A Variational and Computational Framework for Constrained Nonlinear Optimization Problems

Abstract

The paper suggests a single variational and computational model of the solution of constrained nonlinear optimization problems that are usually formulated in the field of engineering and applied sciences. The methodology recast the original optimization problem into a similar variational functional, which allows us to derive optimality conditions systematically by the calculus of variations. To guarantee that the application would be practical, a computational algorithm that is based on combining Lagrangian multipliers and penalty functions along with the gradient-based iterative updates are formulated to address both equality and inequality constraints effectively. The proposed framework offers a structured means to tackle the issues of feasibility and convergence simultaneously through a single formulation, unlike traditional optimization methods. The character of the method is considered with the assistance of a variety of benchmark nonlinear optimization problems both convex and non-convex. Compared to classical gradient-based and interior-point methods, quantitative performance measures (root mean square error (RMSE), convergence rate and computational time) are compared. The findings reveal that the suggested strategy produces a better accuracy (a reduction of up to 50 percent of error), quicker convergence and better constraint satisfaction. These results verify that the framework is robust, as well as scalable to a broad diversity of nonlinear constrained systems. The proposed methodology presents a positive basis on future developments towards large-scale and real-time optimization applications.