Aiming at two typical discrete optimization problems of steel frame，namely, the volume minimization with compliance constraint and the compliance minimization with volume constraint，a linear relaxation approach based on convex combination is proposed. Meanwhile, the linked discreteness of design variables is also relaxed lin? early, and the original nonlinear and nonconvex problems are recast as relaxed convex programming problems. Spe? cifically, the compliance minimization with volume constraint is reestablished as a second-order cone programming, and the volume minimization with compliance constraint is reformulated as a semidefinite programming. The global optimum solutions of two types of convex programming problems can be readily derived using existing mature optimi? zation solvers. These global optimum solutions are also the theoretical lower bound for the discrete optimization prob? lems. An example of a one-bay four-story frame is presented, and the results by the proposed approach are compared with the solutions by complete enumeration and genetic algorithm. The comparison demonstrates that the proposed approach is capable of achieving the theoretical lower bound in an efficient manner.