This paper deals with the multi-objective optimization problems of snap-through instability of geometrically imperfect simply supported helicoidal composite beam with nonlinear orientation schemes. The objective function is to maximize the load required to initiate snap-through and minimize the instability region. It is assumed that the curved bioinspired beam is embedded on nonlinear elastic foundation and is subjected to different types of lateral loads. The presented formulation is based on Euler-Bernoulli theory, nonlinear higher order strain, and imperfect curvature. Bernstein polynomials (BPs) method is employed to solve the nonlinear nonhomogeneous fourth order integro-differential governing equation. Developed model can considered the effect of imperfection, lamination scheme, and elastic foundation on the nonlinear deflection and snap-through behaviors of helicoidal composite structures. Analytical closed form formula of the nonlinear loaddeflection relationship is developed. The analytical relation is validated by comparing it with exact solution and numerical results available in literature. Multi-objective particle swarm optimization (MOPSO) algorithm is employed to determine Pareto-optimal set for limiting load and region of instability. In these analyses, optimal lamination parameter, amplitude of imperfection and elastic foundation constants are found for helicoidal composite beam, and the results are presented as 2D Pareto-optimal design points.