Complexity Analysis of Successive Convex Relaxation Methods for Nonconvex Sets

@article{Kojima2001ComplexityAO,
  title={Complexity Analysis of Successive Convex Relaxation Methods for Nonconvex Sets},
  author={Masakazu Kojima and Akiko Takeda},
  journal={Math. Oper. Res.},
  year={2001},
  volume={26},
  pages={519-542}
}
This paper discusses computational complexity of conceptual successive convex relaxation methods proposed by Kojima and TunA§el for approximating a convex relaxation of a compact subsetof the n-dimensional Euclidean spaceR n . Here,C0 denotes a nonempty compact convex subset ofR n , anda set of finitely or infinitely many quadratic functions. We evaluate the number of iterations which the successive convex relaxation methods require to attain a convex relaxation ofF with a given accuracy e, in… 
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    References

    SHOWING 1-10 OF 19 REFERENCES
    Cones of Matrices and Successive Convex Relaxations of Nonconvex Sets
    TLDR
    The exact equivalence of the SDP relaxation and the semi-infinite convex QOP relaxation proposed originally by Fujie and Kojima is established.
    Successive Convex Relaxation Methods for Nonconvex Quadratic Optimization Problems
    TLDR
    This paper presents practically implementable algorithms, reports numerical results and discusses several important implementation issues of convex relaxation methods for general nonconvex quadratic optimization problems.
    B-347 Towards the Implementation of Successive Convex Relaxation Method for Noncon- vex Quadratic Optimization Problems
    TLDR
    This paper presents practically implementable algorithms, their localized-discretized variants and numerical results for general nonconvex quadratic programs.
    A reformulation-convexification approach for solving nonconvex quadratic programming problems
    TLDR
    This paper considers the class of linearly constrained nonconvex quadratic programming problems, and presents a new approach based on a novel Reformulation-Linearization/Convexification Technique, showing that for many problems, the initial relaxation itself produces an optimal solution.
    A hierarchy of relaxation between the continuous and convex hull representations
    In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into
    Convex analysis and global optimization
    TLDR
    This book presents state-of-the-art results and methodologies in modern global optimization, and has been a staple reference for researchers, engineers, advanced students, and practitioners in various fields of engineering.
    A global optimization algorithm for polynomial programming problems using a Reformulation-Linearization Technique
    This paper is concerned with the development of an algorithm to solve continuous polynomial programming problems for which the objective function and the constraints are specified polynomials. A
    A new reformulation-linearization technique for bilinear programming problems
    TLDR
    This paper is concerned with the development of an algorithm for general bilinear programming problems, and develops a new Reformulation-Linearization Technique (RLT) for this problem, and imbeds it within a provably convergent branch-and-bound algorithm.
    A lift-and-project cutting plane algorithm for mixed 0–1 programs
    We propose a cutting plane algorithm for mixed 0–1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family
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