Improved conditional gradient method for the generalized cone order optimization problem on the local sphere

In this paper, a generalized optimization problem on the local sphere is established by the cone order relation on the tangent space, and solved by an improved conditional gradient method (for short, ICGM). The auxiliary subproblems are constructed by the directed distance function on the tangent space, the iteration step size is updated by the Armijo rule, and the convergence of the ICGM is proved without the convexity of the objective function. Under the assumption of convexity, the clusters of the sequence generated by the ICGM are proved to be the spherical weakly Pareto solutions (also known as weakly efficient solutions) of this problem .