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

Sequential sampling approach to energy‐based multi‐objective design optimization of steel frames with correlated random parameters

“…Moving the left-side terms of the probabilistic constraints in problem (12) to the right side and let…”

“…The number of initial training samples 𝑁 depends on the number of input variables 𝑑, for example, 𝑁 ≥ 15𝑑. [12] Samples of 𝐬 and 𝐫 are generated using Latin-hypercube sampling (LHS) and Equation (19), respectively. Integer samples of 𝐬 are determined by rounding the corresponding real samples by LHS to the nearest integers.…”

“…As the input variable space consists of the spaces of the discrete design variables 𝐬 and continuous random parameters 𝐫, it is natural to divide the exploration of the promising region into two phases. [12] The first phase, discussed in this section, specifies new sampling vectors of 𝐬, denoted as 𝐬 n , which are responsible for the To improve the current solutions, a hypervolume-based approach by Do et al [12] is adopted in this study. Let Ω = {𝐟 1 , … , 𝐟 𝑀 } ∈ ℝ 𝑘 and 𝐟 r ∈ ℝ 𝑘 denote the current set of 𝑀 approximate Pareto-optimal solutions in a space of 𝑘 objective functions and a fixed reference point dominated by all elements of Ω, respectively.…”

“…The literature provides three different methods for such a unified approach, namely, robust design optimization (RDO), [7][8][9][10][11][12] reliability-based design optimization (RBDO), [13][14][15][16] and risk-based optimization (RBO). [17] These methods differs in how unfavorable effects of uncertainty can be managed.…”

“…The GP models for the uncertain LSFs are refined after each optimization iteration by specifying a batch of new sampling points that lie on the Pareto front of a bi-objective deterministic maximization problem formulated for addressing the improvement in the current solutions and the feasibility of the new sampling points simultaneously. This refinement scheme differs from those of other sequential optimization methods, such as Bayesian optimization [12] or a sequential method by the authors, [16] where only one new sampling point is specified after each optimization iteration. As will be demonstrated in a test problem, the new sampling points tend to distribute in the neighborhood of exact solutions, leading to the robustness of the refinement scheme as well as a quick termination of the optimization process.…”