The genetic algorithm with greedy heuristic, initially developed for solving location problems on networks, can be adapted for solving continuous problems such as kmeans. However, the efficiency of such algorithm in case of continuous problems does not allow to use it for solving the large-scale problems. In this paper, authors propose a modification to this algorithm which allows such algorithm to work faster.
In the modern digital economy, optimal decision support systems, as well as machine learning systems, are becoming an integral part of production processes. Artificial neural network training as well as other engineering problems generate such problems of high dimension that are difficult to solve with traditional gradient or conjugate gradient methods. Relaxation subgradient minimization methods (RSMMs) construct a descent direction that forms an obtuse angle with all subgradients of the current minimum neighborhood, which reduces to the problem of solving systems of inequalities. Having formalized the model and taking into account the specific features of subgradient sets, we reduced the problem of solving a system of inequalities to an approximation problem and obtained an efficient rapidly converging iterative learning algorithm for finding the direction of descent, conceptually similar to the iterative least squares method. The new algorithm is theoretically substantiated, and an estimate of its convergence rate is obtained depending on the parameters of the subgradient set. On this basis, we have developed and substantiated a new RSMM, which has the properties of the conjugate gradient method on quadratic functions. We have developed a practically realizable version of the minimization algorithm that uses a rough one-dimensional search. A computational experiment on complex functions in a space of high dimension confirms the effectiveness of the proposed algorithm. In the problems of training neural network models, where it is required to remove insignificant variables or neurons using methods such as the Tibshirani LASSO, our new algorithm outperforms known methods.
The continuous planar facility location problem with the connected region of feasible solutions bounded by arcs is a particular case of the constrained Weber problem. This problem is a continuous optimization problem which has a nonconvex feasible set of constraints. This paper suggests appropriate modifications of four metaheuristic algorithms which are defined with the aim of solving this type of nonconvex optimization problems. Also, a comparison of these algorithms to each other as well as to the heuristic algorithm is presented. The artificial bee colony algorithm, firefly algorithm, and their recently proposed improved versions for constrained optimization are appropriately modified and applied to the case study. The heuristic algorithm based on modified Weiszfeld procedure is also implemented for the purpose of comparison with the metaheuristic approaches. Obtained numerical results show that metaheuristic algorithms can be successfully applied to solve the instances of this problem of up to 500 constraints. Among these four algorithms, the improved version of artificial bee algorithm is the most efficient with respect to the quality of the solution, robustness, and the computational efficiency.
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