Determining solution set of nonlinear inequalities using space-filling curves for finding working spaces of planar robots

• Published in: Journal of global optimization Vol. 89; no. 2; pp. 415 - 434
• Main Authors: Lera, Daniela, Nasso, Maria Chiara, Posypkin, Mikhail, Sergeyev, Yaroslav D.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer; Springer Nature B.V
• Subjects: Algorithms; Approximation; Computer Science; Global optimization; Hilbert curve; Hilbert space; Inequalities; Mathematics; Mathematics and Statistics; Nonlinear systems; Operations Research/Decision Theory; Optimization; Real Functions; Robotics industry; Robots; Technology application; Systems of nonlinear inequalities; Global optimization; Space-filling curves; Derivative-free methods; Working spaces of robots
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-023-01352-2

In this paper, the problem of approximating and visualizing the solution set of systems of nonlinear inequalities is considered. It is supposed that left-hand parts of the inequalities can be multiextremal and non-differentiable. Thus, traditional local methods using gradients cannot be applied in these circumstances. Problems of this kind arise in many scientific applications, in particular, in finding working spaces of robots where it is necessary to determine not one but all the solutions of the system of nonlinear inequalities. Global optimization algorithms can be taken as an inspiration for developing methods for solving this problem. In this article, two new methods using two different approximations of Peano–Hilbert space-filling curves actively used in global optimization are proposed. Convergence conditions of the new methods are established. Numerical experiments executed on problems regarding finding the working spaces of several robots show a promising performance of the new algorithms.