Convergence of the BFGS Method for $LC^1 $ Convex Constrained Optimization

Abstract

This paper proposes a BFGS-SQP method for linearly constrained optimization where the objective function $f$ is required only to have a Lipschitz gradient. The Karush--Kuhn--Tucker system of the problem is equivalent to a system of nonsmooth equations $F(v)=0$. At every step a quasi-Newton matrix is updated if $\|F(v_k)\|$ satisfies a rule. This method converges globally, and the rate of convergence is superlinear when $f$ is twice strongly differentiable at a solution of the optimization problem. No assumptions on the constraints are required. This generalizes the classical convergence theory of the BFGS method, which requires a twice continuous differentiability assumption on the objective function. Applications to stochastic programs with recourse on a CM5 parallel computer are discussed.