aspg-21-3811-1791468709 20261008141149 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group International Journal of Neutrosophic Science IJNS 2692-6148 2690-6805 2025 26 3 Maissam Jdid Faculty of Science, Damascus University, Damascus, Syria; Department of Requirements, International University for Science and Technology, Ghabageb, Syrian Arab Republic Florentin Smarandache University of New Mexico، Mathematics, Physics and Natural Sciences Division 705 Gurley Ave., Gallup, NM 87301, USA Nonlinear programming is one of the most important methods used to obtain the optimal solution to many real-world problems. Given the importance of this method, numerous studies and research have been conducted in recent years with the aim of providing methods that help find the optimal solution. These studies and research have resulted in a basic structure used to find these solutions. This structure initially indicates that the optimal solution can be found at any boundary point in the feasible region, at a point within the feasible region, or at a discontinuity point. In this research, we present some of the important foundations and principles of nonlinear programming and the gradient projection method used in searching for the optimal solution to unrestricted nonlinear programming problems. We will reformulate these foundations and principles using neutrosophic logic concepts as a complement to our previous research, the aim of which is to provide a new vision for some operations research methods, a neutrosophic vision. Our focus will be on the improvement these concepts offer when used in the field of applied mathematics, through the more accurate and comprehensive solutions we obtain, which provide a margin of freedom commensurate with the Given the reality we live in, and the changes that can occur to the data of the actual issue under study, this requires decision makers to prepare many appropriate alternatives for each change. 2025 279 286 3811 https://creativecommons.org/licenses/by/4.0/ 10.54216/IJNS.260320 https://www.americaspg.com/journal/21/article/3811 https://www.americaspg.com/storage/11749069327.pdf