A branch and bound algorithm for solving a class of nonlinear integer programming problems

This article presents a branch and bound method for solving the problem of minimizing a separable concave function over a convex polyhedral set where the variables are restricted to be integer valued. Computational results are reported.     

A class of continuous nonlinear programming problems with time-delayed constraints

A general class of continuous time nonlinear problems is considered. Necessary and sufficient conditions for the existence of solutions are established and optimal solutions are characterized in terms of a duality theorem. The theory is illustrated by means of an example.     

Numerical treatment of a class of semi-infinite programming problems

Many optimization problems occur in both theory and practice when one has to optimize an objective function while an infinite number of constraints must be satisfied. The aim of this paper in to describe methods of handling such problems numerically in an effective manner. We also indicate a number of applications.     

The frequency assignment problem: A solution via nonlinear programming

This paper gives a mathematical programming model for the problem of assigning frequencies to nodes in a communications network. The objective is to select a frequency assignment which minimizes both cochannel and adjacent-channel interference. In addition, a design engineer has the option to designate key links in which the avoidance of jamming due to self interference is given a higher priority. The model has a nonconvex quadratic objective function, generalized upper-bounding constraints, and binary decision variables. We developed a special heuristic algorithm and software for this model and tested it on five test problems which were modifications of a real-world problem. Even though most of the test problems had over 600 binary variables, we were able to obtain a near optimum in less than 12 seconds of CPU time on a CDC Cyber-875.     

An easy solution for a special class of fixed charge problems

The fixed charge problem is a mixed integer mathematical programming problem which has proved difficult to solve in the past. In this paper we look at a special case of that problem and show that this case can be solved by formulating it as a set-covering problem. We then use a branch-and-bound integer programming code to solve test fixed charge problems using the setcovering formulation. Even without a special purpose set-covering algorithm, the results from this solution procedure are dramatically better than those obtained using other solution procedures.     

An algorithm (GIPC2) for solving integer programming problems with separable nonlinear objective functions

This paper presents an algorithm for solving the integer programming problem possessing a separable nonlinear objective function subject to linear constraints. The method is based on a generalization of the Balas implicit enumeration scheme. Computational experience is given for a set of seventeen linear and seventeen nonlinear test problems. The results indicate that the algorithm can solve the nonlinear integer programming problem in roughly the equivalent time required to solve the linear integer programming problem of similar size with existing algorithms. Although the algorithm is specifically designed to solve the nonlinear problem, the results indicate that the algorithm compares favorably with the Branch and Bound algorithm in the solution of linear integer programming problems.     

An algorithm for a class of loading problems

The loading problem we consinder is to assign a set of discrete objects, each having a weight, to a set of boxes, each of which has a capacity limit, in such a way that every object is assigned to a box and the number of boxes used is minimized. A characterization of the assignments is offered and used to develop a set of rules for generating nonredundant assignments. The rules are incorporated into an implicit enumeration algorithm. The algorithm is tested against a very good heuristic. Computational experience shows that the algorithm is highly efficient, solving problems of up to 3600 0-1 variables in a CPU second.     

All-integer linear programming — a new approach via dynamic programming

An exact method for solving all-integer linear-programming problems is presented. Dynamic-programming methodology is used to search efficiently candidate hyperplanes for the optimal feasible integer solution. The explosive storage requirements for high-dimensional dynamic programming are avoided by the development of an analytic representation of the optimal allocation at each stage. Computational results for problems of small to moderate size are also presented.     

Optimality conditions for convex semi-infinite programming problems

This paper gives characterization of optimal Solutions for convex semiinfinite programming problems. These characterizations are free of a constraint qualification assumption. Thus they overcome the deficiencies of the semiinfinite versions of the Fritz John and the Kuhn-Tucker theories, which give only necessary or sufficient conditions for optimality, but not both.     

Algorithmic insights and a convergence analysis for a Karmarkar-type of algorithm for linear programming problems

This paper is concerned with a modification of a recently proposed variant of Karmarkar's algorithm for solving linear programming problems. In analyzing this variant, we exhibit interesting and useful relationships of these types of algorithms with barrier function methods, and subgradient optimization procedures involving space dilation techniques, which subsume the well-known ellipsoidal type of algorithms. Convergence of this variant is established under certain regularity conditions. We also provide remarks on how to obtain dual variables or Lagrange multipliers at optimality.     

旋转飞行器非线性运动稳定性判据

在弹体坐标系和准弹体坐标系中建立了旋转飞行器角运动数学模型。应用李亚普诺夫第一近似理论和劳斯-霍尔维茨方法导出了旋转飞行器的非线性运动稳定性判据,这个判据可应用于有控旋转导弹的运动稳定性分析,也可应用到炮弹和火箭弹上。     

An adaptive dynamic programming algorithm for a stochastic multiproduct batch dispatch problem

We address the problem of dispatching a vehicle with different product classes. There is a common dispatch cost, but holding costs that vary by product class. The problem exhibits multidimensional state, outcome and action spaces, and as a result is computationally intractable using either discrete dynamic programming methods, or even as a deterministic integer program. We prove a key structural property for the decision function, and exploit this property in the development of continuous value function approximations that form the basis of an approximate dispatch rule. Comparisons on single product‐class problems, where optimal solutions are available, demonstrate solutions that are within a few percent of optimal. The algorithm is then applied to a problem with 100 product classes, and comparisons against a carefully tuned myopic heuristic demonstrate significant improvements. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 742–769, 2003.     

An algorithm for separable piecewise convex programming problems

We present a branch and bound algorithm to solve mathematical programming problems of the form: Find x =|(x₁,…x_n) to minimize Σ?_i0(x₁) subject to x?G, l≦x≦L and Σ?_i0(x₁)≦0, j=1,…,m. With l=(l₁,…,l_n) and L=(L₁,…,L_n), each ?_ij is assumed to be lower aemicontinuous and piecewise convex on the finite interval [l_i.L_i]. G is assumed to be a closed convex set. The algorithm solves a finite sequence of convex programming problems; these correspond to successive partitions of the set C={x|l ≦ x ≦L} on the bahis of the piecewise convexity of the problem functions ?_ij. Computational considerations are discussed, and an illustrative example is presented.     

A genetic algorithm methodology for complex scheduling problems

This paper considers the scheduling problem to minimize total tardiness given multiple machines, ready times, sequence dependent setups, machine downtime and scarce tools. We develop a genetic algorithm based on random keys representation, elitist reproduction, Bernoulli crossover and immigration type mutation. Convergence of the algorithm is proved. We present computational results on data sets from the auto industry. To demonstrate robustness of the approach, problems from the literature of different structure are solved by essentially the same algorithm. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 199–211, 1999     

基于动态规划的猎雷具识别路径优化算法

路径长度、海流方向给猎雷具航渡、操控带来的时间消耗,是影响目标识别效率的主要因素。单纯运用动态规划算法只能解决猎雷具最短识别路径的问题,而无法顾全猎雷具在操控方面的时间损耗,从而在提升作战效率上得不偿失。本文基于动态规划算法,优化了识别路径的解算模型,并在模型解算前,提出了目标位置的预处理条件,简化了模型的解算步骤;在模型解算后,提出了识别路径的修正方法,完善了模型的解算结果。     

一类线性规划模型的求解方法

应用矩阵理论知识得到了一类特殊的线性规划模型的相关定理,给出了一种简便求解方法,讨论了求解方法的推广问题.     

Two machine flow shop scheduling problems with sequence dependent setup times: A dynamic programming approach

This paper considers the two different flow shop scheduling problems that arise when, in a two machine problem, one machine is characterized by sequence dependent setup times. The objective is to determine a schedule that minimizes makespan. After establishing the optimally of permutation schedules for both of these problems, an efficient dynamic programming formulation is developed for each of them. Each of these formulations is shown to be comparable, from a computational standpoint, to the corresponding formulation of the traveling salesman problem. Then, the relative merits of the dynamic programming and branch and bound approaches to these two scheduling problems are discussed.     

An algorithm for solving general fractional interval programming problems

A Linear Fractional Interval Programming problem (FIP) is the problem of extremizing a linear fractional function subject to two-sided linear inequality constraints. In this paper we develop an algorithm for solving (FIP) problems. We first apply the Charnes and Cooper transformation on (FIP) and then, by exploiting the special structure of the pair of (LP) problems derived, the algorithm produces an optimal solution to (FIP) in a finite number of iterations.     

Postoptimality analysis in nonlinear integer programming: The right-hand side case

An algorithm is presented to gain postoptimality data about the family of nonlinear pure integer programming problems in which the objective function and constraints remain the same except for changes in the right-hand side of the constraints. It is possible to solve such families of problems simultaneously to give a global optimum for each problem in the family, with additional problems solved in under 2 CPU seconds. This represents a small fraction of the time necessary to solve each problem individually.     

The procurement problem: An integer programming problem well suited to a solution using duality

A procurement problem, as formulated by Murty [10], is that of determining how many pieces of equipment units of each of m types are to be purchased and how this equipment is to be distributed among n stations so as to maximize profit, subject to a budget constraint. We have considered a generalization of Murty's procurement problem and developed an approach using duality to exploit the special structure of this problem. By using our dual approach on Murty's original problem, we have been able to solve large problems (1840 integer variables) with very modest computational effort. The main feature of our approach is the idea of using the current evaluation of the dual problem to produce a good feasible solution to the primal problem. In turn, the availability of good feasible solutions to the primal makes it possible to use a very simple subgradient algorithm to solve the dual effectively.