Application of an optimization problem in Max-Plus algebra to scheduling problems

The problem tackled in this paper deals with products of a finite number of triangular matrices in Max-Plus algebra, and more precisely with an optimization problem related to the product order. We propose a polynomial time optimization algorithm for 2×2 matrices products. We show that the problem under consideration generalizes numerous scheduling problems, like single machine problems or two-machine flow shop problems. Then, we show that for 3×3 matrices, the problem is NP-hard and we propose a branch-and-bound algorithm, lower bounds and upper bounds to solve it. We show that an important number of results in the literature can be obtained by solving the presented problem, which is a generalization of single machine problems, two- and three-machine flow shop scheduling problems. The branch-and-bound algorithm is tested in the general case and for a particular case and some computational experiments are presented and discussed.     

求线性比试和问题的确定性全局优化算法

为求线性比试和问题的全局最优解,本文给出了一个分支定界算法.通过一个等价问题和一个新的线性化松弛技巧,初始的非凸规划问题归结为一系列线性规划问题的求解.借助于这一系列线性规划问题的解,算法可收敛于初始非凸规划问题的最优解.算法的计算量主要是一些线性规划问题的求解.数值算例表明算法是切实可行的.     

The auction algorithm for the transportation problem

The auction algorithm is a parallel relaxation method for solving the classical assignment problem. It resembles a competitive bidding process whereby unassigned persons bid simultaneously for objects, thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder. This paper generalizes the auction algorithm to solve linear transportation problems. The idea is to convert the transportation problem into an assignment problem, and then to modify the auction algorithm to exploit the special structure of this problem. Computational results show that this modified version of the auction algorithm is very efficient for certain types of transportation problems.     

Approximation algorithms for combinatorial fractional programming problems

We are concerned with a combinatorial optimization problem which has the ratio of two linear functions as the objective function. This type of problems can be solved by an algorithm that uses an auxiliary problem with a parametrized linear objective function. Because of its combinatorial nature, however, it is often difficult to solve the auxiliary problem exactly. In this paper, we propose an algorithm which assumes that the auxiliary problems are solved only approximately, and prove that it gives an approximate solution to the original problem, of which the accuracy is at least as good as that of approximate solutions to the auxiliary problems. It is also shown that the time complexity is bounded by the square of the computation time of the approximate algorithm for the auxiliary problem. As an example of the proposed algorithm, we present a fully polynomial time approximation scheme for the fractional 0–1 knapsack problem.     

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.     

Solving Sum of Ratios Fractional Programs via Concave Minimization

This article presents an algorithm for globally solving a sum of ratios fractional programming problem. To solve this problem, the algorithm globally solves an equivalent concave minimization problem via a branch-and-bound search. The main work of the algorithm involves solving a sequence of convex programming problems that differ only in their objective function coefficients. Therefore, to solve efficiently these convex programming problems, an optimal solution to one problem can potentially be used to good advantage as a starting solution to the next problem.     

Cost‐efficient monitoring of continuous‐time stochastic processes based on discrete observations

Planning a cost‐efficient monitoring policy of stochastic processes arises from many industrial problems. We formulate a simple discrete‐time monitoring problem of continuous‐time stochastic processes with its applications to several industrial problems. A key in our model is a doubling trick of the variables, with which we can construct an algorithm to solve the problem. The cost‐efficient monitoring policy balancing between the observation cost and information loss is governed by an optimality equation of a fixed point type, which is solvable with an iterative algorithm based on the Feynman‐Kac formula. This is a new linkage between monitoring problems and mathematical sciences. We show regularity results of the optimization problem and present a numerical algorithm for its approximation. A problem having model ambiguity is presented as well. The presented model is applied to problems of environment, ecology, and energy, having qualitatively different target stochastic processes with each other.     

Minimizing the sum of job completion times on capacitated two-parallel machines

This paper considers a scheduling problem with two identical parallel machines. One has unlimited capacity; the other can only run for a fixed time. A given set of jobs must be scheduled on the two machines with the goal of minimizing the sum of their completion times. The paper proposes an optimal branch and bound algorithm which employs three powerful elements, including an algorithm for computing the upper bound, a lower bound algorithm, and a fathoming condition. The branch and bound algorithm was tested on problems of various sizes and parameters. The results show that the algorithm is quite efficient to solve all the test problems. In particular, the total computation time for the hardest problem is less than 0.1 second for a set of 100 problem instances. An important finding of the tests is that the upper bound algorithm can actually find optimal solutions to a quite large number of problems.     

Preconditioning by Projectors in the Solution of Contact Problems: A Parallel Implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.     

A Computational Study of the Homogeneous Algorithm for Large-scale Convex Optimization

Recently the authors have proposed a homogeneous and self-dual algorithm for solving the monotone complementarity problem (MCP) [5]. The algorithm is a single phase interior-point type method; nevertheless, it yields either an approximate optimal solution or detects a possible infeasibility of the problem. In this paper we specialize the algorithm to the solution of general smooth convex optimization problems, which also possess nonlinear inequality constraints and free variables. We discuss an implementation of the algorithm for large-scale sparse convex optimization. Moreover, we present computational results for solving quadratically constrained quadratic programming and geometric programming problems, where some of the problems contain more than 100,000 constraints and variables. The results indicate that the proposed algorithm is also practically efficient.     

Scatter Search—Wellsprings and Challenges

This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the article presents and validates a number of theoretical properties of these problems. Included among these properties is the result that, under a mild assumption, any globally optimal solution for problem (P1) must belong to the boundary of its feasible region. Also among these properties is a result that shows that problem (P2) can be reformulated as a convex maximization problem. Second, the article presents for the first time an algorithm for globally solving problem (P2). The algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems. Convergence properties of the algorithm are presented, and computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P2), provided that the number of variables is not too large.     

An exact penalty method for solving optimal control problems

This paper discusses an algorithm for solving optimal control problems. An optimal control problem is presented where the final time is unknown. The algorithm consists of an integrator and a minimizer; the latter is an exact penalty function used to solve constrained nonlinear programming problems. Essentially, the optimal control problem is converted to a mathematical programming problem such that a point satisfying the differential equations via the integrator is provided to the minimizer, a lower performance index is obtained, the integrator is reinitiated, etc., until a suitable stopping criterion is satisfied.     

Analyzing the performance of simultaneous generalized hill climbing algorithms

Simultaneous generalized hill climbing (SGHC) algorithms provide a framework for using heuristics to simultaneously address sets of intractable discrete optimization problems where information is shared between the problems during the algorithm execution. Many well-known heuristics can be embedded within the SGHC algorithm framework. This paper shows that the solutions generated by an SGHC algorithm are a stochastic process that satisfies the Markov property. This allows problem probability mass functions to be formulated for particular sets of problems based on the long-term behavior of the algorithm. Such results can be used to determine the proportion of iterations that an SGHC algorithm will spend optimizing over each discrete optimization problem. Sufficient conditions that guarantee that the algorithm spends an equal number of iterations in each discrete optimization problem are provided. SGHC algorithms can also be formulated such that the overall performance of the algorithm is independent of the initial discrete optimization problem chosen. Sufficient conditions are obtained guaranteeing that an SGHC algorithm will visit the globally optimal solution for each discrete optimization problem. Lastly, rates of convergence for SGHC algorithms are reported that show that given a rate of convergence for the embedded GHC algorithm, the SGHC algorithm can be designed to preserve this rate.     

Hyperbolic smoothing function method for minimax problems

In this article, an approach for solving finite minimax problems is proposed. This approach is based on the use of hyperbolic smoothing functions. In order to apply the hyperbolic smoothing we reformulate the objective function in the minimax problem and study the relationship between the original minimax and reformulated problems. We also study main properties of the hyperbolic smoothing function. Based on these results an algorithm for solving the finite minimax problem is proposed and this algorithm is implemented in general algebraic modelling system. We present preliminary results of numerical experiments with well-known nonsmooth optimization test problems. We also compare the proposed algorithm with the algorithm that uses the exponential smoothing function as well as with the algorithm based on nonlinear programming reformulation of the finite minimax problem.     

Global optimization for sum of geometric fractional functions

This paper presents an efficient branch and bound algorithm for globally solving sum of geometric fractional functions under geometric constraints, which arise in various practical problems. By using an equivalent transformation and a new linear relaxation technique, a linear relaxation programming problem of the equivalent problem is obtained. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems. Numerical results are reported to show the feasibility of our algorithm.     

AN EFFECTIVE CONTINUOUS ALGORITHM FOR APPROXIMATE SOLUTIONS OF LARGE SCALE MAX-CUT PROBLEMS

An effective continuous algorithm is proposed to find approximate solutions of NP-hardmax-cut problems.The algorithm relaxes the max-cut problem into a continuous nonlinearprogramming problem by replacing n discrete constraints in the original problem with onesingle continuous constraint.A feasible direction method is designed to solve the resultingnonlinear programming problem.The method employs only the gradient evaluations ofthe objective function,and no any matrix calculations and no line searches are required.This greatly reduces the calculation cost of the method,and is suitable for the solutionof large size max-cut problems.The convergence properties of the proposed method toKKT points of the nonlinear programming are analyzed.If the solution obtained by theproposed method is a global solution of the nonlinear programming problem,the solutionwill provide an upper bound on the max-cut value.Then an approximate solution to themax-cut problem is generated from the solution of the nonlinear programming and providesa lower bound on the max-cut value.Numerical experiments and comparisons on somemax-cut test problems(small and large size)show that the proposed algorithm is efficientto get the exact solutions for all small test problems and well satisfied solutions for mostof the large size test problems with less calculation costs.     

线性分式多乘积规划问题的完全多项式时间近似算法

本文针对线性分式多乘积规划问题,通过Charnes-Cooper转化将原问题转化为一个等价问题,借助此等价问题提出一个获得原问题全局近似最优解的算法,最终证明了算法的收敛性,且提供了算法运算时间的理论分析.     

A simple convex optimization problem with many applications

This paper presents an algorithm for the solution of a simpleconvex optimization problem. This problem is a generalizationof several other optimization problems which have applicationsto resource allocation, optimal capacity expansion, and vehiclescheduling. The algorithm is based on a constraint-relaxationapproach. It is easily implemented and transparent, and canbe used to solve even fairly large problems by hand calculator.     

An algorithm for maximizing a convex function over a simple set

The problem of maximizing a convex function on a so-called simple set is considered. Based on the optimality conditions [19], an algorithm for solving the problem is proposed. This numerical algorithm is shown to be convergent. The proposed algorithm has been implemented and tested on a variety of test problems.     

A neural network approach to facility layout problems

A near-optimum parallel algorithm for solving facility layout problems is presented in this paper where the problem is NP-complete. The facility layout problem is one of the most fundamental quadratic assignment problems in Operations Research. The goal of the problem is to locate N facilities on an N-square (location) array so as to minimize the total cost. The proposed system is composed of N × N neurons based on an artificial two-dimensional maximum neural network for an N-facility layout problem. Our algorithm has given improved solutions for several benchmark problems over the best existing algorithms.