Jesús Martínez-Frutos

Kriging-based infill sampling criterion for constraint handling in multi-objective optimization

This paper proposes a novel infill sampling criterion for constraint handling in multi-objective optimization of computationally expensive black-box functions. To reduce the computational burden, Kriging models are used to emulate the objective and constraint functions. The challenge of this multi-objective optimization problem arises from the fact that the epistemic uncertainty of the Kriging models should be taken into account to find Pareto-optimal solutions in the feasible domain. This is done by the proposed sampling criterion combining the Expected HyperVolume Improvement of the front of nondominated solutions and the Probability of Feasibility of new candidates. The proposed criterion is non-intrusive and derivative-free, and it is oriented to: (1) problems in which the computational cost is mainly from the function evaluation rather than optimization, and (2) problems that use complex in-house or commercial software that cannot be modified. The results using the proposed sampling criterion are compared with the results using Multi-Objective Evolutionary Algorithms. These results show that the proposed sampling criterion permits to identify both the feasible domain and an approximation of the Pareto front using a reduced number of computationally expensive simulations.

Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

This paper proposes an approach for the robust averaged control of random vibrations for the Bernoulli–Euler beam equation under uncertainty in the flexural stiffness and in the initial conditions. The problem is formulated in the framework of optimal control theory and provides a functional setting, which is so general as to include different types of random variables and second-order random fields as sources of uncertainty. The second-order statistical moment of the random system response at the control time is incorporated in the cost functional as a measure of robustness. The numerical resolution method combines a classical descent method with an adaptive anisotropic stochastic collocation method for the numerical approximation of the statistics of interest. The direct and adjoint stochastic systems are uncoupled, which permits to exploit parallel computing architectures to solve the set of deterministic problem that arise from the stochastic collocation method. As a result, problems with a relative large number of random variables can be solved with a reasonable computational cost. Two numerical experiments illustrate both the performance of the proposed method and the significant differences that may occur when uncertainty is incorporated in this type of control problems.

Mathematical Analysis of Optimal Control Problems Under Uncertainty

This chapter is focussed on existence theory for the solutions of robust and risk averse optimal control problems. As a first step, the classical variational theory of deterministic PDEs is extended to the case of random PDEs. This variational theory may be developed by using either the formalism of tensor product of Hilbert spaces or abstract functions, i.e., functions with values in Banach or Hilbert spaces. However, tensor products of Hilbert spaces have the advantage that the numerical approximation of such random PDEs becomes very natural in such a formalism.

Structural Optimization Under Uncertainty

Motivated by its applications in Engineering, since the early’s 70, optimal design of structures has been a very active research topic. Roughly speaking, the structural optimization problem amounts to finding the optimal distribution of material within a design domain such that a certain objective function or mechanical criterion is minimized. To ensure robustness and reliability of the final designs, a number of uncertainties should be accounted for in realistic models. These uncertainties include, among others, manufacturing imperfections, unknown loading conditions and variations of material’s properties.

Miscellaneous Topics and Open Problems

In this final chapter we discuss some related topics to the basic methodology presented in detail in the previous chapters. These topics are related to (i) time-dependent problems, and (ii) physical interpretation of robust and risk-averse optimal controls. We also list a number of challenging problems in the field of control under uncertainty.

Numerical Resolution of Risk Averse Optimal Control Problems

In this chapter, an adaptive, gradient-based, minimization algorithm is proposed for the numerical resolution of the risk averse optimal control problem \((P_{\varepsilon })\), as defined in Sect. 2.2. The numerical approximation of the associated statistics combines an adaptive, anisotropic, non-intrusive, Stochastic Galerkin approach for the numerical resolution of the underlying state and adjoint equations with a standard Monte-Carlo (MC) sampling method for numerical integration in the random domain.

Numerical Resolution of Robust Optimal Control Problems

Both gradient-based methods and methods based on the resolution of first-order optimality conditions may be used for solving numerically the robust optimal control problems presented in the preceding chapters. In both cases, the main difficulty arises in the numerical approximation of statistical quantities of interest associated to solutions of random PDEs. To handle this issue, it is assumed that the random inputs of the underlying PDEs depend on a finite number of random variables.