A relaxation method for nonconvex quadratically constrained quadratic programs

We present an algorithm for finding approximate global solutions to quadratically constrained quadratic programming problems. The method is based on outer approximation (linearization) and branch and bound with linear programming subproblems. When the feasible set is non-convex, the infinite process can be terminated with an approximate (possibly infeasible) optimal solution. We provide error bounds that can be used to ensure stopping within a prespecified feasibility tolerance. A numerical example illustrates the procedure. Computational experiments with an implementation of the procedure are reported on bilinearly constrained test problems with up to sixteen decision variables and eight constraints.This research was supported in part by National Science Foundation Grant DDM-91-14489.     

An implementation of linear and nonlinear multicommodity network flows

This work presents a new code for solving the multicommodity network flow problem with a linear or nonlinear objective function considering additional linear side constraints that link arcs of the same or different commodities. For the multicommodity network flow problem through primal partitioning the code implements a specialization of Murtagh and Saunders' strategy of dividing the set of variables into basic, nonbasic and superbasic. Several tests are reported, using random problems obtained from different network generators and real problems arising from the fields of long and short-term hydrothermal scheduling of electricity generation and traffic assignment, with sizes of up to 150000 variables and 45 000 constraints The performance of the code developed is compared to that of alternative methodologies for solving the same problems: a general purpose linear and nonlinear constrained optimization code, a specialised linear multicommodity network flow code and a primal-dual interior point code.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

A parameter-free multiplier method for constrained minimization problems

This paper is concerned with the development of a parameter-free method, closely related to penalty function and multiplier methods, for solving constrained minimization problems. The method is developed via the quadratic programming model with equality constraints. The study starts with an investigation into the convergence properties of a so-called “primal-dual differential trajectory”, defined by directions given by the direction of steepest descent with respect to the variables x of the problem, and the direction of steepest ascent with respect to the Lagrangian multipliers λ, associated with the Lagrangian function. It is shown that the trajectory converges to a stationary point (x*, λ*) corresponding to the solution of the equality constrained problem. Subsequently numerical procedures are proposed by means of which practical trajectories may be computed and the convergence of these trajectories are analyzed. A computational algorithm is presented and its application is illustrated by means of simple but representative examples. The extension of the method to inequality constrained problems is discussed and a non-rigorous argument, based on the Kuhn—Tucker necessary conditions for a constrained minimum, is put forward on which a practical procedure for determining the solution is based. The application of the method to inequality constrained problems is illustrated by its application to a couple of simple problems.     

Constructing a course schedule by solving a series of assignment type problems

We propose in this paper a new approach for tackling constrained course scheduling problems. The main idea is to decompose the problem into a series of easier subproblems. Each subproblem is an assignment type problem in which items have to be assigned to resources subject to some constraints. By solving a first series of assignment type subproblems, we build an initial solution which takes into account the constraints imposing a structure on the schedule. The total number of overlapping situations is reduced in a second phase by means of another series of assignment type problems. The proposed approach was implemented in practice and has proven to be satisfactory.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

Solving nonlinear integer programs with large-scale optimization software

This paper describes recent experience in tackling large nonlinear integer programming problems using the MINOS large-scale optimization software. A technique is presented for extending the constrained search approach used in MINOS to exploring integer-feasible solutions once a continuous optimal solution is obtained. Computational experience with this approach is described for two classes of problems: quadratic assignment problems and pipeline network design problems.     

An approach for solving a class of transportation scheduling problems

An algorithm is developed for solving a class of transportation scheduling problems. It applies for a variety of problems such as: the Combining Truck Trip problem, the Delivery problem, the School Bus problem, the Assignment of Buses to Schedules, and the Travelling Salesman problem. The objective functions of the above problems differ from each other. Yet, by using the “savings method” proposed by Clarke and Wright, and extended by Gaskell, we are able to define each one of the above problems as a series of assignment problems. The cost matrix entries of each one of the assignment problems are a function of the constraints of the particular routing or scheduling problem. The solution to the assignment problem determines an upper bound of the optimal solution to the original problem. By combining the above procedure with a Branch and Bound procedure, it is possible to obtain the optimal solution in a finite number of steps. In some cases the Branch and Bound process can be eliminated due to the nature of the problem and in those cases the algorithm is efficient.     

Fast automatic differentiation as applied to the computation of second derivatives of composite functions

A technique for deriving formulas for the second derivatives of a composite function with constrained variables is proposed. The original system of constraint equations is associated with a linear system of equations, whose solution is used to determine the Hessian of the function. The resulting formulas are applied to discrete problems obtained by approximating optimal control problems with the use of Runge-Kutta methods of various orders. For a particular optimal control problem, the numerical results obtained by the gradient method and Newton’s method with the resulting formulas are described and analyzed in detail.     

A reliable affine relaxation method for global optimization

An automatic method for constructing linear relaxations of constrained global optimization problems is proposed. Such a construction is based on affine and interval arithmetics and uses operator overloading. These linear programs have exactly the same numbers of variables and inequality constraints as the given problems. Each equality constraint is replaced by two inequalities. This new procedure for computing reliable bounds and certificates of infeasibility is inserted into a classical branch and bound algorithm based on interval analysis. Extensive computation experiments were made on 74 problems from the COCONUT database with up to 24 variables or 17 constraints; 61 of these were solved, and 30 of them for the first time, with a guaranteed upper bound on the relative error equal to \(10^{-8}\). Moreover, this sample comprises 39 examples to which the GlobSol algorithm was recently applied finding reliable solutions in 32 cases. The proposed method allows solving 31 of these, and 5 more with a CPU-time not exceeding 2 min.     

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.     

对等式约束非线性规划问题的Hestenes-Powell增广拉格朗日函数的进一步研究

本文对用无约束极小化方法求解等式约束非线性规划问题的Hestenes-Powell 增广拉格朗日函数作了进一步研究．在适当的条件下,我们建立了Hestenes-Powell增广拉格朗日函数在原问题变量空间上的无约束极小与原约束问题的解之间的关系,并且也给出了Hestenes-Powell增广拉格朗日函数在原问题变量和乘子变量的积空间上的无约束极小与原约束问题的解之间的一个关系．因此,从理论的观点来看,原约束问题的解和对应的拉格朗日乘子值不仅可以用众所周知的乘子法求得,而且可以通过对Hestenes-Powell 增广拉格朗日函数在原问题变量和乘子变量的积空间上执行一个单一的无约束极小化来获得．     

双变量LMEs一种异类约束最小二乘解的MCG算法

借鉴求线性矩阵方程组(LMEs)同类约束最小二乘解的修正共轭梯度法,建立了求双变量LMEs的一种异类约束最小二乘解的修正共轭梯度法,并证明了该算法的收敛性.在不考虑舍入误差的情况下,利用该算法不仅可在有限步计算后得到LMEs的一组异类约束最小二乘解,而且选取特殊初始矩阵时,可求得LMEs的极小范数异类约束最小二乘解.另外,还可求得指定矩阵在该LMEs的异类约束最小二乘解集合中的最佳逼近.算例表明,该算法是有效的.     

New exact method for large asymmetric distance-constrained vehicle routing problem

In this paper we revise and modify an old branch-and-bound method for solving the asymmetric distance–constrained vehicle routing problem suggested by Laporte et al. in 1987. Our modification is based on reformulating distance–constrained vehicle routing problem into a travelling salesman problem, and on using assignment problem as a lower bounding procedure. In addition, our algorithm uses the best-first strategy and new tolerance based branching rules. Since our method is fast but memory consuming, it could stop before optimality is proven. Therefore, we introduce the randomness, in case of ties, in choosing the node of the search tree. If an optimal solution is not found, we restart our procedure. As far as we know, the instances that we have solved exactly (up to 1000 customers) are much larger than the instances considered for other vehicle routing problem models from the recent literature. So, despite of its simplicity, this proposed algorithm is capable of solving the largest instances ever solved in the literature. Moreover, this approach is general and may be used for solving other types of vehicle routing problems.     

The p-Factor-Lagrange Methods for Degenerate Nonlinear Programming

The paper presents a new approach to solving nonlinear programming (NLP) problems for which the strict complementarity condition (SCC), a constraint qualification (CQ), and a second-order sufficient condition (SOSC) for optimality are not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equalities. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient condition (SOSC) for optimality. At the same time, we give examples of the problems where the SOSC does not hold, but our assumption is satisfied. Moreover, no estimation of the set of active constraints is required. The proposed approach can be applied to a variety of problems.     

Near-optimal solution to an employee assignment problem with seniority

I consider the problem of weekly assignment of shifts to operators. The shifts are to be assigned by seniority (or by any other employee hierarchy), but every employee is guaranteed to receive at least one shift. I propose a practical solution method to this problem that guarantees a feasible solution. As an application, the solution method is applied to the data provided in the literature on weekly shift assignment of the operators of the New Brunswick Telephone company. The solution method is also applied to 100 randomly generated problems. The results show that the solution method produces close-to-optimal solutions.     

On diagonally preconditioning the truncated Newton method for super-scale linearly constrained nonlinear programming

We present an algorithm for super-scale linearly constrained nonlinear programming (LCNP) based on Newton's method. In large-scale programming solving the Newton equation at each iteration can be expensive and may not be justified when far from a local solution. For super-scale problems, the truncated Newton method (where an inaccurate solution is computed by using the conjugate-gradient method) is recommended; a diagonal BFGS preconditioning of the gradient is used, so that the number of iterations to solve the equation is reduced. The procedure for updating that preconditioning is described for LCNP when the set of active constraints or the partition of basic, superbasic and nonbasic (structural) variables have been changed.     

A Problem of Assignment with Inter-Related Costs

There is a class of assignment problems where the cost function depends on the assignment of pairs of variables. A method of finding the optimum solution by a systematic exploration of a limited part of the solution space is presented by means of a numerical example. It is not possible to give a formula for the length of the computation since this will depend on the actual costs of the particular problem being solved.