Standard Linear Programming Problem Research Articles

Fuzzy optimal reactive power control

Abstract Optimal reactive power control has been an active research area during the last decade. Where precise data are available, a specific objective with clear-cut constraints was the primary requirement. The sensitivity of a feasible solution in the constraint domain has been reported many times. Fuzziness and imprecision often arise due to poorly defined data, objectives, constraint boundaries and imperfect modeling. In the past few years, fuzzy modeling of such an optimization problem in mathematical programming has been developed and at present is still one of the active research areas in fuzzy set theory applications. In this paper, a fuzzy single-objective function with multiple fuzzy constraints is modeled by fuzzy set theory. Trapezoidal membership functions are defined for the objective function and constraints. Then the fuzzy mathematical programming of the reactive power control is reformulated as a standard linear programming problem and the λ-formulation method is applied for its solution. Ward-Hale 6-bus and IEEE 14-bus test systems are used as illustrative numerical examples and the results of the presented methodology are compared with conventional methods. The results of the computation are optimistic both in terms of minimizing the objective function and of the smaller number of iterations with respect to the previous methods.

The simplex method as a global optimizer: A c-programming perspective

In this paper we give a brief account of the important role that the conventional simplex method of linear programming can play in global optimization, focusing on its collaboration with composite concave programming techniques. In particular, we demonstrate how rich and powerful the c-programming format is in cases where its parametric problem is a standard linear programming problem.

The maximal tolerance analysis of the constraint matrix in linear programming

Abstract This study intends to generalize the concept of maximal volume in the tolerance region for characterizing the perturbations of one column or one row of the constraint matrix in a standard linear programming problem. The Maximal Volume Region, recently proposed by Wang and Huang, is a symmetrically rectangular parallelepiped with the largest volume in a tolerance region over which different parameters with different levels of sensitivity can be conceived and handled simultaneously, independently and with the greatest flexibility. However, the defined Maximal Volume Region does not exist when perturbation occurs in a constraint matrix. Therefore, in this study, we investigate this case from its basic properties and structure with numerical illustrations.

A fuzzy linear programming approach to the reactive power/voltage control problem

The integration of traditional and heuristic techniques is considered for the reactive power/voltage control program. The steady-state reactive power problem is addressed. Fuzzy sets are used to modify the linear programming (LP) approach to voltage control and to incorporate some heuristic concepts of the expert system approach. Multiple objectives and soft constraints are modeled using fuzzy sets. Piecewise linear convex membership functions for the fuzzy sets are defined. Under this definition, the fuzzy optimization problem is reformulated as a standard linear programming problem. The objective function represents the compromise among the original competing objectives and the soft constraints. In addition, discrete constraints are considered. Numerical examples demonstrate the approach. >

Interval Identification and Robust Control Design: A New Perspective

This paper aims initially to present an identification method for constructing an interval model which can be aggregated to robust control design. Mathematical programming is used as a tool to achieve this goal. We, then, formulate necessary conditions for robust stability of continuous and discrete interval plants and seek low order controller candidates for both cases. Decision making strategies based on Kharitonov and Mansour-Kraus results are proposed to effectively construct and solve standard linear programming problems instead of interval ones. The so obtained candidates are cheked for guaranteed robust stabilizers using (generalized) edge or box theorems

The Tolerance Approach to Sensitivity Analysis of Matrix Coefficients in Linear Programming

The tolerance approach to sensitivity analysis allows for simultaneous and independent variations of the elements of a column or a row of the coefficient matrix in a standard linear programming problem. In particular, the approach yields a maximum tolerance percentage within which the elements of a column may all vary simultaneously and independently from their estimated values while still retaining the same set of basic variables in an optimal solution. A similar result is also derived for the perturbations of the elements of a row.

A mofified gub algorithm for solving linear minimax problems

This article is concerned with the minimization of the maximal value of a set of linear functions subject to linear constraints. It is well known that this problem can be transformed into a standard linear programming problem by introducing an additional variable. In case index sets of nonzero coefficients of the variables contained in each function are mutually exclusive, the constraints of the associated LP problem exhibit the almost-GUB structure. We devised a technique which reduces the number of arithmetic operations by exploiting this special structure. Computational results are also presented, which indicates that our method is more efficient than the ordinary revised simplex method.

An implementation of Karmarkar's algorithm for linear programming

This paper describes the implementation of power series dual affine scaling variants of Karmarkar's algorithm for linear programming. Based on a continuous version of Karmarkar's algorithm, two variants resulting from first and second order approximations of the continuous trajectory are implemented and tested. Linear programs are expressed in an inequality form, which allows for the inexact computation of the algorithm's direction of improvement, resulting in a significant computational advantage. Implementation issues particular to this family of algorithms, such as treatment of dense columns, are discussed. The code is tested on several standard linear programming problems and compares favorably with the simplex codeMinos 4.0.

Linear Programming Approach to Solve Geometric Programming Problem

Abstract In this paper the new linear programming algorithm is implemented for higher degree of difficulty geometric programming problems. Geometric programming problem is transformed to a linear programming problem by linearizing the geometric programming problem, using the condensation technique or by posybinomial method. After transforming the problem to a standard linear programming problem the ellipsoid algorithm is used to solve the problem.

Interactive optimization of nonprismatic girders

An efficient and robust optimization algorithm is presented for minimum weight design of continuous multi-span nonprismatic steel plate girders using the General Geometric Programming technique. The nonlinear programming problem is formulated on the basis of the last editon of the American Institute of Steel Construction (AISC) Specification. The design variables are the flange width and thickness, the web depths and thickness, and the dimensions and spacings of the transverse stiffeners for stiffened plate girders. Each span is divided into one uniform and one or two nonuniform finite elements. The stiffness matrix of the nonuniform element is obtained through the Rayleigh-Ritz approach. The element stiffness matrices for each span are subsequently transformed into a superelement stiffness matrix through the condensation of internal degrees of freedom where a change of cross-section occurs. In the optimization algorithm, the nonlinear primal problem is transformed to an equivalent standard linear programming problem via double condensation. The algorithm is general and can be applied to stiffened or unstiffened, homogeneous or hybrid plate girders. The girder may be fully restrained against lateral torsional buckling or may have lateral supports only at selected locations along the length of the girder. The algorithm is implemented in FORTRAN 77 in an interactive computing environment with graphic capabilities.

Exploiting special structure in Karmarkar's linear programming algorithm

We propose methods to take advantage of specially-structured constraints in a variant of Karmarkar's projective algorithm for standard form linear programming problems. We can use these constraints to generate improved bounds on the optimal value of the problem and also to compute the necessary projections more efficiently, while maintaining the theoretical bound on the algorithm's performance. It is shown how various upper-bounding constraints can be handled implicitly in this way. Unfortunately, the situation for network constraints appears less favorable.

Stabilizability conditions using linear programming

A necessary condition for feedback stabilizability of a linear system with a controller of fixed order is proposed. The condition is useful for designing low-order regulators and can be easily calculated in the case of single-input or single-output systems by solving a standard linear-programming problem.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Compact simplex methods

The following documented algorithm solves the standard linear programming problem of optimizing a linear form subject to linear inequality or equality constraints and nonnegativity conditions. The solution procedure incorporates the standard simplex method with no embellishments. There is only one loop corresponding to the basic simplex iteration.

Voltampere reactive compensation using chance-constrained programming

A new technique is developed for solving the voltampere reactive (VAR) compensation problem under uncertain operating conditions. The technique employs chance-constrained programming (CCP), and transforms the problem into a standard linear programming problem. In providing optimal allocation of VAR support, busbars with unacceptably high probability of violating voltage limits are identified and assigned appropriate chance-constraints. Two cases are considered using the new technique. In the first case, capacitive compensation is evaluated for peak load conditions. Inductive compensation is considered in the second case, assuming light load conditions. The method in its general form can be applied in cases where there is uncertainty with respect to equipment cost and/or the coefficients of the voltage-magnitude/reactive power relationships. Normal distributions are not necessary conditions and, if desired, dependent variates can be assumed. The method has been applied to the AEP 30-busbar test system for both heavy and light load conditions, and sample results are presented in the paper.

New criteria for the simplex algorithm

The standard linear programming problem with a finite optimum value is considered. We derive new criteria which guarantee that (i) a non-basic variable of a basic feasible solution will remain a non-basic variable of an optimal basic solution; (ii) a basic variable of a basic feasible solution will remain a basic variable of an optimal basic solution.

A class of methods for linear programming

A class of methods is presented for solving standard linear programming problems. Like the simplex method, these methods move from one feasible solution to another at each iteration, improving the objective function as they go. Each such feasible solution is also associated with a basis. However, this feasible solution need not be an extreme point and the basic solution corresponding to the associated basis need not be feasible. Nevertheless, an optimal solution, if one exists, is found in a finite number of iterations (under nondegeneracy). An important example of a method in the class is the reduced gradient method with a slight modification regarding selection of the entering variable.

On a class of optimization problems for framed structures

A unified treatment is presented for four types of problems on limit analysis of framed structures and related design problems. With the use of the lower bound theorem in plasticity these problems are formulated as standard linear programming problems. Two significant improvements are made. The new formulation of the design problems reduces the number of equations in the resulting linear program. An improved simplex algorithm for large and sparse linear programs is employed. The formulational and algorithmic improvements provide large capacity and high efficiency that are indispensable for computational analysis and design of large and complex structures.

Computing bounds for the optimal value in linear programming

AbstractConsider a standard linear programming problem and suppose that there are bounds available for the decision variables such that those bounds are not violated at an optimal solution of the problem (but they may be violated at some other feasible solutions of the problem). Thus, these bounds may not appear explicitly in the problem, but rather they may have been derived from some prior knowledge about an optimal solution or from the explicit constraints of the problem.In this paper, the bounds on variables are used to compute bounds on the optimal value when the problem is being solved by the simplex method. The latter bounds may then be used as a termination criteria for the simples iterations for the purpose of finding a “sufficiently good” near optimal solution. The bounds proposed are such that the computational effort in evaluating them is insignificant compared to that involved in the simplex iterations. A numerical example is given to demonstrate their performance.

A Linear Programming Analogue, a Duality Theorem, and a Dynamic Algorithm

This paper considers fluid analogues for the standard linear programming problem and for a separable nonlinear programming problem. In the former case the usual duality results are demonstrated using the principle of minimum potential energy. In addition by examining the dynamics of the system a new method, referred to as the R-method, is derived for solving linear programmes, although it is not demonstrated that this has any computational advantages over the standard simplex method, except that degeneracy causes no problems. In the nonlinear case the weak Lagrangian principle is derived. The purpose of the paper is to demonstrate that analogue methods, while being impracticable as a physical method of solving optimisation problems, may give some insight into computational algorithms via dual energy concepts and/or dynamic behaviour.

A compact Fortran sequence for limit analysis

AbstractThis note presents a FORTRAN sequence for solving a particular case of the ‘bounded variables’ problem.1 It is this problem, rather than the standard linear programming problem, which appears when systematic matrix methods are applied to the rigid‐plastic collapse analysis of a plane frame or truss The method uses considerably less storage than a standard simplex approach, and produces the collapse mode as a by‐product of the calculation. The sequence is part of a complete program for the plastic collapse analysis of frameworks, details of which may be obtained from the author.