An efficient computational method for a stochastic dynamic lot-sizing problem under service-level constraints

We provide an efficient computational approach to solve the mixed integer programming (MIP) model developed by Tarim and Kingsman [8] for solving a stochastic lot-sizing problem with service level constraints under the static-dynamic uncertainty strategy. The effectiveness of the proposed method hinges on three novelties: (i) the proposed relaxation is computationally efficient and provides an optimal solution most of the time, (ii) if the relaxation produces an infeasible solution, then this solution yields a tight lower bound for the optimal cost, and (iii) it can be modified easily to obtain a feasible solution, which yields an upper bound. In case of infeasibility, the relaxation approach is implemented at each node of the search tree in a branch-and-bound procedure to efficiently search for an optimal solution. Extensive numerical tests show that our method dominates the MIP solution approach and can handle real-life size problems in trivial time.     

A modified method for constructing efficient solutions structure of MOLP

This paper deals with a recently proposed algorithm for obtaining all weak efficient and efficient solutions in a multi objective linear programming (MOLP) problem. The algorithm is based on solving some weighted sum problems, and presents an easy and clear solution structure. We first present an example to show that the algorithm may fail when at least one of these weighted sum problems has not a finite optimal solution. Then, the algorithm is modified to overcome this problem. The modified algorithm determines whether an efficient solution exists for a given MOLP and generates the solution set correctly (if exists) without any change in the complexity.     

An iterative approach to solve multiobjective linear fractional programming problems

This paper suggests an iterative parametric approach for solving multiobjective linear fractional programming (MOLFP) problems which only uses linear programming to obtain efficient solutions and always converges to an efficient solution. A numerical example shows that this approach performs better than some existing algorithms. Randomly generated MOLFP problems are also solved to demonstrate the performance of new introduced algorithm.     

Implementation of the fixed point method in contact problems with Coulomb friction based on a dual splitting type technique

The paper deals with the numerical solution of the quasi-variational inequality describing the equilibrium of an elastic body in contact with a rigid foundation under Coulomb friction. After a discretization of the problem by mixed finite elements, the duality approach is exploited to reduce the problem to a sequence of quadratic programming problems with box constraints, so that efficient recently proposed algorithms may be applied. A new variant of this method is presented. It combines fixed point with block Gauss–Seidel iterations. The method may be also considered as a new implementation of fixed point iterations for a sequence of problems with given friction. Results of numerical experiments are given showing that the resulting algorithm may be much faster than the original fixed point method and its efficiency is comparable with the solution of frictionless contact problems.     

Range sets for weak efficiency in multiobjective linear programming and a parametric polytopes intersection problem

ABSTRACT

The aim of this paper is to obtain the range set for a given multiobjective linear programming problem and a weakly efficient solution. The range set is the set of all values of a parameter such that a given weakly efficient solution remains efficient when the objective coefficients vary in a given direction. The problem was originally formulated by Benson in 1985 and left to be solved. We formulate an algorithm for determining the range set, based on some hard optimization problems. Due to toughness of these optimization problems, we propose also lower and upper bound approximation techniques. In the second part, we focus on topological properties of the range set. In particular, we prove that a range set is formed by a finite union of intervals and we propose upper bounds on the number of intervals. Our approach to tackle the range set problem is via the intersection problem of parametric polytopes. Thus, our results have much wider area of applicability since the intersection (and separability) problem of convex polyhedra is important in many fields of optimization.     

Maximal and supremal tolerances in multiobjective linear programming

Let a multiobjective linear programming problem and any efficient solution be given. Tolerance analysis aims to compute interval tolerances for (possibly all) objective function coefficients such that the efficient solution remains efficient for any perturbation of the coefficients within the computed intervals. The known methods either yield tolerances that are not the maximal possible ones, or they consider perturbations of weights of the weighted sum scalarization only. We focus directly on perturbations of the objective function coefficients, which makes the approach independent on a scalarization technique used. In this paper, we propose a method for calculating the supremal tolerance (the maximal one need not exist). The main disadvantage of the method is the exponential running time in the worst case. Nevertheless, we show that the problem of determining the maximal/supremal tolerance is NP-hard, so an efficient (polynomial time) procedure is not likely to exist. We illustrate our approach on examples and present an application in transportation problems. Since the maximal tolerance may be small, we extend the notion to individual lower and upper tolerances for each objective function coefficient. An algorithm for computing maximal individual tolerances is proposed.     

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.     

Parametric solution to the joint system identification and optimization problem

This study is concerned with the simultaneous identification and optimization of static systems. The necessity and the advantages of an integrated approach to the identification and optimization of the system model is established theoretically as well as computationally. A parametric approach to the integrated problem is proven to converge to the integrated problem solution. The general methodology of decomposition of large-scale systems is extended by implementingfeasible decomposition of the joint problem. A multilevel approach is then utilized to successfully solve example problems. Handling the system constraints via a penalty-function technique is shown to be an efficient approach when using the parametric formulation of the joint problem. Numerical results for two example problems are presented using the Univac 1108 digital computer, revealing the economic advantages and disadvantages of the integrated approach to the identification and optimization problems.The authors wish to thank Messieurs I. Lefkowitz, L. S. Lasdon, F. Gembicki, and O. B. Olagundoye for their critique, comments, and suggestions during the course of this study.     

An Efficient Algorithm for Min-Max Convex Semi-Infinite Programming Problems

In this article, we consider the convex min-max problem with infinite constraints. We propose an exchange method to solve the problem by using efficient inactive constraint dropping rules. There is no need to solve the maximization problem over the metric space, as the algorithm has merely to find some points in the metric space such that a certain criterion is satisfied at each iteration. Under some mild assumptions, the proposed algorithm is shown to terminate in a finite number of iterations and to provide an approximate solution to the original problem. Preliminary numerical results with the algorithm are promising. To our knowledge, this article is the first one conceived to apply explicit exchange methods for solving nonlinear semi-infinite convex min-max problems.     

Two-Stage Approach for Quantitative Policy Analysis Using Bilevel Programming

Quantitative policy analysis problems with hierarchical decision-making can be modeled as bilevel mathematical programming problems. In general, the solution of these models is very difficult; however, special cases exist in which an optimal solution can be obtained by ordinary mathematical programming techniques. In this paper, a two-stage approach for the formulation, construction, solution, and usage of bilevel policy problem is presented. An outline of an example for analyzing Israel's public expenditure policy is also given.     

On stability in multiobjective programming — A stochastic approach

In this paper we assume that a deterministic multiobjective programming problem is approximated by surrogate problems based on estimations for the objective functions and the constraints. Making use of a large deviations approach, we investigate the behaviour of the constraint sets, the sets of efficient points and the solution sets if the size of the underlying sample tends to infinity. The results are illustrated by applying them to stochastic programming with chance constraints, where (i) the distribution function of the random variable is estimated by the empirical distribution function, (ii) certain parameters have to be estimated.     

New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints

In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series ofprimal andrelaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an -global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and indefinite quadratic programming problems of different sizes. The results show that the properties help to make the algorithm computationally efficient for fairly large problems.     

Branch-and-price for staff rostering: An efficient implementation using generic programming and nested column generation

We present a novel generic programming implementation of a column-generation algorithm for the generalized staff rostering problem. The problem is represented as a generalized set partitioning model, which is able to capture commonly occurring problem characteristics given in the literature. Columns of the set partitioning problem are generated dynamically by solving a pricing subproblem, and constraint branching in a branch-and-bound framework is used to enforce integrality. The pricing problem is formulated as a novel three-stage nested shortest path problem with resource constraints that exploits the inherent problem structure. A very efficient implementation of this pricing problem is achieved by using generic programming principles in which careful use of the C++ pre-processor allows the generator to be customized for the target problem at compile-time. As well as decreasing run times, this new approach creates a more flexible modeling framework that is well suited to handling the variety of problems found in staff rostering. Comparison with a more-standard run-time customization approach shows that speedups of around a factor of 20 are achieved using our new approach. The adaption to a new problem is simple and the implementation is automatically adjusted internally according to the new definition. We present results for three practical rostering problems. The approach captures all features of each problem and is able to provide high-quality solutions in less than 15 minutes. In two of the three instances, the optimal solution is found within this time frame.     

一种改进的GDF多目标优化算法

本文提出一种交互式非线性多目标优化算法，该算法是ＧＤＦ多目标优化算法的改进，具有这样的特点：算法采用了既约设计空间策略，具有良好的收敛性；算法生成的迭代点是有效解；算法具有多种一维搜索准则；对于线性多目标问题，算法只需一次交互迭代即可示出多目标问题的最优解。     

Computational Study of the Relationships Between Feasible and Efficient Sets and an Approximation

The computational difficulty of obtaining the efficient set in multi-objective programming, specially in nonlinear problems, suggest the need of considering an approximation approach to this problem. In this paper, we provide the computational results of the relationships between an approximation to the efficient set and the feasible and efficient sets. Random problem generation is considered for different sizes of the feasible set and we study the implications with respect to the number of objective functions and various kinds of objective functions. Computational experience with this approximation suggests that we obtain a substantial improvement when it increases the number of objective functions.     

A reformulation-convexification approach for solving nonconvex quadratic programming problems

In this paper, we consider the class of linearly constrained nonconvex quadratic programming problems, and present a new approach based on a novel Reformulation-Linearization/Convexification Technique. In this approach, a tight linear (or convex) programming relaxation, or outer-approximation to the convex envelope of the objective function over the constrained region, is constructed for the problem by generating new constraints through the process of employing suitable products of constraints and using variable redefinitions. Various such relaxations are considered and analyzed, including ones that retain some useful nonlinear relationships. Efficient solution techniques are then explored for solving these relaxations in order to derive lower and upper bounds on the problem, and appropriate branching/partitioning strategies are used in concert with these bounding techniques to derive a convergent algorithm. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality). It is shown that for many problems, the initial relaxation itself produces an optimal solution.     

多变量、多约束连续或离散的非线性规划的一个通用算法

利用目标函数对约束函数关于设计变量的一阶微分或差分之比,给出了一个求解非线性规划的通用算法．不论变量和约束有多少,也不论变量是连续的还是离散的,这一算法都比较有效,尤其对离散非线性规划更有效．该方法是一种搜索法,勿需解任何数学方程,只需要计算函数值以及函数对变量的偏微分或差分值．许多数值例题和运筹学中一些经典问题,如1) 一、二维的背包问题;2) 一、二维资源分配问题;3) 复合系统工作可靠性问题;4) 机器负荷问题等,经用此法求解验证均较传统方法更有效和可靠．该方法的主要优点是:1) 不受问题的规模限制;2) 只要在可行域（集）内存在目标函数和约束函数及其一阶导数或差分的值,肯定可以搜索到最优的解,没有不收敛和不稳定的问题．     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

On Steiner trees and minimum spanning trees in hypergraphs

The bottleneck of the state-of-the-art algorithms for geometric Steiner problems is usually the concatenation phase, where the prevailing approach treats the generated full Steiner trees as edges of a hypergraph and uses an LP-relaxation of the minimum spanning tree in hypergraph (MSTH) problem. We study this original and some new equivalent relaxations of this problem and clarify their relations to all classical relaxations of the Steiner problem. In an experimental study, an algorithm of ours which is designed for general graphs turns out to be an efficient alternative to the MSTH approach.