Modified Reduced Gradient With Realizations Sorting for Hard Equality Constraints in Reliability-Based Design Optimization

In this work, the presence of equality constraints in reliability-based design optimization (RBDO) problems is studied. Relaxation of soft equality constraints in RBDO and its challenges are briefly discussed while the main focus is on hard equalities that can not be violated even under uncertainty. Direct elimination of hard equalities to reduce problem dimensions is usually suggested; however, for nonlinear or black-box functions, variable elimination requires expensive root-finding processes or inverse functions that are generally unavailable. We extend the reduced gradient methods in deterministic optimization to handle hard equalities in RBDO. The efficiency and accuracy of the first and the second order predictions in reduced gradient methods are compared. Results show the first order prediction being more efficient when realizations of random variables are available. A gradient-weighted sorting with these random samples is proposed to further improve the solution efficiency of the reduced gradient method. Feasible design realizations subject to hard equality constraints are then available to be implemented with the state-of-the-art sampling techniques for RBDO problems. Numerical and engineering examples show the strength and simplicity of the proposed method.