Simplex-type Algorithm Research Articles

Dynamic trees in exterior-point Simplex-type algorithms for network flow problems

Abstract Recently, a new Dual Network Exterior-Point Simplex Algorithm (DNEPSA) for the Minimum Cost Network Flow Problem (MCNFP) has been developed. In extensive computational studies, DNEPSA performed better than the classical Dual Network Simplex Algorithm (DNSA). In this paper, we present for the first time how to utilize the dynamic trees data structure in the DNEPSA algorithm, in order to achieve an improvement of the amortized complexity per pivot. Our work constitutes a first step towards the development of an efficient implementation of DNEPSA.

A Computational Comparison of Basis Updating Schemes for the Simplex Algorithm on a CPU-GPU System

The computation of the basis inverse is the most time-consuming step in simplex type algorithms. This inverse does not have to be computed from scratch at any iteration, but updating schemes can be applied to accelerate this calculation. In this paper, we perform a computational comparison in which the basis inverse is computed with five different updating schemes. Then, we propose a parallel implementation of two updating schemes on a CPU-GPU System using MATLAB and CUDA environment. Finally, a computational study on randomly generated full dense linear programs is preented to establish the practical value of GPU-based implementation.

On a dual network exterior point simplex type algorithm and its computational behavior

The minimum cost network flow problem, (MCNFP) constitutes a wide category of network flow problems. Recently a new dual network exterior point simplex algorithm (DNEPSA) for the MCNFP has been developed. This algorithm belongs to a special “exterior point simplex type” category. Similar to the classical dual network simplex algorithm (DNSA), this algorithm starts with a dual feasible tree-solution and after a number of iterations, it produces a solution that is both primal and dual feasible, i.e. it is optimal. However, contrary to the DNSA, the new algorithm does not always maintain a dual feasible solution. Instead, it produces tree-solutions that can be infeasible for the dual problem and at the same time infeasible for the primal problem. In this paper, we present for the first time, the mathematical proof of correctness of DNEPSA, a detailed comparative computational study of DNEPSA and DNSA on sparse and dense random problem instances, a statistical analysis of the experimental results, and finally some new results on the empirical complexity of DNEPSA. The analysis proves the superiority of DNEPSA compared to DNSA in terms of cpu time and iterations.

Polynomial Approximations for Continuous Linear Programs

Continuous linear programs have attracted considerable interest due to their potential for modeling manufacturing, scheduling, and routing problems. While efficient simplex-type algorithms have been developed for separated continuous linear programs, crude time discretization remains the method of choice for solving general (nonseparated) problem instances. In this paper we propose a more generic approximation scheme for nonseparated continuous linear programs, where we approximate the functional decision variables (policies) by polynomial and piecewise polynomial decision rules. This restriction results in an upper bound on the original problem, which can be computed efficiently by solving a tractable semidefinite program. To estimate the approximation error, we also compute a lower bound by solving a dual continuous linear program in (piecewise) polynomial decision rules. We establish the convergence of the primal and dual approximations under Slater-type constraint qualifications. We also highlight the...

A labeling algorithm for the sensitivity ranges of the assignment problem

In view of the simplex-type algorithm, the assignment problem is inherently highly degenerate. It may be the optimal basis has changed, but the optimal assignment is unchanged when parameter variation occurs. Degeneracy then makes sensitivity analysis difficult, as well as makes the classical Type I range, which identifies the range the optimal basis unchanged, impractical. In this paper, a labeling algorithm is proposed to identify two other sensitivity ranges – Type II range and Type III range. The algorithm uses the reduced cost matrix, provided in the final results of most solution algorithms for AP, to determine the Type II range which reflects the stability of the current optimal assignment. Thus, the algorithm generates a streamlined situation from searching the optimal solution until performing the sensitivity analysis of the assignment problem. The Type III range, reflecting the flexibility of optimal decision making, can be obtained immediately after the Type II range is determined. Numerical examples are presented to demonstrate the algorithm.

A Shadow Simplex Method for Infinite Linear Programs

We present a simplex-type algorithm—that is, an algorithm that moves from one extreme point of the infinite-dimensional feasible region to another, not necessarily adjacent, extreme point—for solving a class of linear programs with countably infinite variables and constraints. Each iteration of this method can be implemented in finite time, whereas the solution values converge to the optimal value as the number of iterations increases. This simplex-type algorithm moves to an adjacent extreme point and hence reduces to a true infinite-dimensional simplex method for the important special cases of nonstationary infinite-horizon deterministic and stochastic dynamic programs.

A deficient-basis dual counterpart of Paparrizos, Samaras and Stephanides’ primal-dual simplex-type algorithm†

In a recent primal-dual simplex-type algorithm (K. Paparrizos, N. Samaras and G. Stephanides, A new efficient primal dual simplex algorithm, Computers & Operations Research 30 (2003), pp. 1383–1399), its authors show how to take advantage of the knowledge of a primal feasible point and they work with a square basis during the whole process. In this paper we address what could be thought of as its deficient-basis dual counterpart by showing how to take advantage of the knowledge of a dual feasible point in a deficient-basis simplex-type environment. Three small examples are given to illustrate how the specific pivoting rules designed for the proposed algorithm deal with non-unique dual solutions, unbounded dual objectives and a classical exponential example by Goldfarb, thus avoiding some caveats of the dual simplex method. Practical experiments with a new collection of difficult problems for the dual simplex method are reported to justify iteration decrease, and we sketch some details of a sparse projected-gradient implementation in terms of Davis and Hager's CHOLMOD sparse Cholesky factorization (which is row updatable/downdatable) to solve the underlying least-squares subproblems, namely linear least-squares problems and projections onto linear manifolds. †Review (November 2008) of Technical Report MA-07/01 (http://www.matap.uma.es/investigacion/tr.html), Department Applied Mathematics, University of Málaga, September 2007. Talk presented at Joint 2nd Conference on Optimization Methods & Software and 6th EUROPT Workshop on Advances in Continuous Optimization, Prague, Czech Republic, July 2007.

A primal-dual exterior point algorithm for linear programming problems

The aim of this paper is to present a new simplex type algorithm for the Linear Programming Problem. The Primal - Dual method is a Simplex - type pivoting algorithm that generates two paths in order to converge to the optimal solution. The first path is primal feasible while the second one is dual feasible for the original problem. Specifically, we use a three-phase-implementation. The first two phases construct the required primal and dual feasible solutions, using the Primal Simplex algorithm. Finally, in the third phase the Primal - Dual algorithm is applied. Moreover, a computational study has been carried out, using randomly generated sparse optimal linear problems, to compare its computational efficiency with the Primal Simplex algorithm and also with MATLAB's Interior Point Method implementation. The algorithm appears to be very promising since it clearly shows its superiority to the Primal Simplex algorithm as well as its robustness over the IPM algorithm.

A dual exterior point simplex type algorithm for the minimum cost network flow problem

A new dual simplex type algorithm for the Minimum Cost Network Flow Problem (MCNFP) is presented. The proposed algorithm belongs to a special 'exterior- point simplex type' category. Similarly to the classical network dual simplex algorithm (NDSA), this algorithm starts with a dual feasible tree-solution and reduces the primal infeasibility, iteration by iteration. However, contrary to the NDSA, the new algorithm does not always maintain a dual feasible solution. Instead, the new algorithm might reach a basic point (tree-solution) outside the dual feasible area (exterior point - dual infeasible tree).

Worst case examples of an exterior point algorithm for the assignment problem

An efficient exterior point simplex type algorithm for the assignment problem has been developed by Paparrizos [K. Paparrizos, An infeasible (exterior point) simplex algorithm for assignment problems, Math. Program. 51 (1991) 45–54]. This algorithm belongs to the category of forest algorithms and solves an n × n assignment problem in at most n ( n − 1 ) 2 iterations and in at most O ( n 3 ) time. In this paper worst case examples are presented. Specifically, a systematic procedure to construct worst case assignment problems is presented for the exterior point algorithm. The algorithm applied to these examples executes exactly n ( n − 1 ) 2 iterations. This result verifies that the bound O ( n 3 ) is the best possible for the above-mentioned algorithm.

Visual LinProg: A web‐based educational software for linear programming

AbstractVisual LinProg is an educational tool that solves linear problems (LPs), using animation and visualization techniques. The core of the proposed software includes the well‐known class of simplex type algorithms. This tool is a Web‐based software and hence platform independent. Visual LinProg was initially invented to be used in mathematical programming courses to supplement the teaching. The user‐student can solve his own general LPs, view the solution process step‐by‐step, and import or export his own examples in an easy way to read format. The solution process is covered scholastically through textual information and also the necessary steps from the pseudo code are depicted, using multiple views. In this study, we describe Visual LinProg and how it is used in educational purposes. Finally, we present an evaluation of the proposed educational tool. © 2007 Wiley Periodicals, Inc. Comput Appl Eng Educ 15: 1–14, 2007; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20084

A Simplex Approach for Finding Local Solutions of a Linear Bilevel Program by Equilibrium Points

In this paper, a linear bilevel programming problem (LBP) is considered. Local optimality conditions are derived. They are based on the notion of equilibrium point of an exact penalization for LBP. It is described how an equilibrium point can be obtained with the simplex method. It is shown that the information in the simplex tableaux can be used to get necessary and sufficient local optimality conditions for LBP. Based on these conditions, a simplex type algorithm is proposed, which attains a local solution of LBP by moving in equilibrium points. A numerical example illustrates how the algorithm works. Some computational results are reported.

Information raining and optimal link-layer design for mobile hotspots

In this paper, we propose a link layer design for mobile hotspots. We design a novel system architecture that enables high-speed Internet access in railway systems. The proposed design uses a number of repeaters placed along the track and multiple antennas installed on the roof of a vehicle. Each packet is decomposed into smaller fragments and relayed to the vehicle via adjacent repeaters. We also use erasure coding to add parity fragments to original data. This approach is called information raining since fragments are rained upon the vehicle from adjacent repeaters. We investigate two instances of information raining. In blind information raining, all repeaters awaken when they sense the presence of the vehicle. The fragments are then blindly transmitted via awakened repeaters. A vehicle station installed inside the train is responsible for aggregating a large enough number of fragments. In the throughput-optimized information raining, the vehicle station selects a bipartite matching between repeaters and roof-top antennas and activates only a subset of the repeaters. It also dictates the amount of transmission power of each activated repeater. Both the bipartite matching and power allocations are individually shown to be NP-complete. Matching heuristics based on the Hungarian algorithm and Gale-Shapley algorithm are proposed. A simplex-type algorithm is proposed as the power allocation heuristics.

An efficient simplex type algorithm for sparse and dense linear programs

We present a big- M method for solving general linear programming problems with a recently developed exterior point simplex algorithm (EPSA). We also provide intuitive motivations to EPSA. We describe major steps in implementing EPSA for large-scale linear programming. Our implementation was carried out under the MATLAB environment. This algorithm seems to be more efficient than the classical primal simplex algorithm (PSA), employing Dantzig’s rule. Preliminary computational studies on randomly generated sparse and dense small and medium linear programs support this belief. In particular, EPSA is up to 10 times faster than PSA on medium size linear programs. This translates into corresponding savings in CPU time. Although the computational effort required in each step of EPSA requires more time compared to an iteration step of PSA, the improvement of EPSA comes from the fact that it requires adequately less iterations than PSA. Moreover, as the problem size increases and the problem density decreases, EPSA gets relatively faster.

Fractional programming with absolute-value functions

This paper considers a fractional functionals programming problem of the type: maximize z= ∑ 1 nc j|x j|+α ∑ 1 nd j|x j|+β subject to Ax=b, x is unrestricted. In general, adjacent extreme point (simplex-type) methods [Naval Res. Logist. Quart. II (1964) 135] cannot be used to solve this class of problems. However, this work presents the conditions under which simplex-type algorithms can be used to arrive at an optimal solution for a fractional programming problem with an absolute value objective function.

A new efficient primal dual simplex algorithm

The purpose of this paper is to present a revised primal dual simplex algorithm (RPDSA) for linear programming problems. RPDSA has interesting theoretical properties. The advantages of the new algorithm are the simplicity of implementation, low computational overhead and surprisingly good computational performance. The algorithm can be combined with interior point methods to move from an interior point to a basic optimal solution. The new algorithm always proved to be more efficient than the classical simplex algorithm on our test problems. Numerical experiments on randomly generated sparse linear problems are presented to verify the practical value of RPDSA. The results are very promising. In particular, they reveal that RPDSA is up to 146 times faster in terms of number of iterations and 94 times faster in terms of CPU time than the original simplex algorithm (SA) on randomly generated problems of size 1200×1200 and density 2.5%. Scope and purpose In this paper, we present a new simplex-type algorithm for linear programming problems. This algorithm can be viewed as a modification of dual simplex algorithm and it performs sufficiently well in practice, particularly on linear problems of small or medium size. A revised version of the algorithm and solution of general LPs using a modified big- M method are also presented. A computational study on randomly generated sparse linear problems reveals that the new algorithm is surprisingly faster than the original simplex algorithm. In our computational study, we use three different density cases of LPs: 2.5%, 5% and 10%. Compared to the original simplex algorithm, our algorithm's found to be up to 146 times faster in terms of number of iterations and up to 94 times faster in terms of CPU time. Our algorithm can also be seen as a procedure to move from any interior point to an optimal basic solution.

Sparse Regression Ensembles in Infinite and Finite Hypothesis Spaces

We examine methods for constructing regression ensembles based on a linear program (LP). The ensemble regression function consists of linear combinations of base hypotheses generated by some boosting-type base learning algorithm. Unlike the classification case, for regression the set of possible hypotheses producible by the base learning algorithm may be infinite. We explicitly tackle the issue of how to define and solve ensemble regression when the hypothesis space is infinite. Our approach is based on a semi-infinite linear program that has an infinite number of constraints and a finite number of variables. We show that the regression problem is well posed for infinite hypothesis spaces in both the primal and dual spaces. Most importantly, we prove there exists an optimal solution to the infinite hypothesis space problem consisting of a finite number of hypothesis. We propose two algorithms for solving the infinite and finite hypothesis problems. One uses a column generation simplex-type algorithm and the other adopts an exponential barrier approach. Furthermore, we give sufficient conditions for the base learning algorithm and the hypothesis set to be used for infinite regression ensembles. Computational results show that these methods are extremely promising.

PIVOTING ALGORITHMS FOR SOME CLASSES OF STOCHASTIC GAMES: A SURVEY

In this paper, we survey the recent literature on computing the value vector and the associated optimal strategies of the players for special cases of zero-sum stochastic games, or in computing a Nash equilibrium point and the corresponding stationary strategies of the players for special cases of nonzero-sum stochastic games, using finite-step algorithms based on pivoting. Examples of finite-step pivoting algorithms are the various simplex-type algorithms, such as the primal simplex or dual simplex method for solving the linear programming problem or Lemke's or Lemke-Howson's algorithm for solving the linear complementarity problem. Also included are Lemke-type algorithms for solving various generalisations of the linear complementarity problem. The survey also includes a few new results and observations.

Dynamic study of a wind turbine blade with horizontal axis

The study of the dynamic behavior of a wind turbine with horizontal axis can be undertaken by various methods of analysis. The effects of the change of the aerodynamic flow (in the steady and unsteady cases), the variation of parameters of the cinematic movement (angle of attack, pitch angle and yaw angle) and the definition of subsystems characteristics that makes the wind turbine (blade, nacelle and pylon) allow one to characterize the structural dynamic behavior of the wind turbine. It is therefore necessary to develop these items. Once this is done, the structural dynamic behavior of the system can be improved. The term `improve' means the increase of the life duration by mastering the fatigue effects and the reduction of cost without sacrificing the aerodynamic output. The present study aims to examine the behavior of the blade, which is the main part of the wind turbine in that it that transmits forces to all other parts of the structure. The model is based on the theory of three-dimensional beams, under the assumption of variable sections of the type NACA 4415 airfoil, and takes into account membrane, transversal shear, flexion and free torsion effects. With regards to the aerodynamic loads (the lift, the drag and the pitching moment), a validation has been undertaken by considering experimental data and numerical results obtained by a CFD code (Fluent). The forces are obtained by means of a parametric CAD method interpolation of the aerodynamic poles by Bézier patch under geometrical constraints solved by a Simplex type algorithm. The emphasis is put on dynamic aspects by a complete processing of the dynamic equilibrium equation, applied to the wind turbine blade with horizontal axis.

Elasto-Plastic Contact of Rough Surfaces

An isothermal elasto-plastic asperity contact model is developed and presented in this paper, which deals with micro plastic flows of materials and the influence of the elasto-plastic deformation of materials on the behavior of contacting surfaces. The model is solved with the incremental form of a simplex-type algorithm. The von Mises yield criterion is used to determine the onset of the plastic deformation. The effectiveness and validity of the model are studied through analyzing a Hertzian contact problem. Substrate stresses are calculated and differences are observed. Furthermore, the contact pressure, real area of contact, and average gap of real rough surfaces under the elastic, elastic-perfectly-plastic, and the elasto-plastic contact conditions are numerically investigated and the results are compared. Presented at the 56th Annual Meeting Orlando, Florida May 20–24, 2001