Rigorous Global Optimization for Dynamic Systems Subject to Inequality Path Constraints

Abstract

A new approach is described for the rigorous global optimization of dynamic systems subject to inequality path constraints (IPCs). This method employs the sequential (control parameterization) approach and is based on techniques developed for the verified solution of parametric systems of ordinary differential equations. These techniques provide rigorous interval bounds on the state variables, and thus on the path constraints and objective function in the dynamic optimization problem. These techniques also provide explicit analytic representations (Taylor models) of these bounds in terms of the decision variables in the optimization problem. This facilitates the use of constraint propagation techniques that can greatly reduce the domain to be searched for the global optimum. Since IPCs are often related to safety concerns, we adopt a conservative, innerapproximation approach to constraint satisfaction. Through this approach, the search for the global optimum is restricted to a space in which continuous satisfaction of the IPCs is rigorously guaranteed, and an ǫ-global optimum within this space is determined. Examples are presented that demonstrate the potential and computational performance of this approach.

Branch-and-Lift Algorithm for Deterministic Global Optimization in Nonlinear Optimal Control

“…Various strategies have been developed, which determine a convergent lower bound L M (A) without the need for solving this optimization problem exactly. This includes interval analysis and constraint propagation [54,55]; an extension of the αBB method [56] through the use of second-order state sensitivity and/or adjoint information [21,57]; McCormick's relaxation technique [17,22,58]; and, more recently, polyhedral relaxations from Taylor or McCormick-Taylor models [23]. Depending on the expression of the sets F x (t), the feasibility checks that are part of Steps 3 and 4 may be nontrivial to implement as well.…”

Convex Relaxations for Nonlinear Stochastic Optimal Control Problems

“…For optimal control problems with deterministic objectives and constraints, a number of algorithms have recently been developed that can provide guaranteed global solutions [5], [6], [7], [8]. In brief, these methods are predicated on effective algorithms for enclosing the reachable set of the dynamics on subintervals of the decision space.…”

Affine relaxations for the solutions of constrained parametric ordinary differential equations

“…This work demonstrates that constraint information can be obtained from different settings. For instance, in the context of dynamic optimization problems, such constraints are called path constraints; these constraints can be used to tighten the relaxations of the solutions of the parametric ODEs. Thus, if the relaxations are used to construct an overall relaxation of the dynamic optimization problem, the result is a tighter relaxation.…”