A finite steps algorithm for solving convex feasibility problems

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.     

A note on the linearity of Ratliff and Rosenthal's algorithm for optimal picker routing

Ratliff and Rosenthal state that their dynamic programming algorithm for optimal picker routing has linear complexity in the number of aisles. Indeed, solving the dynamic program is linear, but computing the cost coefficients of the dynamic program certainly requires the consideration of all picking positions, whose number is independent of the number of aisles. For a given unsorted sequence of picking positions, our algorithm is linear in the sum of the number of aisles and number of picking positions.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.     

Global optimization algorithm for sum of generalized polynomial ratios problem

In this paper, a global optimization algorithm is proposed for solving sum of generalized polynomial ratios problem (P) which arises in various practical problems. Due to its intrinsic difficulty, less work has been devoted to globally solve the problem (P). For such problems, we present a branch and bound algorithm. In this method, by utilizing exponent transformation and new three-level linear relaxation method, a sequence of linear relaxation programming of the initial nonconvex programming problem (P) are derived which are embedded in a branch and bound algorithm. The proposed method need not introduce new variables and constraints and it is convergent to the global minimum of prime problem by means of the subsequent solutions of a series of linear programming problems. Several numerical examples in the literatures are tested to demonstrate that the proposed algorithm can systematically solve these examples to find the approximate ?-global optimum.     

A strongly polynomial minimum cost circulation algorithm

A new algorithm is presented for the minimum cost circulation problem. The algorithm is strongly polynomial, that is, the number of arithmetic operations is polynomial in the number of nodes, and is independent of both costs and capacities.     

The balanced linear programming problem

The Balanced Linear Programming Problem (BLPP) arises in situations which require equitable distribution of a scarce resource. The BLPP can be transformed to the standard form of the linear programming problem by introducing 2∥N∥ + 2 additional variables and 2∥N∥ additional constraints. This transformation is not desirable from the computational point of view for larger values of ∥N∥ as it increases the problem size substantially. It is also undesirable from a theoretical perspective as it might affect the special structure of the constraint matrix. In this paper, we develop an algorithm for the BLPP which does not require problem enlargement. The algorithm is based on the relationship between the BLPP and the minimax linear programming problem, and solving the latter problem parametrically. Our algorithm, in essence, performs steps that are similar to those performed in the parametric simplex method with parametric right hand side. We then adapt our algorithm for the network flow problem and this specialized algorithm can be applied on the network directly without maintaining the simplex tableau.     

A contraction algorithm for the multiparametric integer linear programming problem

We designed and implemented an algorithm to solve the continuos right hand side multiparametric Integer Linear Programming (ILP) problem, that is to solve a family of ILP problems in which the problems are related by having identical objective and matrix coefficients. Our algorithm works by choosing an appropiate finite sequence of nonparametric Mixed Integer Linear Programming (MILP) problems in order to obtain a complete multiparametrical analysis. The algorithm may be implemented by using any software capable of solving MILP problems.     

A polynomial algorithm for minimum quadratic cost flow problems

Network flow problems with quadratic separable costs appear in a number of important applications such as; approximating input-output matrices in economy; projecting and forecasting traffic matrices in telecommunication networks; solving nondifferentiable cost flow problems by subgradient algorithms. It is shown that the scaling technique introduced by Edmonds and Karp (1972) in the case of linear cost flows for deriving a polynomial complexity bound for the out-of-kilter method, may be extended to quadratic cost flows and leads to a polynomial algorithm for this class of problems. The method may be applied to the solution of singly constrained quadratic programs and thus provides an alternative approach to the polynomial algorithm suggested by Helgason, Kennington and Lall (1980).     

Polynomial algorithms for LP over a subring of the algebraic integers with applications to LP with circulant matrices

We show that a modified variant of the interior point method can solve linear programs (LPs) whose coefficients are real numbers from a subring of the algebraic integers. By defining the encoding size of such numbers to be the bit size of the integers that represent them in the subring, we prove the modified algorithm runs in time polynomial in the encoding size of the input coefficients, the dimension of the problem, and the order of the subring. We then extend the Tardos scheme to our case, obtaining a running time which is independent of the objective and right-hand side data. As a consequence of these results, we are able to show that LPs with real circulant coefficient matrices can be solved in strongly polynomial time. Finally, we show how the algorithm can be applied to LPs whose coefficients belong to the extension of the integers by a fixed set of square roots.     

A new algorithm for Chebyshev minimum-error multiplication of reduced affine forms

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.     

求一类非凸规划问题全局解的确定性算法

本文对一类非凸规划问题(NP)给出一确定性全局优化算法.这类问题包括:在非凸的可行域上极小化有限个带指数的线性函数乘积的和与差,广义线性多乘积规划,多项式规划等.通过利用等价问题和线性化技巧提出的算法收敛到问题(NP)的全局极小.     

Global optimization algorithm for a generalized linear multiplicative programming

This article present a branch and bound algorithm for globally solving generalized linear multiplicative programming problems with coefficients. The main computation involve solving a sequence of linear relaxation programming problems, and the algorithm economizes the required computations by conducting the branch and bound search in R ^p , rather than in R ⁿ , where p is the number of rank and n is the dimension of decision variables. The proposed algorithm will be convergent to the global optimal solution by means of the subsequent solutions of the series of linear relaxation programming problems. Numerical results are given to show the feasibility and effectiveness of the proposed algorithm.     

A TRUST REGION ALGORITHM VIA BILEVEL LINEAR PROGRAMMING FOR SOLVING THE GENERAL MULTICOMMODITY MINIMAL COST FLOW PROBLEMS

This paper proposes a nonmonotonic backtracking trust region algorithm via bilevel linear programming for solving the general multicommodity minimal cost flow problems. Using the duality theory of the linear programming and convex theory, the generalized directional derivative of the general multicommodity minimal cost flow problems is derived. The global convergence and superlinear convergence rate of the proposed algorithm are established under some mild conditions.     

Branch-and-price algorithm for a multicast routing problem

This paper considers a multicast routing problem to find the minimum cost tree where the whole communication link delay on each path(route) of the tree is subject to a given delay allowance. The problem is formulated as an integer programming problem by using path variables. An associated problem reduction property is then characterised to reduce the solution space. Moreover, a polynomial time column generation procedure is exploited to solve the associated linear programming relaxation with such solution space reduced. Therefore a branch-and-price algorithm is derived to obtain the optimal integer solution(tree) for the problem. Computational results show that the algorithm can solve practical size problems in a reasonable length of time.     

A linear programming embedded genetic algorithm for an integrated cell formation and lot sizing considering product quality

Production lot sizing models are often used to decide the best lot size to minimize operation cost, inventory cost, and setup cost. Cellular manufacturing analyses mainly address how machines should be grouped and parts be produced. In this paper, a mathematical programming model is developed following an integrated approach for cell configuration and lot sizing in a dynamic manufacturing environment. The model development also considers the impact of lot sizes on product quality. Solution of the mathematical model is to minimize both production and quality related costs. The proposed model, with nonlinear terms and integer variables, cannot be solved for real size problems efficiently due to its NP-complexity. To solve the model for practical purposes, a linear programming embedded genetic algorithm was developed. The algorithm searches over the integer variables and for each integer solution visited the corresponding values of the continuous variables are determined by solving a linear programming subproblem using the simplex algorithm. Numerical examples showed that the proposed method is efficient and effective in searching for near optimal solutions.     

Closed formulas in local sensitivity analysis for some classes of linear and non-linear problems

This paper presents an integrated approach to sensitivity analysis in some linear and non-linear programming problems. Closed formulas for the sensitivities of the objective function and primal and dual variables with respect to all parameters for some classes of problems are obtained. As particular cases, the sensitivities with respect to all data values, i.e., cost coefficients, constraints coefficients and right hand side terms of the constraints are provided for these classes of problems as closed formulas. The method is illustrated by its application to several examples.      

An error-free algorithm to solve a linear system of polynomial equations

The paper presents an error-free algorithm to solve a system of linear equations with polynomial coefficients. Modular arithmetic in residual polynomial class and in residual numeric class is employed. The algorithm is iterative and well suited for implementation for computers with vector operations and fast and error-free convolutors.     

一类非单调线性互补问题的高阶Dikin型仿射尺度算法

对于一类非单调线性互补问题提出了一个新算法：高阶Dikin型仿射尺度算法，算法的每步迭代．基于线性规划Dikin原始-对偶算法思想来求解一个线性方程组得到迭代方向，再适当选取步长，得到了算法的多项式复杂性。     

Inherited Fuzzy Interpolation Based on the Inherited Lower-upper Factorization

In this paper, a new scheme is proposed to find the fuzzy interpolation polynomial. In this case, the nodes are crisp data and the values are fuzzy numbers. In order to obtain the interpolation polynomial, a linear system is solved with crisp coefficients matrix and fuzzy right hand side. Then, the inherited lower-upper (LU) triangular factorization and inherited interpolation are applied to solve this system. The examples illustrate the applicability, simplicity and efficiency of the proposed method.     

Generalized monotone line search SQP algorithm for constrained minimax problems

In this article, non-linear minimax problems with general constraints are discussed. By means of solving one quadratic programming an improved direction is yielded and a second-order correction direction can also be at hand via one system of linear equations. So a new algorithm for solving the discussed problems is presented. In connection with a special merit function, the generalized monotone line search is used to yield the step size at each iteration. Under mild conditions, we can ensure global and superlinear convergence. Finally, some numerical experiments are operated to test our algorithm, and the results demonstrate that it is promising.