Symbolic Fenchel Conjugation

Borwein, Jonathan M.; Hamilton, Chris

doi:10.1007/s10107-007-0134-4

Cited by 17 publications

(12 citation statements)

References 7 publications

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“…Then, the function g(Ax) is a #-self-concordant barrier on the set A À1 (C) (see [7,Theorem 4.2.3]). Thus, Fðx 1 , x 2 , x 3 Þ :¼ gðAð 1 ðX Þ, 2 ðX ÞÞ T Þ þ gðAð 2 ðX Þ, 1 ðX ÞÞ T Þ is a 2(1 þ 6#)-self-concordant barrier function on À1 (A À1 (C) \ (A À1 (C)) T satisfying (2) and (3), where…”

Section: 22]) Andmentioning

confidence: 95%

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Spectral self-concordant functions in the space of two-by-two symmetric matrices

Peña

Sendov

2011

Optimization

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Section: 22]) Andmentioning

confidence: 95%

“…(For a discussion on conjugate computation, see, e.g. [1,2].) Therefore, our result provides an algorithmic tool for new convex optimization problems with the kind of separable structure encountered in geometric programming and entropy optimization as discussed, e.g.…”

Section: Introductionmentioning

confidence: 94%

Spectral self-concordant functions in the space of two-by-two symmetric matrices

Peña

Sendov

2011

Optimization

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“…For commonly used functions, symbolic computation software allows to perform some computation. Maple implementations were presented in [26] for the one-dimensional case, and in [40] for the multi-dimensional case. Large classes of functions can now be considered and some explicit formulas for the conjugate have been found using these packages.…”

Section: Computer-aided Convex Analysismentioning

confidence: 99%

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

Lucet¹

2010

SIAM Rev.

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Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of Computational Convex Analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing (computing the distance transform, the generalized distance transform, and mathematical morphology operators), partial differential equations (solving Hamilton-Jacobi equations, and using differential equations numerical schemes to compute the convex envelope), max-plus algebra (computing the equivalent of the Fast Fourier Transform), multi-fractal analysis, etc. The fields of applications include, among others, computer vision, robot navigation, thermodynamics, electrical networks, medical imaging, and network communication.

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“…See [Luc10] for historical notes and a survey of numerous applications. While pure symbolic computation was considered [BM06, BH06,BH08], most work in computational convex analysis use a hybrid symbolic-numerical framework that considers a specific class of functions e.g. piecewise linear or piecewise linear-quadratic functions [RW09].…”

Section: Introductionmentioning

confidence: 99%

Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time

Kumar¹,

Lucet²

2018

Set-Valued Var. Anal

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The -subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be performed in logarithmic worst-case time. Furthermore, a new algorithm to compute the entire graph of the -subdifferential in linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined function with little restrictions. This is a pre-print of an article published in Set-Valued and Variational Analysis (SVAA). The final authenticated version is available online at: http://dx.

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Symbolic Fenchel Conjugation

Cited by 17 publications

References 7 publications

Spectral self-concordant functions in the space of two-by-two symmetric matrices

Spectral self-concordant functions in the space of two-by-two symmetric matrices

What Shape Is Your Conjugate? A Survey of Computational Convex Analysis and Its Applications

Computation of the Epsilon-Subdifferential of Convex Piecewise Linear-Quadratic Functions in Optimal Worst-Case Time

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