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Gradient Projection Method for a Class of Optimization Problems with a Constraint in the Form of a Subset of Points of a Smooth Surface

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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.

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REFERENCES

  1. 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).

    MathSciNet  Google Scholar 

  2. Yu. G. Evtushenko and V. G. Zhadan, “Barrier-projective methods for nonlinear programming,” Comput. Math. Math. Phys. 34 (5), 579–590 (1994).

    MathSciNet  MATH  Google Scholar 

  3. 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).

    Article  MathSciNet  Google Scholar 

  4. A. I. Kozlov, “Gradient projection method for finding quasi-solutions of nonlinear irregular operator equations,” Vychisl. Metody Program. 4 (1), 117–125 (2003).

    Google Scholar 

  5. N. Mijajlović and M. Jaćimović, “Some continuous methods for solving quasi-variational inequalities,” Comput. Math. Math. Phys. 58 (2), 190–195 (2018).

    Article  MathSciNet  Google Scholar 

  6. 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).

    Article  MathSciNet  Google Scholar 

  7. 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).

    Article  MathSciNet  Google Scholar 

  8. 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).

    Google Scholar 

  9. 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).

    Article  MathSciNet  Google Scholar 

  10. 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).

    MathSciNet  Google Scholar 

  11. 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).

    Article  MathSciNet  Google Scholar 

  12. 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).

    Article  MathSciNet  Google Scholar 

  13. 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).

    Article  MathSciNet  Google Scholar 

  14. 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).

    MathSciNet  MATH  Google Scholar 

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Correspondence to Yu. A. Chernyaev.

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Translated by I. Ruzanova

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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

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  • DOI: https://doi.org/10.1134/S0965542521020068

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