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    Guaranteed accuracy for conic programming problems in vector lattices
    (2007-07-30)
    Jansson, Christian
    This paper presents rigorous forward error bounds for linear conic optimization problems. The error bounds are formulated in a quite general framework; the underlying vector spaces are not required to be finite-dimensional, and the convex cones defining the partial ordering are not required to be polyhedral. In the case of linear programming, second order cone programming, and semidefinite programming specialized formulas are deduced yielding guaranteed accuracy. All computed bounds are completely rigorous because all rounding errors due to floating point arithmetic are taken into account. Numerical results, applications and software for linear and semidefinite programming problems are described.
    Publicationtype: Journal Article
      34
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    VSDP: Verified SemiDefinite Programming : user's guide ; beta version 0.1 for MATLAB 7.0
    (Optimization Technology Center, 2006)
    Jansson, Christian
    VSDP is a MATLAB software package for rigorously solving semidefinite programming problems. It expresses these problems in a notation closely related to the form given in textbooks and scientific papers. Functions for computing verified forward error bounds of the true optimal value and verified certificates of feasibility and infeasibility are provided. All rounding errors due to floating point arithmetic are taken into account. Computational results are given, including results for the SDPLIB benchmark problems. This package supports interval input data and sparse format.
    Publicationtype: Learning Object
      42
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    Rigorous results in combinatorial optimization
    (2006)
    Jansson, Christian
    Many current deterministic solvers for NP-hard combinatorial optimization problems are based on nonlinear relaxation techniques that use floating point arithmetic. Occasionally, due to solving these relaxations, rounding errors may produce erroneous results, although the deterministic algorithm should compute the exact solution in a finite number of steps. This may occur especially if the relaxations are illconditioned or ill-posed, and if Slater’s constraint qualifications fail. We show how verified results can be obtained by rigorously bounding the optimal value of nonlinear semidefinite relaxations, even in the ill-posed case. All rounding errors due to floating point arithmetic are taken into account.
    Publicationtype: Conference Paper
      283  149
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    Termination and verification for ill-posed semidefinite programming problems
    (2005)
    Jansson, Christian
    We investigate ill-posed semidefinite programming problems for which Slater's constraint qualifications fail, and propose a new reliable termination criterium dealing with such problems. This criterium is scale-independent and provides verified forward error bounds for the true optimal value, where all rounding errors due to floating point arithmetic are taken into account. It is based on a boundedness qualification, which is satisfied for many problems. Numerical experiments, including combinatorial problems, are presented.
    Publicationtype: Working Paper
      310  212
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    A branch and bound algorithm for bound constrained optimization problems without derivatives
    (Springer, 1995-10-01)
    Jansson, Christian
    ;
    Knüppel, Olaf
    In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most "uninteresting" parts of the search domain and concentrates on its "promising" subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given. © 1995 Kluwer Academic Publishers.
    Publicationtype: Journal Article
      50