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Gradient Projection Method for Optimization Problems with a Constraint in the Form of the Intersection of a Smooth Surface and a Convex Closed Set

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Abstract

The gradient projection method is generalized to the case of nonconvex sets of constraints representing the set-theoretic intersection of a smooth surface with a convex closed set. Necessary optimality conditions are studied, 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. F. P. Vasil’ev and A. Nedić, “A version of the regularized gradient projection method,” Comput. Math. Math. Phys. 34 (4), 431–439 (1994).

    MathSciNet  MATH  Google Scholar 

  3. F. P. Vasil’ev and A. Nedić, “A regularized third-order continuous gradient projection method,” Differ. Equations 30 (12), 1869–1877 (1994).

    MathSciNet  MATH  Google Scholar 

  4. A. S. Antipin, “Estimates for the rate of convergence of the gradient projection method,” Russ. Math. 39 (6), 14–22 (1995).

    MathSciNet  Google Scholar 

  5. 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 

  6. Yu. G. Evtushenko and V. G. Zhadan, “Dual barrier-projection and barrier-Newton methods for linear programming problems,” Comput. Math. Math. Phys. 36 (7), 847–859 (1996).

    MathSciNet  MATH  Google Scholar 

  7. A. Z. Ishmukhametov, “Regularized approximate gradient projection and constrained gradient methods with finite-step inner algorithms,” Comput. Math. Math. Phys. 43 (12), 1823–1836 (2003).

    MathSciNet  MATH  Google Scholar 

  8. A. I. Kozlov and M. Yu. Kokurin, “Gradient projection method for stable approximation of quasisolutions to irregular nonlinear operator equations,” Comput. Math. Math. Phys. 49 (10), 1678–1699 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  9. V. I. Zabotin and Yu. A. Chernyaev, “A generalization of the gradient projection method to extremum value problems with preconvex constraints,” Comput. Math. Math. Phys. 41 (3), 340–346 (2001).

    MathSciNet  MATH  Google Scholar 

  10. Yu. A. Chernyaev, “On a numerical algorithm for optimization problems with preconvex constraints,” Comput. Math. Math. Phys. 43 (2), 162–167 (2003).

    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  MATH  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  MATH  Google Scholar 

  13. Yu. A. Chernyaev, “The conditional gradient method for optimization problems with preconvex constraints,” Comput. Math. Math. Phys. 43 (12), 1837–1840 (2003).

    MathSciNet  Google Scholar 

  14. Yu. A. Chernyaev, “Convergence of the conditional gradient method in the class of optimization problems with a constraint in the form of a subset of points of a sphere,” Vestn. Kazan. Gos. Tekh. Univ. im. A.N. Tupoleva, No. 4, 104–110 (2015).

  15. 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 

  16. B. N. Pshenichnyi, Convex Analysis and Optimization Problems (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

  17. F. P. Vasil’ev, Numerical Methods for Solving Extremum Problems (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  18. B. N. Pshenichnyi and Yu. M. Danilin, Numerical Methods in Optimization Problems (Nauka, Moscow, 1975) [in Russian].

    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 Optimization Problems with a Constraint in the Form of the Intersection of a Smooth Surface and a Convex Closed Set. Comput. Math. and Math. Phys. 59, 34–45 (2019). https://doi.org/10.1134/S0965542519010056

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