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Gradient Projection for Separable Convex Sets

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Scalable Algorithms for Contact Problems

Part of the book series: Advances in Mechanics and Mathematics ((AMMA,volume 36))

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Abstract

An important ingredient of our algorithms for solving QP and QCQP problems is the Euclidean projection on the convex set defined by separable convex constraints. To combine the gradient projection with the CG method effectively, it is necessary to have nontrivial bounds on the decrease of f along the projected-gradient path in terms of bounds on the spectrum of its Hessian matrix \({\mathsf {A}}\).

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References

  1. Saad, Y.: Iterative Methods for Large Linear Systems. SIAM, Philadelphia (2002)

    Google Scholar 

  2. Dostál, Z.: Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, 1st edn. Springer, New York (2009)

    MATH  Google Scholar 

  3. Bertsekas, D.P.: Nonlinear Optimization. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  4. Luo, Z.Q., Tseng, P.: Error bounds and convergence analysis of feasible descent methods: a general approach. Annal. Oper. Res. 46, 157–178 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dostál, Z.: On the decrease of a quadratic function along the projected-gradient path. ETNA 31, 25–59 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Bouchala, J., Dostál, Z., Vodstrčil, P.: Separable spherical constraints and the decrease of a quadratic function in the gradient projection. JOTA 157, 132–140 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Schöberl, J.: Solving the Signorini problem on the basis of domain decomposition techniques. Computing 60(4), 323–344 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dostál, Z., Schöberl, J.: Minimizing quadratic functions over non-negative cone with the rate of convergence and finite termination. Comput. Optim. Appl. 30(1), 23–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Schöberl, J.: Efficient contact solvers based on domain decomposition techniques. Comput. Math. Appl. 42, 1217–1228 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kučera, R.: Convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints. SIAM J. Optim. 19, 846–862 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bouchala, J., Dostál, Z., Kozubek, T., Pospíšil, L., Vodstrčil, P.: On the solution of convex QPQC problems with elliptic and other separable constraints. Appl. Math. Comput. 247(15), 848–864 (2014)

    MathSciNet  MATH  Google Scholar 

  12. Barzilai, J., Borwein, J.M.: Two point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Birgin, E.G., Martínez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Birgin, E.G., Raydan, M., Martínez, J.M.: Spectral projected gradient methods: review and perspectives. J. Stat. Softw. 60, 3 (2014)

    Article  Google Scholar 

  15. Dai, Y.H., Fletcher, R.: Projected Barzilai–Borwein methods for large-scale box-constrained quadratic programming. Numer. Math. 100, 21–47 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grippo, L., Lampariello, F., Lucidi, S.: A nonmonotone line search technique for Newton’s method. SIAM J. Numer. Anal. 23(4), 707–716 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pospíšil, L., Dostál, Z.: The projected Barzilai–Borwein method with fall-back for strictly convex QCQP problems with separable constraints (to appear)

    Google Scholar 

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Correspondence to Zdeněk Dostál .

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Dostál, Z. (2016). Gradient Projection for Separable Convex Sets. In: Scalable Algorithms for Contact Problems. Advances in Mechanics and Mathematics, vol 36. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6834-3_6

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