According to the optimality conditions of constrained nonlinear programming theory, the first-order necessary conditions and the second-order sufficient conditions of the weighted least squares solution are derived in inequality constrained partial errors-in-variables model. These conditions are used to design the algorithms and check the optimality of the solution. The nonlinear target function is expanded to the second order at the approximate value with Taylor series and a quadratic programming sub-problem is formed based on the method of sequential quadratic programming. The model parameters and elements of the coefficient matrix are calculated with active set method at the same time. The data simulation and a linear regression example show that the new algorithm is feasible and effective, which is more efficient than the linea- rization method.