Exploring relaxation induced neighborhoods to improve MIP solutions

Given a feasible solution to a Mixed Integer Programming (MIP) model, a natural question is whether that solution can be improved using local search techniques. Local search has been applied very successfully in a variety of other combinatorial optimization domains. Unfortunately, local search relies extensively on the notion of a solution neighborhood, and this neighborhood is almost always tailored to the structure of the particular problem being solved. A MIP model typically conveys little information about the underlying problem structure. This paper considers two new approaches to exploring interesting, domain-independent neighborhoods in MIP. The more effective of the two, which we call Relaxation Induced Neighborhood Search (RINS), constructs a promising neighborhood using information contained in the continuous relaxation of the MIP model. Neighborhood exploration is then formulated as a MIP model itself and solved recursively. The second, which we call guided dives, is a simple modification of the MIP tree traversal order. Loosely speaking, it guides the search towards nodes that are close neighbors of the best known feasible solution. Extensive computational experiments on very difficult MIP models show that both approaches outperform default CPLEX MIP and a previously described approach for exploring MIP neighborhoods (local branching) with respect to several different metrics. The metrics we consider are quality of the best integer solution produced within a time limit, ability to improve a given integer solution (of both good and poor quality), and time required to diversify the search in order to find a new solution.Mathematics Subject Classification (2000):20E28, 20G40, 20C20Acknowledgement We wish to thank the two anonymous referees for their helpful comments.     

Continuous optimization methods for structure alignments

Structural Alignment is an important tool for fold identification of proteins, structural screening on ligand databases, pharmacophore identification and other applications. In the general case, the optimization problem of superimposing two structures is nonsmooth and nonconvex, so that most popular methods are heuristic and do not employ derivative information. Usually, these methods do not admit convergence theories of practical significance. In this work it is shown that the optimization of the superposition of two structures may be addressed using continuous smooth minimization. It is proved that, using a Low Order-Value Optimization approach, the nonsmoothness may be essentially ignored and classical optimization algorithms may be used. Within this context, a Gauss–Newton method is introduced for structural alignments incorporating (or not) transformations (as flexibility) on the structures. Convergence theorems are provided and practical aspects of implementation are described. Numerical experiments suggest that the Gauss–Newton methodology is competitive with state-of-the-art algorithms for protein alignment both in terms of quality and speed. Additional experiments on binding site identification, ligand and cofactor alignments illustrate the generality of this approach. The softwares containing the methods presented here are available at http://www.ime.unicamp.br/∼martinez/lovoalign. This work was supported by PRONEX-Optimization 76.79.1008-00, FAPESP (Grants 01-04597-4 - 02-14203-6 and 05-56773-1) and CNPq     

New approximate P-value of gapped local sequence alignments

We propose a new method to approximate the significativity of gapped local sequence alignments. We focus on short sequences for which standard methods are known to be less accurate since they have been developed under asymptotics. Our approach combines an approximate distribution of ungapped local score of two sequences and a special scoring scheme that allows the insertion of gaps. For a positive integer h, the scoring scheme is defined on h-tuples of the components of the sequences and corresponds to the gapped global score. The influence of h and the accuracy of the p-value are numerically studied. To cite this article: A.M. Fayyaz et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Ab initio Tertiary Structure Prediction of Proteins

A daunting challenge in the area of computational biology has been to develop a method to theoretically predict the correct three-dimensional structure of a protein given its linear amino acid sequence. The ability to surmount this challenge, which is known as the protein folding problem, has tremendous implications. We introduce a novel ab initio approach for the protein folding problem. The accurate prediction of the three-dimensional structure of a protein relies on both the mathematical model used to mimic the protein system and the technique used to identify the correct structure. The models employed are based solely on first principles, as opposed to the myriad of techniques relying on information from statistical databases. The framework integrates our recently proposed methods for the prediction of secondary structural features including helices and strands, as well as -sheet and disulfide bridge formation. The final stage of the approach, which culminates in the tertiary structure prediction of a protein, utilizes search techniques grounded on the foundations of deterministic global optimization, powerful methods which can potentially guarantee the correct identification of a protein's structure. The performance of the approach is illustrated with bovine pancreatic trypsin inhibitor protein and the immunoglobulin binding domain of protein G.     

A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

This paper presents a hybrid of a general heuristic framework and a general purpose mixed-integer programming (MIP) solver. The framework is based on local search and an adaptive procedure which chooses between a set of large neighborhoods to be searched. A mixed integer programming solver and its built-in feasibility heuristics is used to search a neighborhood for improving solutions. The general reoptimization approach used for repairing solutions is specifically suited for combinatorial problems where it may be hard to otherwise design suitable repair neighborhoods. The hybrid heuristic framework is applied to the multi-item capacitated lot sizing problem with setup times, where experiments have been conducted on a series of instances from the literature and a newly generated extension of these. On average the presented heuristic outperforms the best heuristics from the literature, and the upper bounds found by the commercial MIP solver ILOG CPLEX using state-of-the-art MIP formulations. Furthermore, we improve the best known solutions on 60 out of 100 and improve the lower bound on all 100 instances from the literature.     

Protein Structure Prediction Using Bee Colony Optimization Metaheuristic

Predicting the native structure of proteins is one of the most challenging problems in molecular biology. The goal is to determine the three-dimensional structure from the one-dimensional amino acid sequence. De novo prediction algorithms seek to do this by developing a representation of the proteins structure, an energy potential and some optimization algorithm that finds the structure with minimal energy. Bee Colony Optimization (BCO) is a relatively new approach to solving optimization problems based on the foraging behaviour of bees. Several variants of BCO have been suggested in the literature. We have devised a new variant that unifies the existing and is much more flexible with respect to replacing the various elements of the BCO. In particular, this applies to the choice of the local search as well as the method for generating scout locations and performing the waggle dance. We apply our BCO method to generate good solutions to the protein structure prediction problem. The results show that BCO generally finds better solutions than simulated annealing which so far has been the metaheuristic of choice for this problem.     

SCIP: solving constraint integer programs

Constraint integer programming (CIP) is a novel paradigm which integrates constraint programming (CP), mixed integer programming (MIP), and satisfiability (SAT) modeling and solving techniques. In this paper we discuss the software framework and solver SCIP (Solving Constraint Integer Programs), which is free for academic and non-commercial use and can be downloaded in source code. This paper gives an overview of the main design concepts of SCIP and how it can be used to solve constraint integer programs. To illustrate the performance and flexibility of SCIP, we apply it to two different problem classes. First, we consider mixed integer programming and show by computational experiments that SCIP is almost competitive to specialized commercial MIP solvers, even though SCIP supports the more general constraint integer programming paradigm. We develop new ingredients that improve current MIP solving technology. As a second application, we employ SCIP to solve chip design verification problems as they arise in the logic design of integrated circuits. This application goes far beyond traditional MIP solving, as it includes several highly non-linear constraints, which can be handled nicely within the constraint integer programming framework. We show anecdotally how the different solving techniques from MIP, CP, and SAT work together inside SCIP to deal with such constraint classes. Finally, experimental results show that our approach outperforms current state-of-the-art techniques for proving the validity of properties on circuits containing arithmetic.      

Near-optimal MIP solutions for preference based self-scheduling

Making a high quality staff schedule is both difficult and time consuming for any company that has employees working on irregular schedules. We formulate a mixed integer program (MIP) to find a feasible schedule that satisfies all hard constraints while minimizing the soft constraint violations as well as satisfying as many of the employees’ requests as possible. We present the MIP model and show the result from four real world companies and institutions. We also compare the results with those of a local search based algorithm that is designed to emulate the solution strategies when the schedules are created manually. The results show that using near-optimal solutions from the MIP model, with a relative MIP gap of around 0.01–0.1, instead of finding the optimal solution, allows us to find very good solutions in a reasonable amount of time that compare favorably with both the manual solutions and the solutions found by the local search based algorithm.     

Multi-funnel optimization using Gaussian underestimation

In several applications, underestimation of functions has proven to be a helpful tool for global optimization. In protein–ligand docking problems as well as in protein structure prediction, single convex quadratic underestimators have been used to approximate the location of the global minimum point. While this approach has been successful for basin-shaped functions, it is not suitable for energy functions with more than one distinct local minimum with a large magnitude. Such functions may contain several basin-shaped components and, thus, cannot be underfitted by a single convex underestimator. In this paper, we propose using an underestimator composed of several negative Gaussian functions. Such an underestimator can be computed by solving a nonlinear programming problem, which minimizes the error between the data points and the underestimator in the L ¹ norm. Numerical results for simulated and actual docking energy functions are presented.     

SAPI: Statistical Analysis of Propagation of Incidents. A new approach for rescheduling trains after disruptions

In this paper, we present a new approach to solve the railway rescheduling problem. This problem deals with the reparation of a disturbed railway timetable after incidents in such a way to minimize the difference between the original plan and the new provisional plan. We use a mixed integer linear programming (MIP) formulation that models this problem correctly. However, the large number of variables and constraints denies the possibility to solve this problem efficiently using a standard MIP solver. A new approach called SAPI (Statistical Analysis of Propagation of Incidents) has been developed to tackle the problem. The key point of SAPI is to estimate the probability that an event, one step of the itinerary of a train, is affected by a set of incidents. Using these probabilities, the search space is reduced, obtaining very good solutions in a short time. The method has been tested with two different networks located in France and Chile. The numerical results show that our procedure is viable in practice.     

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.     

Progress control in iterated local search for nurse rostering

This paper describes an approach in which a local search technique is alternated with a process which ‘jumps’ to another point in the search space. After each ‘jump’ a (time-intensive) local search is used to obtain a new local optimum. The focus of the paper is in monitoring the progress of this technique on a set of real world nurse rostering problems. We propose a model for estimating the quality of this new local optimum. We can then decide whether to end the local search based on the predicted quality. The fact that we avoid searching these bad neighbourhoods enables us to reach better solutions in the same amount of time. We evaluate the approach on five highly constrained problems in nurse rostering. These problems represent complex and challenging real world rostering situations and the approach described here has been developed during a commercial implementation project by ORTEC bv.     

Mixed integer programming-based solution procedure for single-facility location with maximin of rectilinear distance

In this paper, we study the 1-maximin problem with rectilinear distance. We locate a single undesirable facility in a continuous planar region while considering the interaction between the facility and existing demand points. The distance between facility and demand points is measured in the rectilinear metric. The objective is to maximize the distance of the facility from the closest demand point. The 1-maximin problem has been formulated as an MIP model in the literature. We suggest new bounding schemes to increase the solution efficiency of the model as well as improved branch and bound strategies for implementation. Moreover, we simplify the model by eliminating some redundant integer variables. We propose an efficient solution algorithm called cut and prune method, which splits the feasible region into four equal subregions at each iteration and tries to eliminate subregions depending on the comparison of upper and lower bounds. When the sidelengths of the subregions are smaller than a predetermined value, the improved MIP model is solved to obtain the optimal solution. Computational experiments demonstrate that the solution time of the original MIP model is reduced substantially by the proposed solution approach.     

Product allocation to different types of distribution center in retail logistics networks

We consider the problem of assigning stockkeeping units to distribution centers (DCs) belonging to different DC types of a retail network, e.g., central, regional, and local DCs. The problem is motivated by the real situation of a retail company and solved by an MIP solution approach. The MIP model reflects the interdependencies between inbound transportation, outbound transportation and instore logistics as well as capital tied up in inventories and differences in picking costs between the warehouses. A novel solution approach is developed and applied to a real-life case of a leading European grocery retail chain. The application of the new approach results in cost savings of 6% of total operational costs compared to the present assignment. These savings amount to several million euros per year. In-depth analyses of the results and sensitivity analyses provide insights into the solution structure and the major related issues.     

A MIP-based framework and its application on a lot sizing problem with setup carryover

In this paper we present a framework to tackle mixed integer programming problems based upon a “constrained” black box approach. Given a MIP formulation, a black-box solver, and a set of incumbent solutions, we iteratively build corridors around such solutions by adding exogenous constraints to the original MIP formulation. Such corridors, or neighborhoods, are then explored, possibly to optimality, with a standard MIP solver. An iterative approach in the spirit of a hill climbing scheme is thus used to explore subportions of the solution space. While the exploration of the corridor relies on a standard MIP solver, the way in which such corridors are built around the incumbent solutions is influenced by a set of factors, such as the distance metric adopted, or the type of method used to explore the neighborhood. The proposed framework has been tested on a challenging variation of the lot sizing problem, the multi-level lot sizing problem with setups and carryovers. When tested on 1920 benchmark instances of such problem, the algorithm was able to solve to near optimality every instance of the benchmark library and, on the most challenging instances, was able to find high quality solutions very early in the search process. The algorithm was effective, in terms of solution quality as well as computational time, when compared with a commercial MIP solver and the best algorithm from the literature.     

Local branching

The availability of effective exact or heuristic solution methods for general Mixed-Integer Programs (MIPs) is of paramount importance for practical applications. In the present paper we investigate the use of a generic MIP solver as a black-box ``tactical' tool to explore effectively suitable solution subspaces defined and controlled at a ``strategic' level by a simple external branching framework. The procedure is in the spirit of well-known local search metaheuristics, but the neighborhoods are obtained through the introduction in the MIP model of completely general linear inequalities called local branching cuts. The new solution strategy is exact in nature, though it is designed to improve the heuristic behavior of the MIP solver at hand. It alternates high-level strategic branchings to define the solution neighborhoods, and low-level tactical branchings to explore them. The result is a completely general scheme aimed at favoring early updatings of the incumbent solution, hence producing high-quality solutions at early stages of the computation. The method is analyzed computationally on a large class of very difficult MIP problems by using the state-of-the-art commercial software ILOG-Cplex 7.0 as the black-box tactical MIP solver. For these instances, most of which cannot be solved to proven optimality in a reasonable time, the new method exhibits consistently an improved heuristic performance: in 23 out of 29 cases, the MIP solver produced significantly better incumbent solutions when driven by the local branching paradigm. Mathematics Subject Classification (2000):90C06, 90C10, 90C11, 90C27, 90C59     

A hybrid optimization approach for multi-level capacitated lot-sizing problems

Solving multi-level capacitated lot-sizing problems is still a challenging task, in spite of increasing computational power and faster algorithms. In this paper a new approach combining an ant-based algorithm with an exact solver for (mixed-integer) linear programs is presented. A MAX–MIN ant system is developed to determine the principal production decisions, a LP/MIP solver is used to calculate the corresponding production quantities and inventory levels. Two different local search methods and an improvement strategy based on reduced mixed-integer problems are developed and integrated into the ant algorithm. This hybrid approach provides superior results for small and medium-sized problems in comparison to the existing approaches in the literature. For large-scale problems the performance of this method is among the best.     

Linear Relaxations and Reduced-Cost Based Propagation of Continuous Variable Subscripts

In hybrid solvers for combinatorial optimisation, combining Constraint (Logic) Programming (CLP) and Mixed Integer Programming (MIP), it is important to have tight connections between the two domains. We extend and generalise previous work on automatic linearisations and propagation of symbolic CLP constraints that cross the boundary between CLP and MIP. We also present how reduced costs from the linear programming relaxation can be used for domain reduction on the CLP side. Computational results comparing our hybrid approach with pure CLP and MIP on a configuration problem show significant speed-ups.     

The deterministic multi-item dynamic lot size problem with joint business volume discount

This paper considers the multi-item dynamic lot size model where joint business volume discount is applied for all items purchased whenever the total dollar value of an order reaches a certain level. Multi-item discounts are prevalent in practical applications, yet the literature has only considered limited instances of single-item models. We establish the mathematical formulation and design an effective dynamic programming based heuristic. Computational results disclose our approach obtains high quality solutions that dominate the best known heuristic for the simplified one-item case, and that proves vastly superior to the state-of-the-art CPLEX MIP code for the multi-item case (for which no alternative heuristics have been devised). We obtained significantly better solutions than CPLEX for the more complex problems, while running from 4800 to over 100,000 times faster. Enhanced variants of our method improve these outcomes further. This revised version was published online in June 2006 with corrections to the Cover Date.     

An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints

We provide an efficient computational approach to solve the mixed integer programming (MIP) model developed by Tarim and Kingsman [8] for solving a stochastic lot-sizing problem with service level constraints under the static-dynamic uncertainty strategy. The effectiveness of the proposed method hinges on three novelties: (i) the proposed relaxation is computationally efficient and provides an optimal solution most of the time, (ii) if the relaxation produces an infeasible solution, then this solution yields a tight lower bound for the optimal cost, and (iii) it can be modified easily to obtain a feasible solution, which yields an upper bound. In case of infeasibility, the relaxation approach is implemented at each node of the search tree in a branch-and-bound procedure to efficiently search for an optimal solution. Extensive numerical tests show that our method dominates the MIP solution approach and can handle real-life size problems in trivial time.