation and disaggregation in IP: An application

This paper, using the Unconstrained Shape Matrix Optimization Problem as a test bed, we investigate various aspects of variable aggregation and disaggregation for a class of integer programs that contains binary expansion. We present theoretical and numerical results, and propose an iterative algorithm for exact solutions.     

Bringing order into the neighborhoods: relaxation guided variable neighborhood search

In this article we investigate a new variant of Variable Neighborhood Search (VNS): Relaxation Guided Variable Neighborhood Search. It is based on the general VNS scheme and a new Variable Neighborhood Descent (VND) algorithm. The ordering of the neighborhood structures in this VND is determined dynamically by solving relaxations of them. The objective values of these relaxations are used as indicators for the potential gains of searching the corresponding neighborhoods. We tested this new approach on the well-studied multidimensional knapsack problem. Computational experiments show that our approach is beneficial to the search, improving the obtained results. The concept is, in principle, more generally applicable and seems to be promising for many other combinatorial optimization problems approached by VNS. NICTA is funded by the Australian Government’s Backing Australia’s Ability initiative, in part through the Australian Research Council.The Institute of Computer Graphics and Algorithms is supported by the European RTN ADONET under grant 504438.     

A mixed integer programming model for multiple stage adaptive testing

The last decade has seen paper-and-pencil (P&P) tests being replaced by computerized adaptive tests (CATs) within many testing programs. A CAT may yield several advantages relative to a conventional P&P test. A CAT can determine the questions or test items to administer, allowing each test form to be tailored to a test taker’s skill level. Subsequent items can be chosen to match the capability of the test taker. By adapting to a test taker’s ability, a CAT can acquire more information about a test taker while administering fewer items. A Multiple Stage Adaptive test (MST) provides a means to implement a CAT that allows review before the administration. The MST format is a hybrid between the conventional P&P and CAT formats. This paper presents mixed integer programming models for MST assembly problems. Computational results with commercial optimization software will be given and advantages of the models evaluated.     

A hybrid model of integer programming and variable neighbourhood search for highly-constrained nurse rostering problems

This paper presents a hybrid multi-objective model that combines integer programming (IP) and variable neighbourhood search (VNS) to deal with highly-constrained nurse rostering problems in modern hospital environments. An IP is first used to solve the subproblem which includes the full set of hard constraints and a subset of soft constrains. A basic VNS then follows as a postprocessing procedure to further improve the IP’s resulting solutions. The satisfaction of the excluded constraints from the preceding IP model is the major focus of the VNS. Very promising results are reported compared with a commercial genetic algorithm and a hybrid VNS approach on real instances arising in a Dutch hospital. The comparison results demonstrate that our hybrid approach combines the advantages of both the IP and the VNS to beat other approaches in solving this type of problems. We also believe that the proposed methodology can be applied to other resource allocation problems with a large number of constraints.     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

Linear programming with online learning

We propose online decision strategies for time-dependent sequences of linear programs which use no distributional and minimal geometric assumptions about the data. These strategies are obtained through Vovk's aggregating algorithm which combines recommendations from a given strategy pool. We establish an average-performance bound for the resulting solution sequence.     

A linear programming approach for linear programs with probabilistic constraints

We study a class of mixed-integer programs for solving linear programs with joint probabilistic constraints from random right-hand side vectors with finite distributions. We present greedy and dual heuristic algorithms that construct and solve a sequence of linear programs. We provide optimality gaps for our heuristic solutions via the linear programming relaxation of the extended mixed-integer formulation of Luedtke et al. (2010) [13] as well as via lower bounds produced by their cutting plane method. While we demonstrate through an extensive computational study the effectiveness and scalability of our heuristics, we also prove that the theoretical worst-case solution quality for these algorithms is arbitrarily far from optimal. Our computational study compares our heuristics against both the extended mixed-integer programming formulation and the cutting plane method of Luedtke et al. (2010) [13]. Our heuristics efficiently and consistently produce solutions with small optimality gaps, while for larger instances the extended formulation becomes intractable and the optimality gaps from the cutting plane method increase to over 5%.     

A mixed integer linear programming formulation of the maximum betweenness problem

This paper considers the maximum betweenness problem. A new mixed integer linear programming (MILP) formulation is presented and validity of this formulation is given. Experimental results are performed on randomly generated instances from the literature. The results of CPLEX solver, based on the proposed MILP formulation, are compared with results obtained by total enumeration technique. The results show that CPLEX optimally solves instances of up to 30 elements and 60 triples in a short period of time.     

Reconstructing (0,1)-matrices from projections using integer programming

We study the problem of reconstructing (0,1)-matrices based on projections along a small number of directions. This discrete inverse problem is generally hard to solve for more than 3 projection directions. Building on previous work by the authors, we give a problem formulation with the objective of finding matrices with the maximal number of neighboring ones. A solution approach based on variable splitting and the use of subgradient optimization is given. Further, computational results are given for some structured instances. Optimal solutions are found for instances with up to 10,000 binary variables.     

Reducing the number of variables in integer quadratic programming problem

In this paper, a new variable reduction technique is presented for general integer quadratic programming problem (GP), under which some variables of (GP) can be fixed at zero without sacrificing optimality. A sufficient condition and a necessary condition for the identification of dominated terms are provided. By comparing the given data of the problem and the upper bound of the variables, if they meet certain conditions, some variables can be fixed at zero. We report a computational study to demonstrate the efficacy of the proposed technique in solving general integer quadratic programming problems. Furthermore, we discuss separable integer quadratic programming problems in a simpler and clearer form.     

Integer programming models for the q-mode problem

The q-mode problem is a combinatorial optimization problem that requires partitioning of objects into clusters. We discuss theoretical properties of an existing mixed integer programming (MIP) model for this problem and offer alternative models and enhancements. Through a comprehensive experiment we investigate computational properties of these MIP models. This experiment reveals that, in practice, the MIP approach is more effective for instances containing strong natural clusters and it is not as effective for instances containing weak natural clusters. The experiment also reveals that one of the MIP models that we propose is more effective than the other models for solving larger instances of the problem.     

Linear programming by minimizing distances

To solve the linear program (LP): minimizec ^T l subject toA l+b0, for ann×d-matrixA, ann-vectorb and ad-vectorc, the positive orthantS and the planeE(t) are defined by S={(x₁,x)ⁿ⁺¹ ¦(x₁,x)0}, E(t)={(x₁,x)ⁿ⁺¹¦x₁=–c ^c l+t, x=Al+b}. First a geometric algorithm is given to determine d(E(t),S) for fixedt, where d(·,·) denotes euclidean distance. This algorithm is used to construct a second algorithm to find the minimalt with E(t) S , and thus solve LP. It is shown that all algorithms are finite.     

Solving the plant location problem on a line by linear programming

This paper investigates the simple uncapacitated plant location problem on a line. We show that under general conditions the special structure of the problem allows the optimal solution to be obtained directly from a linear programming relaxation. This result may be extended to the related p-median problem on a line. Thus, the practitioner is now able to use readily available LP codes in place of specialized algorithms to solve these one-dimensional models. The findings also shed some light on the “integer friendliness” of the general problem.     

Integer linear programming models for grid-based light post location problem

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.     

Linear programming with estimatable parameters

Problems of estimating parameters in a linear programming model and using it in production planning are discussed here. An empirical application illustrates the various aspects of the estimation problem. © 1997 by John Wiley & Sons, Ltd.     

Polyhedral results on single node variable upper-bound flow models with allowed configurations

In this paper we investigate the convex hull of single node variable upper-bound flow models with allowed configurations. Such a model is defined by a set , where ρ is one of , = or , and Z{0,1}ⁿ consists of the allowed configurations. We consider the case when Z consists of affinely independent vectors. Under this assumption, a characterization of the non-trivial facets of the convex hull of X_ρ(Z) for each relation ρ is provided, along with polynomial time separation algorithms. Applications in scheduling and network design are also discussed.     

A procedure for one-parametric linear programming

A procedure is proposed for the parametric linear programming problem where all the coefficients are linear or polynomial functions of a scalar parameter. The solution vector and the optimum value are determined explicitly as rational functions of the parameter. In addition to standard linear programming technique, only the determination of eigenvalues is required. The procedure is compared to those by Dinkelbach and Zsigmond, and a numerical example is given.     

A new variable reduction technique for convex integer quadratic programs

Convex integer quadratic programming involves minimization of a convex quadratic objective function with affine constraints and is a well-known NP-hard problem with a wide range of applications. We proposed a new variable reduction technique for convex integer quadratic programs (IQP). Based on the optimal values to the continuous relaxation of IQP and a feasible solution to IQP, the proposed technique can be applied to fix some decision variables of an IQP simultaneously at zero without sacrificing optimality. Using this technique, computational effort needed to solve IQP can be greatly reduced. Since a general convex bounded IQP (BIQP) can be transformed to a convex IQP, the proposed technique is also applicable for the convex BIQP. We report a computational study to demonstrate the efficacy of the proposed technique in solving quadratic knapsack problems.