In practical applications, the problem involving multi criteria decision-making has become a research hotspot. The design choices based on multi-criteria data acquisition schemes to determine the specific parameters of the attack were discussed, and a new attack type was proposed in [
1]. The optimal mapping of hybrid energy systems based on wind and PV (photovoltaic system) was presented by [
2]. In addition, the mixed energy system was also obtained by using HOMER (Hybrid Optimization Model for Multiple Energy Resources) software (National Renewable Energy Laboratory (NREL), CO, Boulder, USA) and the TOPSIS multi criteria algorithm. In [
3], a nonlinear programming (NP) model based on the technique for order preference by similarity to ideal solution (TOPSIS) was developed to solve decision-making problems. Suzdaltsev et al. [
4] developed the genetic, ant colony and bee algorithms for solving the printed circuit board (PCB) design multi criteria optimization problems.
As an extension of linear programming (LP), the multi criteria and multi constraint level linear programming (MC
LP) is a useful tool to handle the decision problems with multiple decision makers and multiple resource constraint levels [
5], which can be seen in many economic situations [
6]. The concept of MC
LP is attractive to practitioners and has been widely applied in many fields such as transportation [
7], data mining [
8,
9], finance [
10], telecommunication management [
11], management information systems [
12,
13], and production planning [
14,
15]. Specifically, the MC
branch-and-partition algorithm and the MC
branch-and-bound algorithm were presented to solve MC
integer linear programs in [
16]. Chen et al. [
17] illustrated that MC
-simplex method was generated by the remarkable LP simplex method, and MC
-interior point method was driven by MC-interior point method, as well as introduced the current status and application areas of MC
LP. A new MC
LP model was proposed based on the structure of MC
LP to correct two types of errors in [
18]. Nonlinear programming as an important branch of operations research is a mathematical programming with nonlinear constraints or objective functions, which is explained in a mathematical terms, that is,
where each
is a mapping from
to
. Traditional methods for solving nonlinear programming include the steepest descent algorithm, Newton method, feasible direction method, function approximation method and trust region method. Aside from those methods, the enhanced Lagrange method is to solve the problem by replacing the original constraint problem with a series of unconstrained sub-problems, [
19] proposed an algorithm for the infeasible constrained nonlinear programming problem based on the large-scale augmented Lagrangian function, and analyzed the global convergence considering the possibility of not being feasible. Sequential quadratic programming (SQP) generates steps by solving quadratic subproblems, which can be applied to small and large problems, as well as problems with important nonlinearity [
20]. Algorithms for feasible SQP (Sequential Quadratic Programming) were designed by Craig and André [
21] to solve optimization problems with nonlinear constraints. In [
22], the original problem was reduced to a bounded-constrained nonlinear optimization problem, with reduced gradient algorithms instead of the penalty method. Regardless of the maturity of MC
LP theory and the further study of general nonlinear programming problems, we should notice that there is not yet very much progress in the research of nonlinear MC
models. In this paper, we will introduce a novel method for MC
NLP problems.
It is well known that the variational inequality theory is a very powerful tool to study the problems arising in nonlinear programming. Mathematical conditions, including a constraint qualification and convexity of the feasible set were shown by Toyasaki et al. [
23], which allowed for characterizing the economic problem by using a variational inequality formulation. An iterative algorithm was suggested by the resolvent operator technique to compute approximate solutions of the system of nonlinear set valued variational inclusion [
24]. The affine variational inequality problems and the polynomial complementary problems were discussed in [
25]; here, it is the extension of the results in [
24]. Then, the authors applied their results to discuss the existence of the solutions of weakly homogeneous nonlinear equations, the domains of which are closed convex cones. Motivated and inspired by [
26,
27], the purpose of this paper is to develop a new iterative algorithm for MC
NLP problems by employing the theory of variational inequalities and the resolvent operator technique. Considering the accuracy of solution for MC
NLP problems, the convergence and stability of the new algorithm are discussed in this paper. The result of this paper is the generalization of Theorem 2A.8 (Lagrange multiplier rule) in [
28].