Smoothing Projected Gradient Method and Its Application to Stochastic Linear Complementarity Problems

A smoothing projected gradient (SPG) method is proposed for the minimization problem on a closed convex set, where the objective function is locally Lipschitz continuous but nonconvex, nondifferentiable. We show that any accumulation point generated by the SPG method is a stationary point associated with the smoothing function used in the method, which is a Clarke stationary point in many applications. We apply the SPG method to the stochastic linear complementarity problem (SLCP) and image restoration problems. We study the stationary point defined by the directional derivative and provide necessary and sufficient conditions for a local minimizer of the expected residual minimization (ERM) formulation of SLCP. Preliminary numerical experiments using the SPG method for solving randomly generated SLCP and image restoration problems of large sizes show that the SPG method is promising.