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.     

Improving benders decomposition using a genetic algorithm

We develop and investigate the performance of a hybrid solution framework for solving mixed-integer linear programming problems. Benders decomposition and a genetic algorithm are combined to develop a framework to compute feasible solutions. We decompose the problem into a master problem and a subproblem. A genetic algorithm along with a heuristic are used to obtain feasible solutions to the master problem, whereas the subproblem is solved to optimality using a linear programming solver. Over successive iterations the master problem is refined by adding cutting planes that are implied by the subproblem. We compare the performance of the approach against a standard Benders decomposition approach as well as against a stand-alone solver (Cplex) on MIPLIB test problems.     

一种具有非线性约束线性规划全局优化算法

本文提出了一种新的适用于处理非线性约束下线性规划问题的全局优化算法。该算法通过构造子问题来寻找优于当前局部最优解的可行解。该子问题可通过模拟退火算法来解决。通过求解一系列的子问题,当前最优解被不断地更新,最终求得全局最优解。最后,本算法应用于几个典型例题,并与罚函数法相比较,数值结果表明该算法是可行的,有效的。     

Fleet assignment and routing with schedule synchronization constraints

This paper introduces a new type of constraints, related to schedule synchronization, in the problem formulation of aircraft fleet assignment and routing problems and it proposes an optimal solution approach. This approach is based on Dantzig–Wolfe decomposition/column generation. The resulting master problem consists of flight covering constraints, as in usual applications, and of schedule synchronization constraints. The corresponding subproblem is a shortest path problem with time windows and linear costs on the time variables and it is solved by an optimal dynamic programming algorithm. This column generation procedure is embedded into a branch and bound scheme to obtain integer solutions. A dedicated branching scheme was devised in this paper where the branching decisions are imposed on the time variables. Computational experiments were conducted using weekly fleet routing and scheduling problem data coming from an European airline. The test problems are solved to optimality. A detailed result analysis highlights the advantages of this approach: an extremely short subproblem solution time and, after several improvements, a very efficient master problem solution time.     

Theory and Computation of Best Strict Constrained Chebyshev Approximation of Discrete Data

Apart from general linear programming algorithms, the publishedtechniques for computing the Chebyshev solution of overdeterminedsystems of linear equations do not allow for linear constraintson the approximation. The necessity for such constraints arisesnaturally when functions have to be fitted to discrete data,and is a subproblem in continuous constrained Chebyshev approximation.We extend the well-known exchange algorithm to solve this problemand use numerically stable techniques to include the cases wherethe matrix of coefficients is rank-deficient and/or ill-conditioned.A simple extension of the algorithm yields the "strict" Chebyshevapproximation. Numerical examples are given which testify tothe robustness of the algorithm.     

A surrogate and Lagrangian approach to constrained network problems

This paper presents the use of surrogate constraints and Lagrange multipliers to generate advanced starting solutions to constrained network problems. The surrogate constraint approach is used to generate a singly constrained network problem which is solved using the algorithm of Glover, Karney, Klingman and Russell [13]. In addition, we test the use of the Lagrangian function to generate advanced starting solutions. In the Lagrangian approach, the subproblems are capacitated network problems which can be solved using very efficient algorithms.The surrogate constraint approach is implemented using the multiplier update procedure of Held, Wolfe and Crowder [16]. The procedure is modified to include a search in a single direction to prevent periodic regression of the solution. We also introduce a reoptimization procedure which allows the solution from thekth subproblem to be used as the starting point for the next surrogate problem for which it is infeasible once the new surrogate constraint is adjoined.The algorithms are tested under a variety of conditions including: large-scale problems, number and structure of the non-network constraints, and the density of the non-network constraint coefficients.The testing clearly demonstrates that both the surrogate constraint and Langrange multipliers generate advanced starting solutions which greatly improve the computational effort required to generate an optimal solution to the constrained network problem. The testing demonstrates that the extra effort required to solve the singly constrained network subproblems of the surrogate constraints approach yields an improved advanced starting point as compared to the Lagrangian approach. It is further demonstrated that both of the relaxation approaches are much more computationally efficient than solving the problem from the beginning with a linear programming algorithm.     

An approach to find redundant objective function(s) and redundant constraint(s) in multi-objective nonlinear stochastic fractional programming problems

Structural redundancies in mathematical programming models are nothing uncommon and nonlinear programming problems are no exception. Over the past few decades numerous papers have been written on redundancy. Redundancy in constraints and variables are usually studied in a class of mathematical programming problems. However, main emphasis has so far been given only to linear programming problems. In this paper, an algorithm that identifies redundant objective function(s) and redundant constraint(s) simultaneously in multi-objective nonlinear stochastic fractional programming problems is provided. A solution procedure is also illustrated with numerical examples. The proposed algorithm reduces the number of nonlinear fractional objective functions and constraints in cases where redundancy exists.     

Trust region subproblem with an additional linear inequality constraint

This paper studies an extended trust region subproblem (eTRS) in which the trust region intersects the unit ball with a single linear inequality constraint. We present an efficient algorithm to solve the problem using a diagonalization scheme that requires solving a simple convex minimization problem. Attainment of the global optimality conditions is discussed. Our preliminary numerical experiments on several randomly generated test problems show that, the new approach is much faster in finding the global optimal solution than the known semidefinite relaxation approach, especially when solving large scale problems.     

An active set strategy for solving optimization problems with up to 200,000,000 nonlinear constraints

Numerical test results are presented for solving smooth nonlinear programming problems with a large number of constraints, but a moderate number of variables. The active set method proceeds from a given bound for the maximum number of expected active constraints at an optimal solution, which must be less than the total number of constraints. A quadratic programming subproblem is generated with a reduced number of linear constraints from the so-called working set, which is internally changed from one iterate to the next. Only for active constraints, i.e., a certain subset of the working set, new gradient values must be computed. The line search is adapted to avoid too many active constraints which do not fit into the working set. The active set strategy is an extension of an algorithm described earlier by the author together with a rigorous convergence proof. Numerical results for some simple academic test problems show that nonlinear programs with up to 200,000,000 nonlinear constraints are efficiently solved on a standard PC.     

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.     

A Hybrid Approach to Scheduling with Earliness and Tardiness Costs

A hybrid technique using constraint programming and linear programming is applied to the problem of scheduling with earliness and tardiness costs. The linear model maintains a set of relaxed optimal start times which are used to guide the constraint programming search heuristic. In addition, the constraint programming problem model employs the strong constraint propagation techniques responsible for many of the advances in constraint programming for scheduling in the past few years. Empirical results validate our approach and show, in particular, that creating and solving a subproblem containing only the activities with direct impact on the cost function and then using this solution in the main search, significantly increases the number of problems that can be solved to optimality while significantly decreasing the search time.     

A partitioning and bounded variable algorithm for linear programming

An interesting new partitioning and bounded variable algorithm (PBVA) is proposed for solving linear programming problems. The PBVA is a variant of the simplex algorithm which uses a modified form of the simplex method followed by the dual simplex method for bounded variables. In contrast to the two-phase method and the big M method, the PBVA does not introduce artificial variables. In the PBVA, a reduced linear program is formed by eliminating as many variables as there are equality constraints. A subproblem containing one ‘less than or equal to’ constraint is solved by executing the simplex method modified such that an upper bound is placed on an unbounded entering variable. The remaining constraints of the reduced problem are added to the optimal tableau of the subproblem to form an augmented tableau, which is solved by applying the dual simplex method for bounded variables. Lastly, the variables that were eliminated are restored by substitution. Differences between the PBVA and two other variants of the simplex method are identified. The PBVA is applied to solve an example problem with five decision variables, two equality constraints, and two inequality constraints. In addition, three other types of linear programming problems are solved to justify the advantages of the PBVA.     

Implementation of a proximal algorithm for linearly constrained nonsmooth optimization problems and computational results

This paper describes an implementation of the so-calledproximal point algorithm for solving convex linearly constrained nonsmooth optimization problems. Contrary to other previous implementations of the same approach (which solve constrained nonsmooth problems as unconstrained problems via exact penalty function techniques), our implementation handles linear constraints explicitly (linear constraints being incorporated into the direction-finding subproblem). The relevance and efficiency of the approach is demonstrated through comparative computational experiments on many classical test problems from the literature, as well as on a series of large constrained dual transportation problems introduced and studied here for the first time.     

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate our approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems.     

A branch-and-price algorithm for a vehicle routing with demand allocation problem

We investigate the vehicle routing with demand allocation problem where the decision-maker jointly optimizes the location of delivery sites, the assignment of customers to (preferably convenient) delivery sites, and the routing of vehicles operated from a central depot to serve customers at their designated sites. We propose an effective branch-and-price (B&P) algorithm that is demonstrated to greatly outperform the use of commercial branch-and-bound/cut solvers such as CPLEX. Central to the efficacy of the proposed B&P algorithm is the development of a specialized dynamic programming procedure that extends works on elementary shortest path problems with resource constraints in order to solve the more complex column generation pricing subproblem. Our computational study demonstrates the efficacy of the proposed approach using a set of 60 problem instances. Moreover, the proposed methodology has the merit of providing optimal solutions in run times that are significantly shorter than those reported for decomposition-based heuristics in the literature.     

Regulation of Overlaps in Technology Development Activities

In this paper, we present an algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems involving (i) 0-1 integer variables, and, (ii) more than one parameter, bounded between lower and upper bounds, present on the right hand side (RHS) of constraints. The solution is approached by decomposing the mp-MILP into two subproblems and then iterating between them. The first subproblem is obtained by fixing integer variables, resulting in a multiparametric linear programming (mp-LP) problem, whereas the second subproblem is formulated as a mixed integer linear programming (MILP) problem by relaxing the parameters as variables.     

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.     

Solving the subproblem in the lagrangian dual of separable discrete programs with linear constraints

This note presents an efficient method for the routine solution of the subproblem associated with the Lagrangian dual of discrete programming problems having separable non-linear objective function and linear constraints. An additional advantage for subgradient methods is described.     

A simple SLP algorithm for solving a class of nonlinear programs

Successive linear programming (SLP) algorithms solve nonlinear optimization problems via a sequence of linear programs. We present an approach for a special class of nonlinear programming problems, which arise in multiperiod coal blending. The class of nonlinear programming problems and the solution approach considered in this paper are quite different from previous work. The algorithm is very simple, easy to apply and can be applied to as large a problem as the linear programming code can handle. The quality of solution, produced by the proposed algorithm, is discussed and the results of some test problems, in the real world environment, are provided.     

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with cone constraints, and its dual, which has fewer variables than the original one, is a semismoothly differentiable (SC ¹) convex programming problem with both a linear inequality constraint and a linear second-order cone constraint. We demonstrate the global convergence of the augmented Lagrangian method with an exact solution to the subproblem and prove that the convergence rate of primal iterates, generated by the augmented Lagrangian method, is proportional to 1/r, and the rate of multiplier iterates is proportional to $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. Furthermore, a semismooth Newton method with Armijo line search is constructed to solve the subproblems in the augmented Lagrangian approach. Finally, numerical results are reported to show the effectiveness of the augmented Lagrangian method with both an exact solution and an inexact solution to the subproblem for solving the inverse second-order cone quadratic programming problem.