Convergence Analysis of Difference-of-Convex Algorithm with Subanalytic Data View Full Text


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Article Info

DATE

2018-10

AUTHORS

Hoai An Le Thi, Van Ngai Huynh, Tao Pham Dinh

ABSTRACT

Difference-of-Convex programming and related algorithms, which constitute the backbone of nonconvex programming and global optimization, were introduced in 1985 by Pham Dinh Tao and have been extensively developed by Le Thi Hoai An and Pham Dinh Tao since 1994 to become now classic and increasingly popular. That algorithm is a descent method without linesearch and every limit point of its generated sequence is a critical point of the related Difference-of-Convex program. Determining its convergence rate is a challenging problem. Its knowledge is crucial from both theoretical and practical points of view. In this work, we treat this problem for the class of Difference-of-Convex programs with subanalytic data by using the nonsmooth form of the Lojasiewicz inequality. We have successfully proved that the whole sequence is convergent, if it is bounded, provided that the objective function is subanalytic continuous on its domain and one of the two Difference-of-Convex components is differentiable with locally Lipschitz derivative. We also established a result on the convergence rate, which depended on the Lojasiewicz exponent of the objective function. Finally, for both classes of trust-region subproblems and nonconvex quadratic programs, we showed that the Lojasiewicz exponent was one half, and thereby, our proposed algorithms applied to these Difference-of-Convex programs were Root-linearly convergent. More... »

PAGES

103-126

References to SciGraph publications

  • 1988-01. Semianalytic and subanalytic sets in PUBLICATIONS MATHÉMATIQUES DE L'IHÉS
  • 2012-06. Behavior of DCA sequences for solving the trust-region subproblem in JOURNAL OF GLOBAL OPTIMIZATION
  • 2000-05. An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints in MATHEMATICAL PROGRAMMING
  • 1997-10. Solving a Class of Linearly Constrained Indefinite Quadratic Problems by D.C. Algorithms in JOURNAL OF GLOBAL OPTIMIZATION
  • 1998-03. A Combined D.C. Optimization—Ellipsoidal Branch-and-Bound Algorithm for Solving Nonconvex Quadratic Programming Problems in JOURNAL OF COMBINATORIAL OPTIMIZATION
  • 1998-09. A Branch and Bound Method via d.c. Optimization Algorithms and Ellipsoidal Technique for Box Constrained Nonconvex Quadratic Problems in JOURNAL OF GLOBAL OPTIMIZATION
  • 2009-01. On the convergence of the proximal algorithm for nonsmooth functions involving analytic features in MATHEMATICAL PROGRAMMING
  • 2013-02. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods in MATHEMATICAL PROGRAMMING
  • 2009-01. Error bounds for systems of lower semicontinuous functions in Asplund spaces in MATHEMATICAL PROGRAMMING
  • 1997. Geometry of Subanalytic and Semialgebraic Sets in NONE
  • 2013-02. Convergence of Pham Dinh–Le Thi’s algorithm for the trust-region subproblem in JOURNAL OF GLOBAL OPTIMIZATION
  • 2012-09. Convergence Rate of the Pham Dinh–Le Thi Algorithm for the Trust-Region Subproblem in JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
  • 2005-01. The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems in ANNALS OF OPERATIONS RESEARCH
  • 1998. Variational Analysis in NONE
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    http://scigraph.springernature.com/pub.10.1007/s10957-018-1345-y

    DOI

    http://dx.doi.org/10.1007/s10957-018-1345-y

    DIMENSIONS

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