The reason for both are technical challenges in making the features stable and fast enough for us to run reliably for everyone. I realise this may disappoint many of you.
It is very disappointing for us as well. However, we have learned a lot from your feedback and usage of both features, and plan to bring them back in improved and stable forms in the not too distant future. Thank you for your feedback and enthusiasm for the year Discogs project: the best music database and marketplace in the world. The optimal mapping of hybrid energy systems based on wind and PV photovoltaic system was presented by [ 2 ].
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. As an extension of linear programming LP , the multi criteria and multi constraint level linear programming MC 2 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 2 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 2 branch-and-partition algorithm and the MC 2 branch-and-bound algorithm were presented to solve MC 2 integer linear programs in [ 16 ]. Chen et al. 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, min f x s. 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. An iterative algorithm was suggested by the resolvent operator technique to compute approximate solutions of the system of nonlinear set valued variational inclusion [ 24 ].
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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.
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Motivated and inspired by [ 26 , 27 ], the purpose of this paper is to develop a new iterative algorithm for MC 2 NLP problems by employing the theory of variational inequalities and the resolvent operator technique. Considering the accuracy of solution for MC 2 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. We should notice that the condition above is necessary and the differentiability assumptions on f and g are not always satisfied, so the Lagrange multiplier rule may not always hold.
Thus we need some more generalized method. In this section, an iterative algorithm for problem 3 will be presented, and the related properties will be discussed. We can give the algorithm for problem 3 as follows:.
Thus, we get the corollary of Theorem 2. Then, we will use Algorithm 1 to construct a convergent sequence. It is well known that linear problems can be seen as special cases of nonlinear problems, so the following application of the oilfield production distribution optimization can be showed based on our results. The historical data of eight sub-measures and the corresponding influencing factors of a oilfield which were in the mid-later stage from to are presented in Table 1. These historical data reflect to some extent the law of dynamic change of its development [ 30 ].
The new model of the output distribution of the oilfield in is max CX , s.
Table 5 shows the results of measure production and total production, and Figure 1 shows the convergence of the solution for each measure. By comparing with M C 2 -simplex method and M C 2 -interior point method, some differences can be found in number of iterations and computation complexity. Table 6 has shown that the computation complexity of the three algorithms differs slightly in this problem. This is because the decision variables, objective functions and constraint levels involved in this problem are relatively small.
However, for large-scale problems, the obvious differences in number of iterations and computation complexity could be discovered. Although the number of iterations of Algorithm 1 and M C 2 -simplex method are the same, we still believe that our method would be efficient if the model turns to a nonlinear case. This paper proposed Algorithm 1 for MC 2 NLP problems, and its novelty lies in the application of the theory of variational inequalities and the resolvent operator technique. The convergence of the generated iterative sequence was analyzed by Example 1, and the stability of Algorithm 1 was also verified by error propagation.
In Section 4 , Algorithm 1 was utilized to solve the application problem 17 ; the comparison with two other algorithms showed that this method performs efficiently for the nonlinear problems. The multi objective multi criteria programming is an important research topic in operations research and management science, not only because of the multi criteria and multi constraint level nature of most real-world decision problems, but also because there are still many open questions in this area.
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Algorithm 1 as a new tool to solve multi criteria and multi objective problems in decision systems provides algorithm support for decision makers; the basic structural ideas of this algorithm can be extended to practical linear and nonlinear problems in the future, such as project evaluation, program decision-making, engineering, industrial sector development sequencing and rational allocation of resources, etc. In addition, the algorithm proposed will be improved in actual application. Author Contributions Conceptualization, C. Figure 1. Convergence result of Algorithm 1.