Optimal design problems with probabilistic constraints, often referred to as Reliability-Based Design Optimization (RBDO) problems, have been the subject of extensive recent studies. Solution methods to date have focused more on improving efficiency rather than accuracy and the global convergence behavior of the solution. A new strategy utilizing an adaptive sequential linear programming (SLP) algorithm is proposed as a promising approach to balance accuracy, efficiency, and convergence. The strategy transforms the nonlinear probabilistic constraints into equivalent deterministic ones using both first order and second order approximations, and applies a filter-based SLP algorithm to reach the optimum. Simple numerical examples show promise for increased accuracy without sacrificing efficiency.