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A linear programming problem seeks for a non-negative column vector, x, that maximizes a linear objective function, uTx. subject to Ax 5 b, where A is a given matrix, and b and u are given column vectors. Using the same dpta, the dual problem to the primal seeks for a non-negative column vector, y, to min~mize a linear objective function, b y, subject to 2 u. The surrogate methods exploit the Duality Theory to combine the two problems ~nto one'system of lrnear inequalities that treats the sign-restr~cied variables and the objective functions as constraints. Because the set of constraints in linear programming problems is sometimes a of inequality and equality constraints, this paper modifies the surrogate methods and comes up with hybrids of the ones designed for a system of linear ineq~.alitie$ and those for a system of linear equations. The paper also proves that a feas~ble solution to the resulting l~near inequality problem is made u j~ of the primal and dual optimal solutions for the given primal problem and its associated dual. It goes further to prwe the dual theorem as it relates to the surrogate methods.. ,

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