Integer programming formulations of vehicle routing problems

Various mathematical formulations are available for situations represented by vehicle routing problems. The assignment-based integer programming formulations of these problems are more common and easy to understand. Such formulations are discussed in this paper and a much simpler formulation for the vechicle routing problem is presented for the case, when all the vehicles have the same load capacity and maximum allowable cost per route.     

Generalized Nonlinear Lagrangian Formulation for Bounded Integer Programming

Several nonlinear Lagrangian formulations have been recently proposed for bounded integer programming problems. While possessing an asymptotic strong duality property, these formulations offer a success guarantee for the identification of an optimal primal solution via a dual search. Investigating common features of nonlinear Lagrangian formulations in constructing a nonlinear support for nonconvex piecewise constant perturbation function, this paper proposes a generalized nonlinear Lagrangian formulation of which many existing nonlinear Lagrangian formulations become special cases.     

The equivalence of semidefinite relaxations of polynomial 0-1 and ± 1 programs via scaling

Many combinatorial optimization problems can be modelled as polynomial-programming problems in binary variables that are all 0-1 or ±1. A sufficient condition under which a common method for obtaining semidefinite-programming relaxations of the two models of the same problem gives equivalent relaxations is established.     

Linear programming in tector criterion markov and semi-Markov decision processes

Optimality problems in infinite horizon, discrete time, vector criterion Markov and semi-Markov decision processes are expressed as standard problems of multiobjective linear programming. Processes with discounting, absorbing processes and completely ergodie processes without discounting are investigated. The common properties and special structure of derived multiobjective linear programming problems are overviewed. Computational simplicities associated with these problems in comparison with general multiobjective linear programming problems are discussed. Methods for solving these problems are overviewed and simple numerical examples are given.     

You want them to remember? Then make it memorable! Means for enhancing operations research education

In this article I summarize the main points I made in the keynote presentation of the same title I gave at the EURO XXIV conference in Lisbon, Portugal in July of 2010. Each of these points deals in some way with making communications between an operations research professional (academic or practitioner) and a student, client, subordinate, supervisor, or colleague more effective. Furthermore, each point is directly related to some realization (or epiphany) that I have had with regard to communication since I began teaching ORMS in 1984. It is noteworthy that these communications share a common objective; we are trying to facilitate learning. Since I have spent most of my career in academia, my primary emphasis is on communication with students (particularly those enrolled in introductory ORMS courses). However, I have also spent a great deal of time working on operations research problems outside of academia, either as an employee in private industry or as an operations research consultant to corporations and not-for-profit organizations, and I hope as a consequence my discussion is also relevant to those working in the practice of ORMS.     

The one-dimensional cutting stock problem with due dates

The one-dimensional cutting stock problem is the problem of cutting stock material into shorter lengths, in order to meet demand for these shorter lengths while minimizing waste. In industrial cutting operations, it may also be necessary to fill the orders for these shorter lengths before a given due date. We propose new optimization models and solution procedures which solve the cutting stock problem when orders have due dates. We evaluate our approach using data from a large manufacturer of reinforcement steel and show that we are able to solve industrial-size problems, while also addressing common cutting considerations such as aggregation of orders, multiple stock lengths and cutting different types of material on the same machine. In addition, we evaluate operational performance in terms of resulting waste and tardiness of orders using our model in a rolling horizon framework.     

指数型增广拉格朗日函数在广义半无限规划中的应用

本文利用指数型增广拉格朗日函数将一类广义半无限极大极小问题在一定条件下转化为标准的半无限极大极小问题,使它们具有相同的局部与全局最优解.我们给出了两个转化条件:一个是充分与必要条件,另一个是在实际中易于验证的充分条件.通过这种转化,我们给出了广义半无限极大极小问题的一个新的一阶最优性条件.     

A new proof of the strong duality theorem for semidefinite programming

Semidefinite programs are convex optimization problems arising in a wide variety of applications and are the extension of linear programming. Most methods for linear programming have been generalized to semidefinite programs. Just as in linear programming, duality theorem plays a basic and an important role in theory as well as in algorithmics. Based on the discretization method and convergence property, this paper proposes a new proof of the strong duality theorem for semidefinite programming, which is different from other common proofs and is more simple.     

Interactive TOPSIS algorithms for solving multi-level non-linear multi-objective decision-making problems

This paper extended the concept of the technique for order preference by similarity to ideal solution (TOPSIS) to develop a methodology for solving multi-level non-linear multi-objective decision-making (MLN-MODM) problems of maximization-type. Also, two new interactive algorithms are presented for the proposed TOPSIS approach for solving these types of mathematical programming problems. The first proposed interactive TOPSIS algorithm includes the membership functions of the decision variables for each level except the lower level of the multi-level problem. These satisfactory decisions are evaluated separately by solving the corresponding single-level MODM problems. The second proposed interactive TOPSIS algorithm lexicographically solves the MODM problems of the MLN-MOLP problem by taking into consideration the decisions of the MODM problems for the upper levels. To demonstrate the proposed algorithms, a numerical example is solved and compared the solutions of proposed algorithms with the solution of the interactive algorithm of Osman et al. (2003) [4]. Also, an example of an application is presented to clarify the applicability of the proposed TOPSIS algorithms in solving real world multi-level multi-objective decision-making problems.     

Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems

This paper presents a review of advances in the mathematical programming approach to discrete/continuous optimization problems. We first present a brief review of MILP and MINLP for the case when these problems are modeled with algebraic equations and inequalities. Since algebraic representations have some limitations such as difficulty of formulation and numerical singularities for the nonlinear case, we consider logic-based modeling as an alternative approach, particularly Generalized Disjunctive Programming (GDP), which the authors have extensively investigated over the last few years. Solution strategies for GDP models are reviewed, including the continuous relaxation of the disjunctive constraints. Also, we briefly review a hybrid model that integrates disjunctive programming and mixed-integer programming. Finally, the global optimization of nonconvex GDP problems is discussed through a two-level branch and bound procedure.     

A branch-and-reduce approach to global optimization

This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.     

A Parallel Interior Point Method and Its Application to Facility Location Problems

We present a parallel interior point algorithm to solve block structured linear programs. This algorithm can solve block diagonal linear programs with both side constraints (common rows) and side variables (common columns). The performance of the algorithm is investigated on uncapacitated, capacitated and stochastic facility location problems. The facility location problems are formulated as mixed integer linear programs. Each subproblem of the branch and bound phase of the MIP is solved using the parallel interior point method. We compare the total time taken by the parallel interior point method with the simplex method to solve the complete problems, as well as the various costs of reoptimisation of the non-root nodes of the branch and bound. Computational results on two parallel computers (Fujitsu AP1000 and IBM SP2) are also presented in this paper.     

Generating Convex Polynomial Inequalities for Mixed 0–1 Programs

We develop a method for generating valid convex quadratic inequalities for mixed0–1 convex programs. We also show how these inequalities can be generated in the linear case by defining cut generation problems using a projection cone. The basic results for quadratic inequalities are extended to generate convex polynomial inequalities.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

A survey on the continuous nonlinear resource allocation problem

Our problem of interest consists of minimizing a separable, convex and differentiable function over a convex set, defined by bounds on the variables and an explicit constraint described by a separable convex function. Applications are abundant, and vary from equilibrium problems in the engineering and economic sciences, through resource allocation and balancing problems in manufacturing, statistics, military operations research and production and financial economics, to subproblems in algorithms for a variety of more complex optimization models. This paper surveys the history and applications of the problem, as well as algorithmic approaches to its solution. The most common techniques are based on finding the optimal value of the Lagrange multiplier for the explicit constraint, most often through the use of a type of line search procedure. We analyze the most relevant references, especially regarding their originality and numerical findings, summarizing with remarks on possible extensions and future research.     

On the Quasiconcave Bilevel Programming Problem

Bilevel programming involves two optimization problems where the constraint region of the first-level problem is implicitly determined by another optimization problem. In this paper, we consider the case in which both objective functions are quasiconcave and the constraint region common to both levels is a polyhedron. First, it is proved that this problem is equivalent to minimizing a quasiconcave function over a feasible region comprised of connected faces of the polyhedron. Consequently, there is an extreme point of the polyhedron that solves the problem. Finally, it is shown that this model includes the most important case where the objective functions are ratios of concave and convex functions     

On the rate of convergence of some feasible direction algorithms

This paper is concerned with first-order methods of feasible directions. Pironneau and Polak have recently proved theorems which show that three of these methods have a linear rate of convergence for certain convex problems in which the objective functions have positive definite Hessians near the solutions. In the present note, it is shown that these theorems on rate of convergence can be extended to larger classes of problems. These larger classes are determined in part by certain second-order sufficiency conditions, and they include many nonconvex problems. The arguments used here are based on the finite-dimensional version of Hestenes' indirect sufficiency 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.     

Bilevel Decision with Generalized Semi-infinite Optimization for Fuzzy Mappings as Lower Level Problems

In this paper, we introduce the bilevel decision problems with parametric generalized semi-infinite optimization for fuzzy mappings as the lower-level problem, and its corresponding MPEC problems. For these problems, we establish two models which are different in the feasible region setting of lower-level problems. Some new existence results are obtained in rather weak conditions.