Properties of indefinite matrix constraint for linear programming in optimal solution - Sam, Mei Lee (2018)
Finding the optimum solution in engineering and science is a common problem where one wishes to get the objective under certain constraints. This situation is also a typical issue in manufacturing industries where maximum profit and minimum cost are a common objective under certain constraints on the available resources. One approach to solve optimization is to use formulation problem in linear form and subjects to linear constraints, the problem can be deliberated as linear programming problem. The linear constraints can be in a form of a matrix. There are limited researches that discuss the effect of the properties of matrix constraint to the solution. In fact, the matrix constraint has significant influence to the existent of the optimal solution to the optimization problem. This research focused on the investigation of characteristics of non-symmetric indefinite square matrices of linear programming problems which represent the constraints of linear programming problems. The non-symmetric indefinite square matrices are generated randomly by the MATLAB simulation software and its indefinite properties are verified through the principal minor test, quadratic form test and eigenvalues test. The solutions of the primal and dual linear programming problem are simulated and discussed. Optimization software, LINGO, is used to validate the solutions to assure the reliability of the simulated solutions in the MATLAB software. Based on the simulation results, some of the non-symmetric indefinite random matrices found duality gap and those matrices could not provide optimal solution to the problem. Whereas, some indefinite matrices with certain characteristics could achieve optimal solution and no duality gap presented. An indefinite random matrix with all positive off-diagonal entries and the determinant of leading principal minors with positive sign at odd orders and negative sign at even orders surely deliver the optimal solution to the linear programming problems. This research may contribute to the advancement of linear programming solution particularly when the constraints form an indefinite matrix.