Local Convergence of Sequential Convex Programming for Nonconvex Optimization

where c ∈ R, g : R → R is non-linear and smooth on its domain, and Ω is a nonempty closed convex subset in R. This paper introduces sequential convex programming (SCP), a local optimization method for solving the nonconvex problem (P). We prove that under acceptable assumptions the SCP method locally converges to a KKT point of (P) and the rate of convergence is linear. Problems in the form of (P) conveniently formulate many problems of interest such as least squares problems, quadratically constrained quadratic programming, nonlinear semidefinite programming (SDP), and nonlinear second order cone programming problems (see, e.g., [1, 2, 5, 6, 10]). In nonlinear optimal control, by using direct transcription methods, the resulting problem is usually formulated as an optimization problem of the form (P) where the equality constraint g(x) = 0 originates from the dynamic system of an optimal control problem. The main difficulty of the problem (P) is concentrated in the nonlinear constraint g(x) = 0 that can be overcome by linearizing it around the current