A Solution For A Kind Of Linear System Of Strict Inequalities

Linear system of inequalities is widely used in many fields such as mathematics, physics,computer application and management science; many problems in practice can use or betransformed to linear system of inequalities such as the problem of determining the viabilityfor a class of nonlinear control systems on a region with nonsmooth boundary in [1] and theproblem of optimal transmitter power control in cellular radio systems in [2]. Linear systemof inequalities is usually solved as a linear programming problem by simplex method andinterior-point method. But when referring to the linear system of strict inequalities, the linearprogramming method is not proper, as its result is often the boundary point instead of thesolution of the problem. This paper considers the numerical solution of linear system of strictinequalities, especially for the problem formed as(?) which rises from the quadraticinverse eigenvalue problem. We transform the special kind of linear system of strictinequalities into three kinds of minimax problems, and then smooth the object functions withthe aggregate function method. Then we solve the three minimax problems with steepestdescent method with Armijo inexact line search rule, the aggregate function method withadaptive parameter adjustment and the method of reduced gradient. At last the numericalexperiment is done under the environment of Matlab. The result shows that the transformationfrom linear system of strict inequalities to minimax problem avoids the shortage of getting aboundary point solution and is satisfying.