Computational experience with a globally convergent primal—dual predictor—corrector algorithm for linear programming

Kojima, Megiddo, and Mizuno proved global convergence of a primal—dual algorithm that corresponds to methods used in practice. Here, the numerical efficiency of a predictor—corrector extension of that algorithm is tested. Numerical results are extremely positive, indicating that the safety of a globally convergent algorithm can be obtained at little computational cost. The algorithm is tested on infeasible problems with less success. Finally, the algorithm is applied to a warm started problem, with very encouraging preliminary results.Corresponding author. The research of this author is sponsored by the Air Force Office of Scientific Research, Air Force System Command under Grant AFOSR-92-J0046. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notations thereon.     

非线性整数规划的一个近似算法

利用连续总体优化填充函数法的思想，本文设计了非线性整数规划的一个近似算法．首先，给出了非线性整数规划问题离散局部极小解的定义，设计了找离散局部极小解的局部搜索算法；其次，用所设计的局部搜索算法极小化填充函数来找比当前离散局部极小解好的解．本文的近似算法是直接法，且与连续总体优化的填充函数法相比，本文填充函数中的参数易于选取．数值试验表明，本文的近似算法是有效的．     

Alternative group relaxation of integer programming problems

The classical group approach to integer linear programming problems (IP) can be generalized in order to obtain group minimization problems with different computational load and different relaxation.The aim of this work is to analyze some group problems, associated to the same (IP), both from the point of view of the relaxation of the (IP) and of the complexity of the group solution algorithm; evaluation criteria for these group problems are pointed out.     

Computational experience with Davidon's least-square algorithm

Computational results are presented for Davidon's new least-square algorithm. Computational experience with this algorithm is reported which motivated the development of a production code version of the algorithm. Several heuristic modifications, which have been added, are described. Fifteen zero-residual test problems have been used in comparing the new production code version with two established versions of the Levenberg-Marquardt algorithm. The production code version of Davidon's least-square algorithm performed faster and used less function evaluations than the Levenberg-Marquardt algorithm in almost every case of the test problems.It is a pleasure to acknowledge and thank M. Thomas, R. & I. Consultant, Western Illinois University Computer Center, for writing the timing routine and taking the time to run the comparison tests on the IBM 360/50. Part of this work was also performed at the Applied Mathematics Division of Argonne National Laboratory under the auspices of the US Energy Research and Development Administration.     

An algorithm for a separable integer programming problem with cumulatively bounded variables

An algorithm is presented for obtaining the optimal solution of an integer programming problem in which the nested constraints represent the cumulative bounding of the variables and the objective function is a sum of concave functions of one variables. The algorithm requires O(m log(m)log²(b_m/m)) time, where m is the number of variables and b_m is an upper bound on the sum of the m variables, b_m ⩾m⩾1. It is also demonstrated that a special case of identical concave functions is solvable in O(m). Both results significantly improve the previous bounds for these problems.     

Computational experience with penalty-barrier methods for nonlinear programming

It was recently shown that modified barrier methods are not only theoretically but also computationally superior to classic barrier methods when applied to general nonlinear problems. In this paper, a penalty-barrier function is presented that was designed to overcome particular problems associated with modified log-barrier functions. A quadratic extrapolation of logarithmic terms as well as handling simple bounds separately are utilized. The resulting penalty-barrier method is outlined and compared to previous methods. The conclusion drawn from the computational tests is that the revised method exhibits superior performance on the test set of this study and consequently holds promise as a viable technique for general nonlinear programming.     

A polynomial algorithm for integer programming covering problems satisfying the integer round-up property

LetA be a non-negative matrix with integer entries and no zero column. The integer round-up property holds forA if for every integral vectorw the optimum objective value of the generalized covering problem min{1y: yA w, y 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrixA and right hand side vectorw, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrixA does not have the integer round-up property.     

Computational comparison on the partitioning strategies in multiple choice integer programming

In this paper we review the theoretical background of two partitioning strategies, the Weighted-Mean-Method (WMM) and the Reformulation-and-Transformation-Technique (RTT), incorporated in the special ordered set branch-and-bound procedures leading to the global optimum in Multiple Choice Integer Programming (MCIP). Procedure flow is developed with a numerical example presented to illustrate the effect of these two partitioning strategies in the algorithm. This procedure flow is then coded using IBM APL2. A total of 24 test problems are solved by this APL2 code for MCIP to analyze the performance of WMM and RTT in MCIP. The preliminary computational results indicate that, on the average, RTT produces smaller branching trees than does WMM. Its performance tends to be better for those MCIP problems having more multiple choice sets and a fewer average number of variables in each set. However, the average CPU time consumed by using RTT does not differ much from that consumed by using WMM.     

A linear algorithm for integer programming in the plane

We show that a 2-variable integer program, defined by m constraints involving coefficients with at most bits, can be solved with O(m+) arithmetic operations on rational numbers of size O().     

An effective branch-and-bound algorithm for convex quadratic integer programming

We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids. Experiments show that our approach is very fast on both unconstrained problems and problems with box constraints. The main reason is that all expensive calculations can be done in a preprocessing phase, while a single node in the enumeration tree can be processed in linear time in the problem dimension.     

A partial enumeration algorithm for pure nonlinear integer programming

In this paper, a partial enumeration algorithm is developed for a class of pure IP problems. Then, a computational algorithm, named PE_SPEEDUP (partial enumeration speedup), has been developed to use whatever explicit linear constraints are present to speedup the search for a solution. The method is easy to understand and implement, yet very effective in dealing with many pure IP problems, including knapsack problems, reliability optimization, and spare allocation problems. The algorithm is based on monotonicity properties of the problem functions, and uses function values only; it does not require continuity or differentiability of the problem functions. This allows its use on problems whose functions cannot be expressed in closed algebraic form. The reliability and efficiency of the proposed algorithm and the PE_SPEEDUP algorithm has been demonstrated on some integer optimization problems taken from the literature.     

A trust-region-based derivative free algorithm for mixed integer programming

A trust-region-based derivative free algorithm for solving bound constrained mixed integer nonlinear programs is developed in this paper. The algorithm is proven to converge to a local minimum after a finite number of function evaluations. In addition, an improved definition of local minima of mixed integer programs is proposed. Computational results showing the effectiveness of the derivative free algorithm are presented.     

An improved partial enumeration algorithm for integer programming problems

In this paper, we present an improved Partial Enumeration Algorithm for Integer Programming Problems by developing a special algorithm, named PE_SPEEDUP (partial enumeration speedup), to use whatever explicit linear constraints are present to speedup the search for a solution. The method is easy to understand and implement, yet very effective in dealing with many integer programming problems, including knapsack problems, reliability optimization, and spare allocation problems. The algorithm is based on monotonicity properties of the problem functions, and uses function values only; it does not require continuity or differentiability of the problem functions. This allows its use on problems whose functions cannot be expressed in closed algebraic form. The reliability and efficiency of the proposed PE_SPEEDUP algorithm has been demonstrated on some integer optimization problems taken from the literature.     

Computational experience with the Chow—Yorke algorithm

The Chow—Yorke algorithm is a nonsimplicial homotopy type method for computing Brouwer fixed points that is globally convergent. It is efficient and accurate for fixed point problems. L.T. Watson, T.Y. Li, and C.Y. Wang have adapted the method for zero finding problems, the nonlinear complementarity problem, and nonlinear two-point boundary value problems. Here theoretical justification is given for applying the method to some mathematical programming problems, and computational results are presented.This work was partially supported by NSF Grant MCS 7821337.     

Stochastic programming with integer variables

 Including integer variables into traditional stochastic linear programs has considerable implications for structural analysis and algorithm design. Starting from mean-risk approaches with different risk measures we identify corresponding two- and multi-stage stochastic integer programs that are large-scale block-structured mixed-integer linear programs if the underlying probability distributions are discrete. We highlight the role of mixed-integer value functions for structure and stability of stochastic integer programs. When applied to the block structures in stochastic integer programming, well known algorithmic principles such as branch-and-bound, Lagrangian relaxation, or cutting plane methods open up new directions of research. We review existing results in the field and indicate departure points for their extension. Received: December 2, 2002 / Accepted: April 23, 2003 Published online: May 28, 2003 Mathematics Subject Classification (2000): 90C15, 90C11, 90C06, 90C57     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

An algorithm for the multiple objective integer linear programming problem

A technique is presented for solving the multiple objective integer linear programming problem. The technique can be used to identify some or all efficient solutions. While the technique is applicable with any integer programming algorithm, it is well suited for implementation using integer postoptimality techniques. Such an implementation, based on Balas' Additive algorithm, is described for problems with zero-one variables.     

Computational experience in sensitivity analysis for nonlinear programming

A method for sensitivity analysis in nonlinear programming has recently been developed using the sequential unconstrained minimization technique. It is applied here to perform sensitivity analyses on four example problems to demonstrate the computational feasibility and characteristics of the approach. The sensitivity analysis is conducted along the minimizing trajectory for each problem. The convergence characteristics of the first partial derivatives of the variables and objective function with respect to the parameters in the sensitivity analysis are illustrated.