## 1 Introduction
It is important to note that algorithms (4) and (5) only converge weakly in infinite dimensional spaces. However, in machine learning and CT reconstruction, strong convergence is more desirable in infinite dimensional spaces [12]. Therefore, it is necessary to modify (3), such that it can achieve strong convergence in real Hilbert spaces. In the last two decades, so many modifications of the forward-backward method have been constructed to obtain strong convergence results in real Hilbert spaces; see [11; 12; 15; 16] and the references in them. In recent years, the construction of inertial-based algorithms has attracted massive interest from researchers. The idea of including inertial terms in iterative methods for solving optimization problems was initiated by [PERSON] [17] and it has been confirmed by numerous authors that the inclusion of an inertial term in a method acts as a boost to the convergence speed of the method. A common feature of the inertial-type algorithm is that the next iteration depends on the combination of two previous iterates; for more details, see [3; 18; 19]. Many inertial-type algorithms have been studied and numerical tests have demonstrated that the inertial effects on these methods greatly improve their performances; see [1; 3; 20]. Recently, [PERSON] and [PERSON] [17] introduced and studied the following inertial FBA to solve the MIP (2):  
\[ \begin{cases}w_{k}=p_{k}+\theta_{k}(p_{k}-p_{k-1}),\\ q_{k}=(I+\lambda_{k}E)^{-1}(I-\lambda_{k}D)p_{k}.\end{cases}  \tag{6}\]  
Note that method (6) only convergences weakly in real Hilbert spaces; numerical tests by the authors proved that their method outperforms several existing methods without inertial terms. Several mathematical problems, such as variational inequality problems, equilibrium problems, split feasibility problems, and split minimization problems, are all special MIP cases. These problems have been applied to solve diverse real-world problems, such as modeling inverse problems arising from phase retrieval, modeling intensity-modulated radiation therapy planning, sensor networks in computerized and data compression, optimal control problems, and image/signal processing problems [21; 22; 23]. The bilevel programming problem is a constrained optimization problem in which the constrained set is a solution set of another optimization problem. This problem is enriched with many applications in modeling Stackelberg games, the convex feasibility problem, determination in Wardrop equilibria for network flow, domain decomposition methods for PDEs, optimal control problems, and image/signal processing problems [23]. When the first-level problem is a VIP and the second-level problem is a fixed point set of a mapping, then the bilevel problem is known as the hierarchical variational inequality problem. In [24; 25; 26], [PERSON] introduced the following method, called the hybrid steepest-descent iterative method, to solve the hierarchical VIP:  
\[p_{k+1}=(I-\alpha_{k}\mathcal{G}\mathcal{F})\mathcal{S}p_{k}, \tag{7}\]  
where \(\mathcal{F}\) is a strongly monotone-Lipschitz continuous operator and \(\mathcal{S}\) is a nonexpansive mapping. In this paper, we consider the problem of solving a VIP over the solution set of the MIP in a real Hilbert space. This problem is formulated as follows:  
\[\text{Find }p^{\star}\in(D+E)^{-1}(0)\text{ such that }\langle\mathcal{G}p^{\star},q-p^{ \star}\rangle\geq 0,\forall q\in(D+E)^{-1}(0), \tag{8}\]  
where \(\mathcal{F}\) is a strongly monotone-Lipschitz continuous operator, \(D\) is a monotone-Lipschitz continuous operator, and \(E\) is a maximal monotone operator. Inspired by the inertial technique, the Tseng splitting algorithm, projection, and contraction method, and hybrid steepest decent method, we introduce two efficient iterative algorithms to solve problem (8). We prove the strong convergence results of the suggested method under some standard assumptions on the control parameters. Also, the strong convergence results are achieved without prior knowledge of the operator norm.