Simplex-type Method Research Articles

On the number of pivots of Dantzig's simplex methods for linear and convex quadratic programs

Refining and extending works by Ye and Kitahara-Mizuno, this paper presents new results on the number of pivots of simplex-type methods for solving linear programs of the Leontief kind, certain linear complementarity problems of the P kind, and nonnegatively constrained convex quadratic programs. Our results contribute to the further understanding of the complexity and efficiency of simplex-type methods for solving these problems. Two applications of the quadratic programming results are presented.

A New Approach for Solving Linear Fractional Programming Problems with Duality Concept

Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

Simplex-type algorithm for second-order cone programmes via semi-infinite programming reformulation

To solve the (linear) second-order cone programmes (SOCPs), the primal–dual interior-point method has been studied extensively so far and said to be the most efficient method by many researchers. On the other hand, the simplex-type method for SOCP is much less spotlighted, while it still keeps an important position for linear programmes. In this paper, we apply the dual–simplex primal-exchange (DSPE) method, which was originally developed for solving linear semi-infinite programmes, to the SOCP by reformulating the second-order cone constraint as an infinite number of linear inequality constraints. Then, we show that the sequence generated by the DSPE method converges to the SOCP optimum under certain assumptions. In the numerical experiments, we consider the situation to solve multiple SOCPs with similar structures successively. Then we observe that our simplex-type method can be more efficient than the existing interior-point method when we apply the so-called ‘hot start’ technique.

Enhanced Features for Design of Traveling Wave Tubes Using CHRISTINE-1D

Traveling wave tubes (TWTs) are vacuum devices invented in the early 1940s for amplification of radio frequency power. These devices are critical for radar, communications, and electronic warfare missions in the military, as well as in commercial applications. The physics-based design and simulation code CHRISTINE-1D was used in the past to explore different TWT circuit designs and to automate the process of parameter estimation. However, the current capability of CHRISTINE-1D allows optimization of only helix TWT designs and includes a limited number of optimization goal functions. In addition, the current optimizer in CHRISTINE-1D employs a modified steepest descent method to carry out the optimization process. The objectives of this paper are threefold: (1) to investigate optimization techniques that may be better suited for this problem (for example, simplex type methods such as Nelder-Mead and Dividing Rectangles); (2) to allow optimization of nonhelix TWTs; and (3) to implement new optimization goal functions. Finally, to show the feasibility of our approach, we apply our optimization algorithms to the problem of designing a folded waveguide slow-wave circuit.

Neural response profile design: Reducing epileptogenic activity by modifying neuron responses to synchronized input using novel potassium channels obtained by parameter search optimization

Neurons obtain their dynamical electrical characteristics by a set of ion channels. These properties may not only affect the function of the neuron and the local network it forms part of, but it may also eventually affect behavior. We were interested to study whether epileptogenic activity could be reduced by adding an ion channel. In this work, we used computational search techniques to optimize ion channel properties for the goal of modifying neural response characteristics. Our results show that this type of parameter search is possible and reasonably efficient. Successful searches were generated using the direct method PRAXIS, and by systematic searches in low-dimensional sub-spaces. We also report on unsuccessful searches using a simplex-type method, a gradient-based method, and attempts to reduce goal function evaluation time. Importantly, using this search strategy, our study has shown that it is possible to change a neuron's characteristics selectively with regard to response to degree of synchronicity in synaptic input.

Parametric Approach to Some Linearly Constrained Optimization Problems Using Simplex-Type Methods

This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

PARAMETRIC APPROACH TO SOME LINEARLY CONSTRAINED OPTIMIZATION PROBLEMS USING SIMPLEX-TYPE METHODS

ABSTRACT This paper establishes a framework for solving some optimization problems with linear constraints using simplex-type methods. The problems include those found in linear programs, linear fractional programs, and generalized linear fractional programs. In this study, these problems refer to a standard form of minimizing a single parameter subject to parameterized linear equations. Based on the analysis of parameterized basis-based solutions, a unified simplex-type approach is proposed. The adaptability of the parameterized model and that of the solution procedure are discussed. In particular, the proposed algorithm can prevent cycling when compared with the conventional simplex method used for solving linear programs.

Systematic construction of examples for cycling in the simplex method

We present systematic procedures to construct examples of linear programs that cycle when the simplex method is applied. Cycling examples are constructed for diverse variants of pivot selection strategies: most negative reduced-cost and steepest-edge rule for the entering variable, and smallest ratio rule for the leaving variable (where ties are broken according to the least-index or the largest coefficient criterion, respectively). The results are of theoretical interest since only a limited number of cycling examples have been presented in the literature up to date. Constructed cycling examples may also serve as test problems to evaluate the practical performance of anticycling procedures or new variants of simplex type methods.

Piecewise linear programming via interior points

The main aim of this work is to provide some basis for the development of interior point algorithms to minimize piecewise linear objective functions. Specifically, we study a piecewise linear separable and convex objective function, subject to linear constraints. The available methods in the literature for this class of problem are of the Simplex type, except for specific cases, such as linear fitting in the sense of L 1- norm. A common practice for the resolution of piecewise linear programs consists of transforming them into equivalent linear programs and exploring their structure. This strategy is suitable for simplex-type methods, but inadequate for interior point methods. We show how to extend known interior point methods devised for linear programming to piecewise linear programming without resorting to equivalent linear programs. We also show that the generated interior points for the original piecewise linear program are not interior points for the equivalent linear program. Finally, some computational experiments are presented. Scope and purpose Engineering and Sciences, in general, are commonly faced with problems which have various feasible solutions, but a criterion can be devised that make it possible to compare such solutions, indicating that there exist solutions better than others, i.e., optimal solutions. These problems are called optimization problems and their important classes can be mathematically modeled, where an objective function (representing the criterion of comparing solutions) should be minimized, and restricted to a set of constraints (system of equations or inequalities that defines the feasible solutions). Linear programming (or linear optimization) is one of the most studied class of optimization problems and has efficient algorithms, such as, the simplex method (published by George Dantzig in 1947) and its variants, or the interior point method (published by Narenda Karmarkar in 1984) and its variants. The special class of piecewise linear programming problems, focused in this work, arises in production planning, expansion of telephone network, fitting curves, etc., where the objective function to be minimized is nonlinear, but behaves linearly in subregions of its domain. A common practice for the solution of piecewise linear programs consists in transforming them into equivalent linear programs, by redefining variables, for which there are efficient packages. This transformation, however, increases the problem dimension slowing the convergence. On the other hand, it is possible to extend the methods of linear programming to work directly with the original problem. Extensions of the simplex method can be found in the book of Golstein and Youdine published in 1966 in French Language ( Problèmes particuliers de la programation linéare) and more recently in the papers of Robert Fourer published in Mathematical Programming (1985, 1988, 1992). On the other hand, only a few papers have been published extending interior point methods to solve special piecewise linear programming problems, such as L 1 and L ∞ solutions of overdetermined systems. In this work we provide some basis for the extension of interior point algorithms to a wider class of piecewise linear programming problems.

A semidefinite framework for trust region subproblems with applications to large scale minimization

Primal-dual pairs of semidefinite programs provide a general framework for the theory and algorithms for the trust region subproblem (TRS). This latter problem consists in minimizing a general quadratic function subject to a convex quadratic constraint and, therefore, it is a generalization of the minimum eigenvalue problem. The importance of (TRS) is due to the fact that it provides the step in trust region minimization algorithms. The semidefinite framework is studied as an interesting instance of semidefinite programming as well as a tool for viewing known algorithms and deriving new algorithms for (TRS). In particular, a dual simplex type method is studied that solves (TRS) as a parametric eigenvalue problem. This method uses the Lanczos algorithm for the smallest eigenvalue as a black box. Therefore, the essential cost of the algorithm is the matrix-vector multiplication and, thus, sparsity can be exploited. A primal simplex type method provides steps for the so-called hard case. Extensive numerical tests for large sparse problems are discussed. These tests show that the cost of the algorithm is 1 +ź(n) times the cost of finding a minimum eigenvalue using the Lanczos algorithm, where 0<ź(n)<1 is a fraction which decreases as the dimension increases.

Optimization of control parameters of a hot cold controller by means of Simplex type methods

This paper describes a hot/cold controller for regulating crystallization operations. The system was identified with a common method (the Broida method) and the parameters were obtained by the Ziegler-Nichols method. The paper shows that this empirical method will only allow a qualitative approach to regulation and that, in some instances, the parameters obtained are unreliable and therefore cannot be used to cancel variations between the set point and the actual values. Optimization methods were used to determine the regulation parameters and solve this identcation problem. It was found that the weighted centroid method was the best one.

A simplex type method using multiple updatings for specially structured linear programs

Many important large-scale linear programs with special structures lead to special computational procedures which are more efficient than the ordinary procedure of the generalized methods. Problems and their solvers taking advantage of multiple updatings of the basis in the dual simplex type method are presented. Computational results run by the efficient algorithm and a standard code MINOS for the test problems are compared and analyzed. It is shown that the amount of work for the optimal solution for the problem can be reduced by the new algorithm.

A polynomial-time simplex type method for solving linear systems of inequalities

A finite polynomial-time method is proposed for solving linear systems of inequalities. This method is a combination of Karmarkar's method [1] and the simplex method. Numerical experiments show that in some instances it is more efficient than the implementation of the simplex method in the MPR2 package.

The Maximization of a Quadratic Function of Variables Subject to Linear Inequalities

A simplex-type method for finding a local maximum of [Formula: see text] subject to [Formula: see text] and [Formula: see text] is proposed. At a local maximum, the objective function (1), can be expressed, in terms of the non-basic variables λ0, as [Formula: see text] and the vector of partial derivatives of (13), with respect to the non-basic variables may be written, [Formula: see text] This allows calculation of the maximum values of the non-basic variables, increased one at a time, consistent with ∇Z ≧ 0. A “cutting plane” a**λ′ ≧ 1 is then defined which excludes the local optimum, and many lower values (but no higher values) of (1). The form of the square matrix C is immaterial.