Most of the structural behavior constraints involved in structural robust design and optimization are nonconvex in nature. Therefore if local optimality criteria based optimization algorithms are employed to find the worst case structural responses that are used for examining the feasibility of a given design, it is highly possible that the optimization process will get stuck in a local optimum. If this is the case, the reliability of a ''robust" design cannot be guaranteed, at least theoretically. The aim of the present paper is to develop some new formulations and the corresponding numerical algorithms to find the confidence optimal design under non-probabilistic stiffness and load uncertainties. To this end, two Bi-level program formulations for confidence robust design are proposed. In order to ensure the strict feasibility of the optimal solution, in the lower-level of program, the constraints are imposed on the confidence upper bounds of the structural responses, which can be obtained efficiently by solving some convex linear semi-definite programs (LSDPs). Based on the sensitivity analysis of the lower-level LSDP problem, the upper level programs are then solved by employing the classical gradient-based nonlinear optimization algorithms. Furthermore, for the case of stiffness uncertainty, a single-level nonlinear semi-definite programming (NSDP) formulation is also proposed and its mathematical properties are analyzed. Numerical examples show that confidence robust optimal design can be obtained via the proposed approaches effectively without resorting to too many computational efforts.