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    On verified numerical computations in convex programming
    (Springer, 2009-10)
    Jansson, Christian
    This survey contains recent developments for computing verified results of convex constrained optimization problems, with emphasis on applications. Especially, we consider the computation of verified error bounds for non-smooth convex conic optimization in the framework of functional analysis, for linear programming, and for semidefinite programming. A discussion of important problem transformations to special types of convex problems and convex relaxations is included. The latter are important for handling and for reliability issues in global robust and combinatorial optimization. Some remarks on numerical experiences, including also large-scale and ill-posed problems, and software for verified computations concludes this survey.
    Publicationtype: Journal Article
      46
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    A classical interpretation of quantum mechanics and the measurement problem
    (2008-01-21)
    Jansson, Christian
    In this paper a didactic approach is described which immediately leads to an understanding of those postulates of quantum mechanics used most frequently in quantum computation. Moreover, an interpretation of quantum mechanics is presented which is motivated by retaining the point of view of classical mechanics as much as possible, and which is consistent with relativity theory. Everything can be written down in terms of well-known mathematical formulations that can be found in every textbook about quantum mechanics. Therefore, in this version, almost no formulas are used.
    Publicationtype: Journal Article
      31
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    Rigorous error bounds for the optimal value in semidefinite programming
    (Society for Industrial & Applied Mathematics (SIAM), 2007-12-01)
    Jansson, Christian
    ;
    Chaykin, Denis
    ;
    Keil, Christian
    A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many ill-conditioned and real-life problems.
    Publicationtype: Journal Article
      43
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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 : a MATLAB software package for verified semidefinite programming
    (Institute of Electronics, Information and Communication Engineers, 2006)
    Jansson, Christian
    VSDP is a MATLAB software package for solving rigorously semidefinite programming problems. 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.
    Publicationtype: Journal Article
      80
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    On verification of Ill-posed optimization problems
    (Lappeenranta Univ. of Technology, 2006)
    Jansson, Christian
    Publicationtype: Journal Article
      24
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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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    On verifed computation in combinatorial optimization
    (2005)
    Jansson, Christian
    Publicationtype: Conference Paper
      36
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    Item type:Publication, Access status: with files ,
    Rigorous error bounds for the optimal value in semidefinite programming
    (2005)
    Jansson, Christian
    ;
    Keil, Christian
    A wide variety of problems in global optimization, combinatorial optimization as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library; these libraries contain many ill-conditioned and real life problems.
    Publicationtype: Working Paper
      803  742