Geodesic
A new method of linear programming
Numerical methods and programming, Tome 24 (2023) no. 4, pp. 408-429.

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The article presents a new iterative method of linear programming, called the surface movement method. This method builds a path from the initial boundary point to the point at which the optimal value of the objective function is achieved on the surface of a polytope that restricts the feasible region of linear programming problem. The movement vector defines the direction of the maximum increase/decrease in the value of the objective function. A formal description of the algorithm implementing the surface movement method is presented. The convergence theorem is proved. The described method involves the use of a feed forward deep neural network to determine the direction of movement along the edges of a feasible polytope. To do this, a multidimensional local image of the linear programming problem is constructed at the point of the current approximation, which is fed to the input of the neural network. The set of labeled precedents necessary for training a neural network can be obtained using the apex method.
Keywords: linear programming; surface motion method; iterative method; convergence theorem; deep neural network. 
@article{VMP_2023_24_4_a4,
     author = {N. A. Olkhovsky and L. B. Sokolinskii},
     title = {A new method of linear programming},
     journal = {Numerical methods and programming},
     pages = {408--429},
     publisher = {mathdoc},
     volume = {24},
     number = {4},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMP_2023_24_4_a4/}
}
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N. A. Olkhovsky; L. B. Sokolinskii. A new method of linear programming. Numerical methods and programming, Tome 24 (2023) no. 4, pp. 408-429. http://geodesic.mathdoc.fr/item/VMP_2023_24_4_a4/