Modified Reduced Gradient With Realization 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 cannot 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 second-order predictions in reduced gradient methods are compared. Results show that the first-order prediction is more efficient when realizations of random variables are available. 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 state-of-the-art sampling techniques for RBDO problems. Numerical and engineering examples show the strength and simplicity of the proposed method.

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.

A comparative study on optimization methods for the constrained nonlinear programming problems

Constrained nonlinear programming problems often arise in many engineering applications. The most well-known optimization methods for solving these problems are sequential quadratic programming methods and generalized reduced gradient methods. This study compares the performance of these methods with the genetic algorithms which gained popularity in recent years due to advantages in speed and robustness. We present a comparative study that is performed on fifteen test problems selected from the literature.

REDUCED GRADIENT METHODS AND THEIR RELATION TO REACTION PATHS

The reaction path is an important concept in theoretical chemistry. We discuss different definitions, their merits as well as their drawbacks: IRC (steepest descent from saddle), reduced gradient following (RGF), gradient extremals, and some others. Many properties and problems are explained by two-dimensional figures. This paper is both a review and a pointer to future research. The branching points of RGF curves are valley-ridge inflection (VRI) points of the potential energy surface. These points may serve as indicators for bifurcations of the reaction path. The VRI points are calculated with the help of Branin's method. All the important features of the potential energy surface are independent of the coordinate system. Besides the theoretical definitions, we also discuss the numerical use of the methods.