An ordinary differential equation approach for nonlinear programming and nonlinear complementary problem

Abstract

We consider an ordinary differential equation (ODE) approach for solving non- linear programming (NLP) and nonlinear complementary problem (NCP). The Karush- Kuhn Tucker (KKT) optimality conditions can be converted to NCP. Based on the Fischer-Burmeister (FB) function and the Natural-Residual (NR) function are obtained the new NCP-functions. A special technique is employed to reformulate of the NCP as the system of nonlinear algebraic equations (NAEs) later on reformulated once more by force. of an original time-like function into an ODE. Afterwards, a group preserving scheme (GPS) is a package to reformulate an ODE into the new numerical equation in a way the ODEs system is designed into a nonlinear dynamical system (NDS) and is continued to a discovery the new numerical equation through activating the Lorentz group SO0(n, 1) and its Lie algebra so(n, 1). Lastly, the fictitious time integration method (FTIM) is utilized into this new numerical equation to determine an approximation solution at the numerical experiments area.

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

“…Because of the crucial role of the nonlinear problems, the researchers provided different numerical methods to obtain the solutions of nonlinear PDEs, which are properly transformed into the nonlinear algebraic equations (NAEs) by using numerically discretized methods. As mentioned in [1][2][3][4], nonlinear optimization problem, nonlinear programming, nonlinear sloshing problem, and nonlinear complementarity problem (NCP) can be recast to the issues to solve NAEs, with the help of NCP-functions, for example, the Fischer-Burmeister NCP [5]. Indeed, the numerical solution of NAEs is one of the main problems in computational mathematics.…”