9 research outputs found
Economic inexact restoration for derivative-free expensive function minimization and applications
The Inexact Restoration approach has proved to be an adequate tool for
handling the problem of minimizing an expensive function within an arbitrary
feasible set by using different degrees of precision in the objective function.
The Inexact Restoration framework allows one to obtain suitable convergence and
complexity results for an approach that rationally combines low- and
high-precision evaluations. In the present research, it is recognized that many
problems with expensive objective functions are nonsmooth and, sometimes, even
discontinuous. Having this in mind, the Inexact Restoration approach is
extended to the nonsmooth or discontinuous case. Although optimization phases
that rely on smoothness cannot be used in this case, basic convergence and
complexity results are recovered. A derivative-free optimization phase is
defined and the subproblems that arise at this phase are solved using a
regularization approach that take advantage of different notions of
stationarity. The new methodology is applied to the problem of reproducing a
controlled experiment that mimics the failure of a dam
Douglas-Rachford Algorithm for Control- and State-constrained Optimal Control Problems
We consider the application of the Douglas-Rachford (DR) algorithm to solve
linear-quadratic (LQ) control problems with box constraints on the state and
control variables. We split the constraints of the optimal control problem into
two sets: one involving the ODE with boundary conditions, which is affine, and
the other a box. We rewrite the LQ control problems as the minimization of the
sum of two convex functions. We find the proximal mappings of these functions
which we then employ for the projections in the DR iterations. We propose a
numerical algorithm for computing the projection onto the affine set. We
present a conjecture for finding the costates and the state constraint
multipliers of the optimal control problem, which can in turn be used in
verifying the optimality conditions. We carry out numerical experiments with
two constrained optimal control problems to illustrate the working and the
efficiency of the DR algorithm compared to the traditional approach of direct
discretization.Comment: 20 pages, 3 figures, 3 table
Assessing the reliability of general-purpose Inexact Restoration methods
Inexact Restoration methods have been proved to be effective to solve constrained optimization problems in which some structure of the feasible set induces a natural way of recovering feasibility from arbitrary infeasible points. Sometimes natural ways of dealing with minimization over tangent approximations of the feasible set are also employed. A recent paper Banihashemi and Kaya (2013)] suggests that the Inexact Restoration approach can be competitive with well-established nonlinear programming solvers when applied to certain control problems without any problem-oriented procedure for restoring feasibility. This result motivated us to revisit the idea of designing general-purpose Inexact Restoration methods, especially for large-scale problems. In this paper we introduce affordable algorithms of Inexact Restoration type for solving arbitrary nonlinear programming problems and we perform the first experiments that aim to assess their reliability. Initially, we define a purely local Inexact Restoration algorithm with quadratic convergence. Then, we modify the local algorithm in order to increase the chances of success of both the restoration and the optimization phase. This hybrid algorithm is intermediate between the local algorithm and a globally convergent one for which, under suitable assumptions, convergence to KKT points can be proved28
Inexact restoration method for derivative-free optimization with smooth constraints
Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)A new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not. The method is based on the inexact restoration framework, by means of which each iteration is divided in two phases. In the first phase one considers only the constraints, in order to improve feasibility. In the second phase one minimizes a suitable objective function subject to a linear approximation of the constraints. The second phase must be solved using derivative-free methods. An algorithm introduced recently by Kolda, Lewis, and Torczon for linearly constrained derivative-free optimization is employed for this purpose. Under usual assumptions, convergence to stationary points is proved. A computer implementation is described and numerical experiments are presented.A new method is introduced for solving constrained optimization problems in which the derivatives of the constraints are available but the derivatives of the objective function are not. The method is based on the inexact restoration framework, by means of23211891213CNPQ - CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOFAPERJ - FUNDAÇÃO CARLOS CHAGAS FILHO DE AMPARO À PESQUISA DO ESTADO DO RIO DE JANEIROFAPESP - FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULOConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [E-26/171.164/2003-APQ1]FAPESP [FAPESP 2011-51305-0]FAPESP [03/09169-6, 06/53768-0, 07/06663-0, 08/00468-4]E-26/171.164/2003–APQ12011-51305-0; 03/09169-6; 06/53768-0; 07/06663-0; 08/00468-4sem informaçãoWe are indebted to associate editor Prof. Margaret Wright and two anonymous referees for many useful comments and remarks that led to significant improvement of this pape
Inexact Restoration for Euler Discretization of Box-Constrained Optimal Control Problems
The Inexact Restoration method for Euler discretization of state and control constrained optimal control problems is studied. Convergence of the discretized (finite-dimensional optimization) problem to an approximate solution using the Inexact Restoration method and convergence of the approximate solution to a continuous-time solution of the original problem are established. It is proved that a sufficient condition for convergence of the Inexact Restoration method is guaranteed to hold for the constrained optimal control problem. Numerical experiments employing the modelling language AMPL and optimization software Ipopt are carried out to illustrate the robustness of the Inexact Restoration method by means of two computationally challenging optimal control problems, one involving a container crane and the other a free-flying robot. The experiments interestingly demonstrate that one might be better-off using Ipopt as part of the Inexact Restoration method (in its subproblems) rather than using Ipopt directly on its own. © 2012 Springer Science+Business Media, LLC