A Bayesian approach to constrained single- and multi-objective optimization

This article addresses the problem of derivative-free (single- or multi-objective) optimization subject to multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, non-linear and expensive to evaluate. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited, as in complex industrial design optimization problems. The method we propose to overcome this difficulty has its roots in both the Bayesian and the multi-objective optimization literatures. More specifically, an extended domination rule is used to handle objectives and constraints in a unified way, and a corresponding expected hyper-volume improvement sampling criterion is proposed. This new criterion is naturally adapted to the search of a feasible point when none is available, and reduces to existing Bayesian sampling criteria---the classical Expected Improvement (EI) criterion and some of its constrained/multi-objective extensions---as soon as at least one feasible point is available. The calculation and optimization of the criterion are performed using Sequential Monte Carlo techniques. In particular, an algorithm similar to the subset simulation method, which is well known in the field of structural reliability, is used to estimate the criterion. The method, which we call BMOO (for Bayesian Multi-Objective Optimization), is compared to state-of-the-art algorithms for single- and multi-objective constrained optimization

Adaptive Ranking Based Constraint Handling for Explicitly Constrained Black-Box Optimization

A novel explicit constraint handling technique for the covariance matrix adaptation evolution strategy (CMA-ES) is proposed. The proposed constraint handling exhibits two invariance properties. One is the invariance to arbitrary element-wise increasing transformation of the objective and constraint functions. The other is the invariance to arbitrary affine transformation of the search space. The proposed technique virtually transforms a constrained optimization problem into an unconstrained optimization problem by considering an adaptive weighted sum of the ranking of the objective function values and the ranking of the constraint violations that are measured by the Mahalanobis distance between each candidate solution to its projection onto the boundary of the constraints. Simulation results are presented and show that the CMA-ES with the proposed constraint handling exhibits the affine invariance and performs similarly to the CMA-ES on unconstrained counterparts.Comment: 9 page

Solving the G-problems in less than 500 iterations: Improved efficient constrained optimization by surrogate modeling and adaptive parameter control

Constrained optimization of high-dimensional numerical problems plays an important role in many scientific and industrial applications. Function evaluations in many industrial applications are severely limited and no analytical information about objective function and constraint functions is available. For such expensive black-box optimization tasks, the constraint optimization algorithm COBRA was proposed, making use of RBF surrogate modeling for both the objective and the constraint functions. COBRA has shown remarkable success in solving reliably complex benchmark problems in less than 500 function evaluations. Unfortunately, COBRA requires careful adjustment of parameters in order to do so. In this work we present a new self-adjusting algorithm SACOBRA, which is based on COBRA and capable to achieve high-quality results with very few function evaluations and no parameter tuning. It is shown with the help of performance profiles on a set of benchmark problems (G-problems, MOPTA08) that SACOBRA consistently outperforms any COBRA algorithm with fixed parameter setting. We analyze the importance of the several new elements in SACOBRA and find that each element of SACOBRA plays a role to boost up the overall optimization performance. We discuss the reasons behind and get in this way a better understanding of high-quality RBF surrogate modeling

A constrained multi-objective surrogate-based optimization algorithm

Surrogate models or metamodels are widely used in the realm of engineering for design optimization to minimize the number of computationally expensive simulations. Most practical problems often have conflicting objectives, which lead to a number of competing solutions which form a Pareto front. Multi-objective surrogate-based constrained optimization algorithms have been proposed in literature, but handling constraints directly is a relatively new research area. Most algorithms proposed to directly deal with multi-objective optimization have been evolutionary algorithms (Multi-Objective Evolutionary Algorithms -MOEAs). MOEAs can handle large design spaces but require a large number of simulations, which might be infeasible in practice, especially if the constraints are expensive. A multi-objective constrained optimization algorithm is presented in this paper which makes use of Kriging models, in conjunction with multi-objective probability of improvement (PoI) and probability of feasibility (PoF) criteria to drive the sample selection process economically. The efficacy of the proposed algorithm is demonstrated on an analytical benchmark function, and the algorithm is then used to solve a microwave filter design optimization problem

Comparative study on the application of evolutionary optimization techniques to orbit transfer maneuvers

Orbit transfer maneuvers are here considered as benchmark cases for comparing performance of different optimization techniques in the framework of direct methods. Two different classes of evolutionary algorithms, a conventional genetic algorithm and an estimation of distribution method, are compared in terms of performance indices statistically evaluated over a prescribed number of runs. At the same time, two different types of problem representations are considered, a first one based on orbit propagation and a second one based on the solution of Lambert’s problem for direct transfers. In this way it is possible to highlight how problem representation affects the capabilities of the considered numerical approaches