Solving inverse optimal control problems via value functions to global optimality

Abstract

In this paper, we show how a special class of inverse optimal control problems of elliptic partial differential equations can be solved globally. Using the optimal value function of the underlying parametric optimal control problem, we transfer the overall hierarchical optimization problem into a nonconvex single-level one. Unfortunately, standard regularity conditions like Robinson's CQ are violated at all the feasible points of this surrogate problem. It is, however, shown that locally optimal solutions of the problem solve a Clarke-stationarity-type system. Moreover, we relax the feasible set of the surrogate problem iteratively by approximating the lower level optimal value function from above by piecewise affine functions. This allows us to compute globally optimal solutions of the original inverse optimal control problem. The global convergence of the resulting algorithm is shown theoretically and illustrated by means of a numerical example. KeywordsBilevel optimal control • Global optimization • Inverse optimal control • Optimality conditions • Solution algorithm This work is supported by the DFG Grant Analysis and Solution Methods for Bilevel Optimal Control Problems (Grant Nos. DE 650/10-1 and WA3636/4-1) within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).

A non-convex denoising model for impulse and Gaussian noise mixture removing using bi-level parameter identification

“…The situation in our case is highly challenged as the lower problem (10b) is non-smooth, non-convex and implicitly depends on the variable u γ,v , which is the solution of the fractional nonlinear PDE (9). Our strategy against these difficulties is to reformulate the bi-level optimization problem into a single optimization one via the optimal value formulation [24,48,54,55,59,70]. Indeed, this formulation allows to reformulate equivalently the bi-level optimization problem (10a)-(10b) into a single optimization one subject to partially calmness constraints [59,61], even if the lower problem is non-convex.…”

“…This naturally leads to solve the so-called bi-level optimization problem [4,25]. The bi-level optimization approach has been successfully used to solve many inverse problems [24,48,54,59,70].…”

“…In [24,48,54,59,70], the authors exploited this idea to infer optimality conditions in the context of bi-level optimal control. The main advantage of this approach is that we do not need to assume that the lower level problem is convex.…”

“…Second, is there an iterative algorithm that generates a sequence converging to a stationary point of ( 50)? To answer the first question, we note that necessary optimality conditions of Clarkestationarity-type can be derived via a relaxation approach, see [24,48]. Moreover, previous works assume that the constraint qualification is of the Calmness-type [59].…”

A non-convex non-smooth bi-level parameter learning for impulse and Gaussian noise mixture removing

“…implicitly depends on the variable u γ,v which is the solution of the proposed fractional nonlinear PDE. Our strategy to face these difficulties is to reformulate the bi-level optimization problem into a single optimization one via the optimal value formulation [13,21,24,27]. Indeed, this formulation allows to reformulate equivalently the bi-level optimization problem (1.10a)-(1.10b) into a single optimization one subject to partially calmness constraints [21], even while the lower problem is non-convex.…”