A storm of feasibility pumps for nonconvex MINLP

One of the foremost difficulties in solving Mixed-Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed-Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed-Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pump algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pump algorithms is that, unlike most previous successive projection methods from the literature, projection is ??naturally?? taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performance of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.     

A multi-period network design problem for cellular telecommunication systems

Mathematical Programming models for multi-period network design problems, which arise in cellular telecommunication systems are presented. The underlying network topologies range from a simple star to complex multi-layer Steiner-like networks. Linear programming, Lagrangian relaxation, and branch-and-cut heuristics are proposed and a polynomial-bounded heuristic based on an interior point linear programming implementation is described. Extensive computational results are presented on a number of randomly generated problem sets and the performance of the heuristic(s) are compared with an optimal branch-and-bound algorithm.     

Optimum Allocation in Two-Stage Stratified Randomized Response Model

In the present paper a two-stage stratified Warner’s randomized response model is used to determine the optimum allocation in the presence of non-response. The problem is formulated as a Nonlinear Programming Problem. A complete method of solution of the formulated problem is proposed. Two numerical examples are worked out to illustrate the computational details of the proposed method.     

Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming

In this paper, we present a new hybrid algorithm for convex Mixed Integer Nonlinear Programming (MINLP). The proposed hybrid algorithm is an improved version of the classical nonlinear branch-and-bound (BB) procedure, where the enhancements are obtained with the application of the outer approximation algorithm on some nodes of the enumeration tree. The two methods are combined in such a way that each one collaborates to the convergence of the other. Computational experiments with benchmark instances of the MINLP problem show the good performance of the proposed algorithm, which is compared to the outer approximation algorithm, the nonlinear BB algorithm and the hybrid algorithm implemented in the solver Bonmin.     

Integrating SQP and Branch-and-Bound for Mixed Integer Nonlinear Programming

This paper considers the solution of Mixed Integer Nonlinear Programming (MINLP) problems. Classical methods for the solution of MINLP problems decompose the problem by separating the nonlinear part from the integer part. This approach is largely due to the existence of packaged software for solving Nonlinear Programming (NLP) and Mixed Integer Linear Programming problems.In contrast, an integrated approach to solving MINLP problems is considered here. This new algorithm is based on branch-and-bound, but does not require the NLP problem at each node to be solved to optimality. Instead, branching is allowed after each iteration of the NLP solver. In this way, the nonlinear part of the MINLP problem is solved whilst searching the tree. The nonlinear solver that is considered in this paper is a Sequential Quadratic Programming solver.A numerical comparison of the new method with nonlinear branch-and-bound is presented and a factor of up to 3 improvement over branch-and-bound is observed.     

Minimizing total completion time on a single machine with step improving jobs

Production systems often experience a shock or a technological change, resulting in performance improvement. In such settings, job processing times become shorter if jobs start processing at, or after, a common critical date. This paper considers a single machine scheduling problem with step-improving processing times, where the effects are job-dependent. The objective is to minimize the total completion time. We show that the problem is NP-hard in general and discuss several special cases which can be solved in polynomial time. We formulate a Mixed Integer Programming model and develop an LP-based heuristic for the general problem. Finally, computational experiments show that the proposed heuristic yields very effective and efficient solutions.     

Two linear approximation algorithms for convex mixed integer nonlinear programming

We present two new algorithms for convex Mixed Integer Nonlinear Programming (MINLP), both based on the well known Extended Cutting Plane (ECP) algorithm proposed by Weterlund and Petersson. Our first algorithm, Refined Extended Cutting Plane (RECP), incorporates additional cuts to the MILP relaxation of the original problem, obtained by solving linear relaxations of NLP problems considered in the Outer Approximation algorithm. Our second algorithm, Linear Programming based Branch-and-Bound (LP-BB), applies the strategy of generating cuts that is used in RECP, to the linear approximation scheme used by the LP/NLP based Branch-and-Bound algorithm. Our computational results show that RECP and LP-BB are highly competitive with the most popular MINLP algorithms from the literature, while keeping the nice and desirable characteristic of ECP, of being a first-order method.

     

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional big-M formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.     

Jointly pricing and ordering for a multi-product multi-constraint newsvendor problem with supplier quantity discounts

We present an extension to the multi-product newsvendor problem by incorporating the retailer’s pricing decision as well as considering supplier quantity discount. The objective is to maximize the expected profit of the retailer through jointly determining the ordering quantities and selling prices for the products, subject to multiple capacity constraints. We formulate the problem as a Generalized Disjunctive Programming (GDP) model and develop a Lagrangian heuristic approach for its solution. Randomly produced instances involving up to 1000 products are used to test the proposed approach. Computational results show that the Lagrangian heuristic approach can present very good solutions to all instances in reasonable time.     

A hybrid algorithm for the Heterogeneous Fleet Vehicle Routing Problem

This paper deals with the Heterogeneous Fleet Vehicle Routing Problem (HFVRP). The HFVRP generalizes the classical Capacitated Vehicle Routing Problem by considering the existence of different vehicle types, with distinct capacities and costs. The objective is to determine the best fleet composition as well as the set of routes that minimize the total costs. The proposed hybrid algorithm is composed by an Iterated Local Search (ILS) based heuristic and a Set Partitioning (SP) formulation. The SP model is solved by means of a Mixed Integer Programming solver that interactively calls the ILS heuristic during its execution. The developed algorithm was tested in benchmark instances with up to 360 customers. The results obtained are quite competitive with those found in the literature and new improved solutions are reported.     

An evolutionary heuristic for quadratic 0–1 programming

In this paper we present a heuristic algorithm for the well-known Unconstrained Quadratic 0–1 Programming Problem. The approach is based on combining solutions in a genetic paradigm and incorporates intensification algorithms used to improve solutions and speed up the method. Extensive computational experiments on instances with up to 500 variables are presented and we compare our approach both with powerful heuristic and exact algorithms from the literature establishing the effectiveness of the method in terms of solutions quality and computing time.     

Improved MILP models for two-machine flowshop with batch processing machines

In this paper, we consider the problem of scheduling jobs in a flowshop with two batch processing machines such that the makespan is minimized. Batch processing machines are frequently encountered in many industrial environments such as heat treatment operations in a steel foundry and chemical processes performed in tanks or kilns. Improved Mixed Integer Linear Programming (MILP) models are presented for the flowshop problem with unlimited or zero intermediate storage. An MILP-based heuristic is also developed for the problem. Computational experiments show that the new MILP models can significantly improve the original ones. Also, the heuristic can obtain the optimal solutions for all the test problem instances.     

Exact and heuristic approaches for the index tracking problem with UCITS constraints

Index tracking aims at determining an optimal portfolio that replicates the performance of an index or benchmark by investing in a smaller number of constituents or assets. The tracking portfolio should be cheap to maintain and update, i.e., invest in a smaller number of constituents than the index, have low turnover and low transaction costs, and should avoid large positions in few assets, as required by the European Union Directive UCITS (Undertaking for Collective Investments in Transferable Securities) rules. The UCITS rules make the problem hard to be satisfactorily modeled and solved to optimality: no exact methods but only heuristics have been proposed so far. The aim of this paper is twofold. First, we present the first Mixed Integer Quadratic Programming (MIQP) formulation for the constrained index tracking problem with the UCITS rules compliance. This allows us to obtain exact solutions for small- and medium-size problems based on real-world datasets. Second, we compare these solutions with the ones provided by the state-of-art heuristic Differential Evolution and Combinatorial Search for Index Tracking (DECS-IT), obtaining information about the heuristic performance and its reliability for the solution of large-size problems that cannot be solved with the exact approach. Empirical results show that DECS-IT is indeed appropriate to tackle the index tracking problem in such cases. Furthermore, we propose a method that combines the good characteristics of the exact and of the heuristic approaches.     

Models and algorithms for combinatorial optimization problems arising in railway applications

This is a summary of the author’s PhD thesis supervised by Alberto Caprara and Paolo Toth and defended on 29 May 2007 at the Università di Bologna. The thesis is written in English and is available from the author upon request. This work deals with Railway Optimization, and in particular it focuses on the Train Timetabling Problem (in the basic version on a corridor and in the extension to a railway network), and on the Train Unit Assignment Problem. Integer Linear Programming (ILP) formulations are proposed for both problems, and their continuous and Lagrangian relaxations are used to obtain optimal and heuristic solutions to real-world instances.      

A new line search inexact restoration approach for nonlinear programming

A new general scheme for Inexact Restoration methods for Nonlinear Programming is introduced. After computing an inexactly restored point, the new iterate is determined in an approximate tangent affine subspace by means of a simple line search on a penalty function. This differs from previous methods, in which the tangent phase needs both a line search based on the objective function (or its Lagrangian) and a confirmation based on a penalty function or a filter decision scheme. Besides its simplicity the new scheme enjoys some nice theoretical properties. In particular, a key condition for the inexact restoration step could be weakened. To some extent this also enables the application of the new scheme to mathematical programs with complementarity constraints.     

Siting and Sizing of Facilities under Probabilistic Demands

In this paper a discrete location model for non-essential service facilities planning is described, which seeks the number, location, and size of facilities, that maximizes the total expected demand attracted by the facilities. It is assumed that the demand for service is sensitive to the distance from facilities and to their size. It is also assumed that facilities must satisfy a threshold level of demand (facilities are not economically viable below that level). A Mixed-Integer Nonlinear Programming (MINLP) model is proposed for this problem. A branch-and-bound algorithm is designed for solving this MINLP and its convergence to a global minimum is established. A finite procedure is also introduced to find a feasible solution for the MINLP that reduces the overall search in the binary tree generated by the branch-and-bound algorithm. Some numerical results using a GAMS/MINOS implementation of the algorithm are reported to illustrate its efficacy and efficiency in practice.     

Two-stage vehicle routing problem with arc time windows: A mixed integer programming formulation and a heuristic approach

In this paper, we introduce a new variant of the Vehicle Routing Problem (VRP), namely the Two-Stage Vehicle Routing Problem with Arc Time Windows (TS_VRP_ATWs) which generally emerges from both military and civilian transportation. The TS_VRP_ATW is defined as finding the vehicle routes in such a way that each arc of the routes is available only during a predefined time interval with the objective of overall cost minimization. We propose a Mixed Integer Programming (MIP) formulation and a heuristic approach based on Memetic Algorithm (MA) to solve the TS_VRP_ATW. The qualities of both solution approaches are measured by using the test problems in the literature. Experimental results show that the proposed MIP formulation provides the optimal solutions for the test problems with 25 and 50 nodes, and some test problems with 100 nodes. Results also show that the proposed MA is promising quality solutions in a short computation time.     

The smoothed number of Pareto-optimal solutions in bicriteria integer optimization

Mathematical Programming - A well-established heuristic approach for solving bicriteria optimization problems is to enumerate the set of Pareto-optimal solutions. The heuristics following this...     

Optimal allocation of stock levels and stochastic customer demands to a capacitated resource

This paper considers a new class of stochastic resource allocation problems that requires simultaneously determining the customers that a capacitated resource must serve and the stock levels of multiple items that may be used in meeting these customers’ demands. Our model considers a reward (revenue) for serving each assigned customer, a variable cost for allocating each item to the resource, and a shortage cost for each unit of unsatisfied customer demand in a single-period context. The model maximizes the expected profit resulting from the assignment of customers and items to the resource while obeying the resource capacity constraint. We provide an exact solution method for this mixed integer nonlinear optimization problem using a Generalized Benders Decomposition approach. This decomposition approach uses Lagrangian relaxation to solve a constrained multi-item newsvendor subproblem and uses CPLEX to solve a mixed-integer linear master problem. We generate Benders cuts for the master problem by obtaining a series of subgradients of the subproblem’s convex objective function. In addition, we present a family of heuristic solution approaches and compare our methods with several MINLP (Mixed-Integer Nonlinear Programming) commercial solvers in order to benchmark their efficiency and quality.