Bayesian optimization for robust design of steel frames with joint and individual probabilistic constraints

This work proposes a Bayesian optimization (BO) method for solving multi-objective robust design optimization (RDO) problems of steel frames under aleatory uncertainty in external loads and material properties. Joint and individual probabilistic constrained RDO problems are formulated to consider two different ways the frame reaches its collapse state. Each problem involves three conflicting objective functions, namely, the total mass of the frame, the mean and variance of the maximum inter-story drift. Since the uncertain objective and probabilistic constraint functions of both problems are implicit within a finite element analysis program and the computation of the probabilistic constraints is an NP-hard problem, BO is used to guide the optimization process toward better solutions after it completes an iteration and offers a set of near Pareto-optimal solutions when it terminates. Specifically, Bayesian regression models called Gaussian processes (GPs) serve as surrogates for the structural responses. Two acquisition functions are then developed for the two RDO problems and a maximization problem of these functions is formulated as a mixed-integer nonlinear programming (MINLP) problem. A new random search coupled with simulated annealing is devised to solve the MINLP problem, thereby locating the most promising point in the input variable space at which the current solutions maximize their chance to be improved and the GP models are refined before the BO starts a new iteration. A test problem and two design examples show that exact or good Pareto-optimal solutions to the RDO problems can be found by the proposed method with 20 iterations.