A semidefinite programming-based optimization method is proposed for lightweight design of boom and stay bars of container cranes. The boom and stay bars are welded together by steel plates, and in engineering practices the cross-sectional dimensions of boom and stay bars are of discrete values. Thus, the lightweight design of boom and stay bars is a discrete sizing optimization problem. The cross-section of boom is irregular and after optimization the boom section may become distorted, which may negatively affect the assembling of boom with other parts. In view of this defect, a design method is proposed by transforming the discrete sizing problem into a section-type selection problem, which means selections of sections from a predefined set of available sections. To relax the discrete problem as a continuous problem, a linear relaxation approach based on the convex hull of discrete points is proposed, with which a linearized stiffness matrix is derived. Furthermore, a new method is proposed by implicitly containing the stress constraint within a narrowed compliance constraint. In this way, the original discrete optimization problem with stress and stiffness constraints is simplified as a compliance-constrained problem, which can be further reformulated as a relaxed semidefinite programming problem. With existing optimization solvers, the global optimum solution for the relaxed semidefinite programming problem can be quickly achieved. Based on the global optimum solution, a discrete feasible solution is derived through section rounding. Finally a numerical example of boom and stay bars of a certain contain crane is presented, and the result validates the effectiveness of the proposed method.