A non-Convex Optimization Approach for Process Parameter Control Using Variational Analysis

Abstract

The problem of controlling process parameters in case of non-convex processes represents a mathematically difficult one since the process operator can ignore global monotonicity while the feasible domain of the process parameters may have a non-convex structure. In this paper, an approach to solving such problems based on variational optimization is developed. The process-control problem is formulated as a non-convex variational inequality and equivalently represented through the inclusion 0 ∈ ????(????∗) + ????????(????∗). To eliminate the ambiguity of the projection into a non-convex set, the feasible set is supposed to be locally r-prox-regular, thus guaranteeing a unique metric projection along the optimization path. The weak monotonicity constant for the function is calculated from the symmetric Jacobian, hence the initial regularization coefficient is defined. The proposed method introduces ????????(????) = ????(????) + ????(???? −????0) and progressively reduces ???? toward zero. Lyapunov descent is used to prove that a certain type of resummability is attained and that the accumulation points converge to the state of variational stationarity, under the described local regularity assumptions. The results are confirmed by numerical tests, conducted for a nonlinear three parameter process model, where a convergence for different starting points is achieved with the variational gap being less than 10−6. The locally minimal stationary states are separated from other stationary points based on the analysis of the reduced Hessian. The unregularized projected gradient stress benchmark shows improved stability.