Pointwise residual method for solving primal and dual ill-posed linear programming problems with approximate data

AbstractWe propose a variation of the pointwise residual method for solving primal and dual ill-posed linear programming with approximate data, sensitive to small perturbations. The method leads to an auxiliary problem, which is also a linear programming problem. Theorems of existence and convergence of approximate solutions are established and optimal estimates of approximation of initial problem solutions are achieved.

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Approximate solutions and error bounds for a class of continuous-time linear programming problems

Optimization ◽

10.1080/02331934.2011.562292 ◽

2012 ◽

Vol 61 (2) ◽

pp. 163-185 ◽

Cited By ~ 7

Author(s):

Ching-Feng Wen ◽

Yung-Yih Lur ◽

Hang-Chin Lai

Keyword(s):

Linear Programming ◽

Error Bounds ◽

Continuous Time ◽

Approximate Solutions ◽

Linear Programming Problems ◽

Time Linear

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A Mathematical Model for Solving the Linear Programming Problems Involving Trapezoidal Fuzzy Numbers via Interval Linear Programming Problems

Journal of Mathematics ◽

10.1155/2021/5564598 ◽

2021 ◽

Vol 2021 ◽

pp. 1-17

Author(s):

Ladji Kané ◽

Daouda Diawara ◽

Lassina Diabaté ◽

Moussa Konaté ◽

Souleymane Kané ◽

...

Keyword(s):

Linear Programming ◽

Fuzzy Numbers ◽

Strong Duality ◽

Optimal Solutions ◽

Interval Numbers ◽

Complementary Slackness ◽

Trapezoidal Fuzzy Numbers ◽

Linear Programming Problems ◽

Primal And Dual Problems ◽

Primal And Dual

We define linear programming problems involving trapezoidal fuzzy numbers (LPTra) as the way of linear programming problems involving interval numbers (LPIn). We will discuss the solution concepts of primal and dual linear programming problems involving trapezoidal fuzzy numbers (LPTra) by converting them into two linear programming problems involving interval numbers (LPIn). By introducing new arithmetic operations between interval numbers and fuzzy numbers, we will check that both primal and dual problems have optimal solutions and the two optimal values are equal. Also, both optimal solutions obey the strong duality theorem and complementary slackness theorem. Furthermore, for illustration, some numerical examples are used to demonstrate the correctness and usefulness of the proposed method. The proposed algorithm is flexible, easy, and reasonable.

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Regularization of ill-posed problems by using stabilizers in the form of the total variation of a function and its derivatives

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2015-0050 ◽

2016 ◽

Vol 24 (2) ◽

Author(s):

Vladimir V. Vasin

Keyword(s):

Linear Operator ◽

Total Variation ◽

Normed Space ◽

Regularization Method ◽

Approximation Scheme ◽

Approximate Solutions ◽

Tikhonov Regularization Method ◽

Variation Of A Function ◽

Ill Posed ◽

Convergence Of Approximate Solutions

AbstractUnder the assumption that the solution of a linear operator equation is presented in the form of a sum of several components with various smoothness properties, a modified Tikhonov regularization method is studied. The stabilizer of this method is the sum of three functionals, where each one corresponds to only one component. Each such functional is either the total variation of a function or the total variation of its derivative. For every component, the convergence of approximate solutions in a corresponding normed space is proved and a general discrete approximation scheme for the regularizing algorithm is justified.

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Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients

Optimizing, Innovating, and Capitalizing on Information Systems for Operations ◽

10.4018/978-1-4666-2925-7.ch003 ◽

2013 ◽

pp. 40-63

Author(s):

Ali Ebrahimnejad ◽

Seyed Hadi Nasseri ◽

Sayyed Mehdi Mansourzadeh

Keyword(s):

Linear Programming ◽

Auxiliary Problem ◽

Simplex Algorithm ◽

Upper Bounds ◽

Bounded Linear ◽

Fuzzy Variables ◽

Decision Variables ◽

Bounded Variables ◽

Linear Programming Problems ◽

Fuzzy Cost

In most practical problems of linear programming problems with fuzzy cost coefficients, some or all variables are restricted to lie within lower and upper bounds. In this paper, the authors propose a new method for solving such problems called the bounded fuzzy primal simplex algorithm. Some researchers used the linear programming problem with fuzzy cost coefficients as an auxiliary problem for solving linear programming with fuzzy variables, but their method is not efficient when the decision variables are bounded variables in the auxiliary problem. In this paper the authors introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus.

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An approach to the construction of approximate solutions of Boolean linear programming problems

Cybernetics ◽

10.1007/bf01069525 ◽

1988 ◽

Vol 24 (1) ◽

pp. 48-52

Author(s):

I. V. Sergienko ◽

V. P. Shilo

Keyword(s):

Linear Programming ◽

Approximate Solutions ◽

Linear Programming Problems

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Bounded Primal Simplex Algorithm for Bounded Linear Programming with Fuzzy Cost Coefficients

International Journal of Operations Research and Information Systems ◽

10.4018/joris.2011010105 ◽

2011 ◽

Vol 2 (1) ◽

pp. 96-120 ◽

Cited By ~ 24

Author(s):

Ali Ebrahimnejad ◽

Seyed Hadi Nasseri ◽

Sayyed Mehdi Mansourzadeh

Keyword(s):

Linear Programming ◽

Auxiliary Problem ◽

Simplex Algorithm ◽

Lower And Upper Bounds ◽

Bounded Linear ◽

Fuzzy Variables ◽

Decision Variables ◽

Bounded Variables ◽

Linear Programming Problems ◽

Fuzzy Cost

In most practical problems of linear programming problems with fuzzy cost coefficients, some or all variables are restricted to lie within lower and upper bounds. In this paper, the authors propose a new method for solving such problems called the bounded fuzzy primal simplex algorithm. Some researchers used the linear programming problem with fuzzy cost coefficients as an auxiliary problem for solving linear programming with fuzzy variables, but their method is not efficient when the decision variables are bounded variables in the auxiliary problem. In this paper the authors introduce an efficient approach to overcome this shortcoming. The bounded fuzzy primal simplex algorithm starts with a primal feasible basis and moves towards attaining primal optimality while maintaining primal feasibility throughout. This algorithm will be useful for sensitivity analysis using primal simplex tableaus.

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