The efficient solution of large-scale linear programming problems—some algorithmic techniques and computational results

In the development of industrial digital twins, the optimization problem of technological and business processes often arises. In many cases, this problem can be reduced to a largescale linear programming (LP) problem. The article is devoted to the new method for solving large-scale LP problems. This method is called the “apex-method”. The apex-method uses the predictor-corrector framework. The predictor step calculates a point belonging to the feasible region of LP problem. The corrector step calculates a sequence of points converging to the exact solution of the LP problem. The article gives a formal description of the apex-method and provides information about its parallel implementation in C++ language by using the MPI library. The results of large-scale computational experiments on a cluster computing system to study the scalability of the apex method are presented.

This paper proposes a compromise model, based on the technique for order preference through similarity ideal solution (TOPSIS) methodology, to solve the multi-objective large-scale linear programming (MOLSLP) problems with block angular structure involving fuzzy parameters. The problem involves fuzzy parameters in the objective functions and constraints. This compromise programming method is based on the assumption that the optimal alternative is closer to fuzzy positive ideal solution (FPIS) and at the same time, farther from fuzzy negative ideal solution (FNIS).An aggregating function that is developed from LP- metric is based on the particular measure of ‘‘closeness” to the ‘‘ideal” solution.An efficient distance measurement is utilized to calculate positive and negative ideal solutions. The solution process is as follows: first, the decomposition algorithm is used to divide the large-dimensional objective space into a two-dimensional space. A multi-objective identical crisp linear programming is derived from the fuzzy linear model for solving the problem. Then, a single-objective large-scale linear programming problem is solved to find the optimal solution. Finally, to illustrate the proposed method, an illustrative example is provided.

In this paper, we focus on large-scale multiobjective fuzzy linear programming problems with the block angular structure and examine the efficiency of the Dantzig-Wolfe decomposition method in the interactive fuzzy satisficing method proposed by M. Sakawa et al. After overviewing the Dantzig-Wolfe decomposition method (1961) and the interactive fuzzy satisficing method, three-objective linear programming problems with 15 coupling constraints are considered in order to show the efficiency of the Dantzig-Wolfe decomposition method over the revised simplex method. Through a lot of computational experiments on workstation for numerical examples with both 50 and 200 variables, the advantages of the Dantzig-Wolfe decomposition method are discussed with respect to processing time and required memory storage.

The theory of the support vector machine (SVM) algorithm is based on the statistical learning theory. Training of SVMs leads to either a quadratic programming (QP) problem, or linear programming (LP) problem. This depends on the specific norm that is used when the distance between the convex hulls of two classes are computed. The l/sub 1/ norm distance leads to a large scale linear programming problem in the case where the sample size is very large. We propose to apply the Benders decomposition technique to the resulting LP for the regression case. Preliminary results show that this technique is much faster than the QP formulation.

In this paper we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM). The resulting algorithm (IP-PMM) is interpreted as a primal-dual regularized IPM, suitable for solving linearly constrained convex quadratic programming problems. We apply few iterations of the interior point method to each sub-problem of the proximal method of multipliers. Once a satisfactory solution of the PMM sub-problem is found, we update the PMM parameters, form a new IPM neighbourhood and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under standard assumptions. To our knowledge, this is the first polynomial complexity result for a primal-dual regularized IPM. The algorithm is guided by the use of a single penalty parameter; that of the logarithmic barrier. In other words, we show that IP-PMM inherits the polynomial complexity of IPMs, as well as the strict convexity of the PMM sub-problems. The updates of the penalty parameter are controlled by IPM, and hence are well-tuned, and do not depend on the problem solved. Furthermore, we study the behavior of the method when it is applied to an infeasible problem, and identify a necessary condition for infeasibility. The latter is used to construct an infeasibility detection mechanism. Subsequently, we provide a robust implementation of the presented algorithm and test it over a set of small to large scale linear and convex quadratic programming problems. The numerical results demonstrate the benefits of using regularization in IPMs as well as the reliability of the method.

Degeneracy can cause enormous problems when solving large scale linear programming problems. This is not only because there is a possibility that the problem can cycle, but also because a large number of iterations can be executed that do not improve the objective. In this article a procedure which utilises derived reduced costs is discussed. The derived reduced cost of a non- basic variable is defined in such a way that it makes the introduction to the non-basic variable into the basis unattractive if such a decision fails to improve the objective. The procedure deliberately strives to combat degeneracy using derived reduced costs, but it also utilises the advantageous properties of the classical gradient methods.

Decomposition methods are used for solving large-scale linear and convex programming problems in order to save time by reducing the number of references to the external memory of a computer. Such methods convert the solution of the original problem into the solution of a series of problems of lower dimension (blocks). They are particularly efficient if the structure of each block permits the use of special, fast solution methods, or the structure of multiprocessor computer permits to make computations simultaneously for many blocks. We shall consider the use of the subgradient-type methods in iterative algorithms for solving linear, convex and some mixed programming problems with the aid of decomposition schemes.

Online advertising has been the mainstream monetization approach for internet-based companies, in which bid optimization plays a crucial role in enhancing advertising performance. Currently, the bid optimization problem has narrowed down to two specific forms: Budget-constrained bidding (BCB) and Multi-constraint bidding (MCB). Existing solutions try to solve BCB/MCB via linear programming solvers, learning methods, or feedback control. However, in large-scale complex e-commerce, they still suffer from inefficiency, poor convergence, or slow adaptation to the changing market. This research presents an online receding optimization method as a solution for practical bid optimization problems. We conduct a theoretical analysis of the optimal bidding strategy's structure. Further, an online receding optimization process is designed based on open-loop feedback control, which periodically updates a constructed optimal bid formulation that can be solved by linear programming. Then, considering large-scale linear programming problems, we propose an efficient down sampling scheme. Besides, a neural-network-based auction scale prediction is used to adapt to the changing market. Finally, a series of online A/B experiments on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Taobao Sponsored Search</i> compare our work to industrial methods and state-of-the-art from several aspects. The proposed method has been implemented on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Taobao</i>, a billion-scaled online advertising business, for over a year.

The class of splitting preconditioners for the iterative solution of linear systems arising from Mehrotra's predictor-corrector method for large scale linear programming problems needs to find a ba...

Dantzig–Wolfe and block coordinate-descent decomposition in large-scale integrated refinery-planning

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of these solvers depends critically on the sparsity of the problem data and advanced matrix factorization techniques. For a large scale LP problem with data matrix $A$ that is dense (possibly structured) or whose corresponding normal matrix $AA^T$ has a dense Cholesky factor (even with reordering), these solvers may require excessive computational cost and/or extremely heavy memory usage in each interior-point iteration. Unfortunately, the natural remedy, i.e., the use of iterative methods based IPM solvers, although it can avoid the explicit computation of the coefficient matrix and its factorization, is often not practically viable due to the inherent extreme ill-conditioning of the large scale normal equation arising in each interior-point iteration. While recent progress has been made to alleviate the ill-conditioning issue via sophisticated preconditioning techniques, the difficulty remains a challenging one. To provide a better alternative choice for solving large scale LPs with dense data or requiring expensive factorization of its normal equation, we propose a semismooth Newton based inexact proximal augmented Lagrangian (Snipal) method. Different from classical IPMs, in each iteration of Snipal, iterative methods can efficiently be used to solve simpler yet better conditioned semismooth Newton linear systems. Moreover, Snipal not only enjoys a fast asymptotic superlinear convergence but is also proven to enjoy a finite termination property. Numerical comparisons with Gurobi have demonstrated encouraging potential of Snipal for handling large-scale LP problems where the constraint matrix $A$ has a dense representation or $AA^T$ has a dense factorization even with an appropriate reordering. For a few large LP instances arising from correlation clustering, our algorithm can be up to 20--100 times faster than the barrier method implemented in Gurobi for solving the problems to the accuracy of $10^{-8}$ in the relative KKT residual. However, when tested on some large sparse LP problems available in the public domain, our algorithm is not yet practically competitive against the barrier method in Gurobi, especially when the latter can compute the Schur complement matrix and its sparse Cholesky factorization in each iteration cheaply.

In this study, calculations necessary to solve the large scale linear progra mming problems in two operating systems, Linux and Windows 7 (Win), are compared using two different methods. Relying on the interior-point methods, linear-programming interior point solvers (LIPSOL) software was used for the first method and re lying on an augmented Lagrangian method-based algorithm, the second method used the generalized derivative. The performed calculations for various problems show the produced random in the Linux operating system (OS) and Win OS indicate the efficiency of the perfo rmed calculations in the Linux OS in terms of the accuracy and using of the optimum memory.

Here we discuss the issue of planning a telemarketing campaign which will promote new services and use databases of the current customers. The databases can contain data of hundreds of thousands up to a few million customers. To ensure the best possible efficiency of the marketing action, certain conditions have to be satisfied: among other things, the campaign has to be planned so as to present at most one offer to each prospective customer. The problem discussed here can be treated as a vector maximization problem. The consecutive components of the vector criterion function are the numbers of the services sold of each kind. The campaign may have as its objective the creation of a plan which maximizes the profit or the total number of the services sold. The problem can be also regarded as hierarchical; one can also apply other known scalarization approaches, which will be described here. The problems to be solved are large-scale binary linear programming problems. We present a possible solution of these problems using the R package.

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.

Benchmark is a method of measuring performance, and we can obtain continuous performance improvement of programming algorithm through Benchmark validation. In order to solve large-scale linear programming problems, this paper proposes an integrated Benchmark validation which integrates theoretical Benchmark analysis with advance language-based Benchmark. Through the integrated Benchmark validation, we can continuously improve an optimizing algorithm, and validate whether the new optimizing algorithm achieves the performance objectives. The results of experiments show the proposed integrated Benchmark validation is an effective method for developing large-scale linear programming algorithms.