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.

The aim of this research is focused on the mathematical optimization in computational systems biological models based on find the good method that gives converges faster. Our strategy is to use Bundle Method instead of using Subgradient Methods. Also, we improve the convergence of theoretical properties of the methods. The basic idea of approximation is the subdifferential of the objective function by using subgradients from previous iterations of a bundle method.

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