Analytical approach for linear programming using barrier and penalty function methods

Parwadi Moengin, . Analytical approach for linear programming using barrier and penalty function methods. 39-. ISSN 0128-6072

In order to solve the primal linear programming problems (and its dual) some methods have been used such as simplex method geometric approach and interior points methods. None of these methods used Lagrangian function as a tool to solve the problems. Thus in this research we study and analyze how the behaviour and performance of barrier function and penalty function methods for solving the linear programming problems work. All of these functions are in Lagrangian form.With logarithmic barrier function method we introduce three types of function that is primal dual and primal-dual logarithmic functions. There are two main results obtained from the logarithmic function method. First we prove that for every value of the barrier parameter the logarithmic barrier function for the problem has unique minimizer and then if the sequence of the values of barrier parameters tends to zero then the sequence of the minimizers converges to a minimizer of the problem. From these properties we construct an algorithm for solving the problem using the logarithmic barrier function methods. Second we give the equivalences between the interior points set the primal the dual the primal-dual logarithmic barrier function and the system of linear equations associated with these functions. In this research we also investigate the behavior and performance of the penalty function methods for linear programming problems.In this research we also investigate the behavior and performance of the penalty function methods for linear programming problems. It includes polynomial penalty functions (such as primal and dual penalty functions) and exponential penalty function methods.The main result of these methods is that we can solve the problem by taking a sequence of values of the penalty parameters that tends to infinity and then the sequence of the minimizers of the penalty functions associated with the value of penalty parameter will tend to the minimizer for the problem. With this we can formulate an algorithm for solving the problem using penalty function methods. The study also includes the exponential barrier function. The result obtained is similar with the result of the exponential penalty function methods. We also investigate the higher-order derivatives of the linear programming and the system of linear equations associated with the problem. From this method we are able and successful in formulating an algorithm for solving the problem. Finally we give an analysis of sensitivity for the linear programming using interior point approach introduced by Karmarkar using logarithmic barrier function. It encompasses cases of changing the right-hand sides of the constrained changing the cost vector of the objective function upper bounds lower bounds of variables and ranges of the constraints.

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Abstract

In order to solve the primal linear programming problems (and its dual) some methods have been used such as simplex method geometric approach and interior points methods. None of these methods used Lagrangian function as a tool to solve the problems. Thus in this research we study and analyze how the behaviour and performance of barrier function and penalty function methods for solving the linear programming problems work. All of these functions are in Lagrangian form.With logarithmic barrier function method we introduce three types of function that is primal dual and primal-dual logarithmic functions. There are two main results obtained from the logarithmic function method. First we prove that for every value of the barrier parameter the logarithmic barrier function for the problem has unique minimizer and then if the sequence of the values of barrier parameters tends to zero then the sequence of the minimizers converges to a minimizer of the problem. From these properties we construct an algorithm for solving the problem using the logarithmic barrier function methods. Second we give the equivalences between the interior points set the primal the dual the primal-dual logarithmic barrier function and the system of linear equations associated with these functions. In this research we also investigate the behavior and performance of the penalty function methods for linear programming problems.In this research we also investigate the behavior and performance of the penalty function methods for linear programming problems. It includes polynomial penalty functions (such as primal and dual penalty functions) and exponential penalty function methods.The main result of these methods is that we can solve the problem by taking a sequence of values of the penalty parameters that tends to infinity and then the sequence of the minimizers of the penalty functions associated with the value of penalty parameter will tend to the minimizer for the problem. With this we can formulate an algorithm for solving the problem using penalty function methods. The study also includes the exponential barrier function. The result obtained is similar with the result of the exponential penalty function methods. We also investigate the higher-order derivatives of the linear programming and the system of linear equations associated with the problem. From this method we are able and successful in formulating an algorithm for solving the problem. Finally we give an analysis of sensitivity for the linear programming using interior point approach introduced by Karmarkar using logarithmic barrier function. It encompasses cases of changing the right-hand sides of the constrained changing the cost vector of the objective function upper bounds lower bounds of variables and ranges of the constraints.

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Item Type:	Article
AGROVOC Term:	LINEAR PROGRAMMING
AGROVOC Term:	FORMULATIONS
AGROVOC Term:	MATHEMATICAL MODELS
AGROVOC Term:	ANALYTICAL METHODS
AGROVOC Term:	MALAYSIA PROGRAMACION LINEAL
AGROVOC Term:	FORMULACIONES
AGROVOC Term:	MODELOS MATEMATICOS
AGROVOC Term:	TECNICAS ANALITICAS
AGROVOC Term:	MALASIA
Depositing User:	Ms. Suzila Mohamad Kasim
Last Modified:	24 Apr 2025 06:26
URI:	http://webagris.upm.edu.my/id/eprint/21168

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