The aim of this paper is to clarify several important points, including a brief and adequate explanation of loving improvement, as well as laying out a number of important mathematical formulas that we need, supported with graphs.

In this paper we present a study on the problem of NP hardness and their applications in computer science. In addition, our study sheds light on the most important applications present in our daily life and how the problem of NP stiffness is an important primary focus in it. Moreover, the aim was to search for further modifications in order to obtain optimal methods for their adoption. Finally, the aim of this study is to seek to solve the decision problem of NP hardness to achieve the desired goals with the optimal result.

In this paper we will introduce a new approach for solving K-cluster problem which is one of the NP-hardness problem, in combinatorial optimization problems. In addition, P is NP-hardness if and only if the polynomial time of each NP problem is reduced to P. Actually, our study was focused on the two methods which is Penalty and Augmented Lagrangian methods base on the numerical result. Moreover, we tested the K-cluster problem and found the Augmented Lagrangian Method faster than Penalty method. Finally, our research is not just focus on the numerical computational but also improving the theoretical converges properties.

A new approach has been introduced for solving NP-hardness problem in combinatorial optimization problems. Actully, our study focused on the relationship between the Lagrange method and Penalty method ,this paper  introduce a new relaxation of the fesible region.Furthermore, NP hard problem has been tested and showed that the Augmented Lagrangian Approach outperformed the Penalty method. Finally, our study focuses on enhancing the theoretical convergence features as well as numerical computing.

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