Abstract
The gradient projection method is generalized to nonconvex sets of constraints representing the set-theoretic difference of a set of points of a smooth surface and the union of a finite number of convex open sets. Necessary optimality conditions are examined, and the convergence of the method is analyzed.
Similar content being viewed by others
REFERENCES
F. P. Vasil’ev and A. Nedić, “A three-step regularized gradient projection method for solving minimization problems with inexact initial data,” Russ. Math. 37 (12), 34–43 (1993).
Yu. G. Evtushenko and V. G. Zhadan, “Barrier-projective methods for nonlinear programming,” Comput. Math. Math. Phys. 34 (5), 579–590 (1994).
A. S. Antipin, B. A. Budak, and F. P. Vasil’ev, “A regularized continuous extragradient method of the first order with a variable metric for problems of equilibrium programming,” Differ. Equations 38 (12), 1683–1693 (2002).
A. I. Kozlov, “Gradient projection method for finding quasi-solutions of nonlinear irregular operator equations,” Vychisl. Metody Program. 4 (1), 117–125 (2003).
N. Mijajlović and M. Jaćimović, “Some continuous methods for solving quasi-variational inequalities,” Comput. Math. Math. Phys. 58 (2), 190–195 (2018).
M. Yu. Kokurin, “Solution of ill-posed nonconvex optimization problems with accuracy proportional to the error in input data,” Comput. Math. Math. Phys. 58 (11), 1748–1760 (2018).
N. Du, H. Wang, and W. Liu, “A fast gradient projection method for a constrained fractional optimal control,” J. Sci. Comput. 68 (1), 1–20 (2016).
Z. Tang, J. Qin, J. Sun, and B. Geng, “The gradient projection algorithm with adaptive mutation step length for non-probabilistic reliability index,” Teh. Vjesnik 24 (1), 53–62 (2017).
J. Preininger and P. T. Vuong, “On the convergence of the gradient projection method for convex optimal control problems with bang-bang solutions,” Comput. Optim. Appl. 70 (1), 221–238 (2018).
V. I. Zabotin and Yu. A. Chernyaev, “Convergence of an iterative method for a programming problem with constraints in the form of a convex smooth surface,” Comput. Math. Math. Phys. 44 (4), 575–578 (2004).
Yu. A. Chernyaev, “An extension of the gradient projection method and Newton’s method to extremum problems constrained by a smooth surface,” Comput. Math. Math. Phys. 55 (9), 1451–1460 (2015).
Yu. A. Chernyaev, “Convergence of the gradient projection method and Newton’s method as applied to optimization problems constrained by intersection of a spherical surface and a convex closed set,” Comput. Math. Math. Phys. 56 (10), 1716–1731 (2016).
Yu. A. Chernyaev, “Gradient projection method for optimization problems with a constraint in the form of the intersection of a smooth surface and a convex closed set,” Comput. Math. Math. Phys. 59 (1), 34–45 (2019).
A. M. Dulliev and V. I. Zabotin, “Iteration algorithm for projecting a point on a nonconvex manifold in a normed linear space,” Comput. Math. Math. Phys. 44 (5), 781–784 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by I. Ruzanova
Rights and permissions
About this article
Cite this article
Chernyaev, Y.A. Gradient Projection Method for a Class of Optimization Problems with a Constraint in the Form of a Subset of Points of a Smooth Surface. Comput. Math. and Math. Phys. 61, 368–375 (2021). https://doi.org/10.1134/S0965542521020068
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542521020068