Piecewise linear approximations in nonconvex nonsmooth optimization

Gaudioso, Manlio; Gorgone, Enrico; Monaco, Maria Flavia

doi:10.1007/s00211-009-0228-4

Cited by 12 publications

(4 citation statements)

References 30 publications

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“…Of particular interest in this sense is the fact that aggregation allows restricting the number of quadratic pieces to any fixed value (as low as two), which may ease concerns about dealing with many dense quadratic constraints in the master problem. We also believe that the use of quadratic models could be usefully extended to bundle methods designed to jointly deal with nonconvexity and nonsmoothness [34,12,14,15].…”

Section: Discussionmentioning

confidence: 99%

Piecewise-quadratic Approximations in Convex Numerical Optimization

Astorino¹,

Frangioni²,

Gaudioso³

et al. 2011

SIAM J. Optim.

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Abstract. We present a bundle method for convex nondifferentiable minimization where the model is a piecewise-quadratic convex approximation of the objective function. Unlike standard bundle approaches, the model only needs to support the objective function from below at a properly chosen (small) subset of points, as opposed to everywhere. We provide the convergence analysis for the algorithm, with a general form of master problem which combines features of trust region stabilization and proximal stabilization, taking care of all the important practical aspects such as proper handling of the proximity parameters and the bundle of information. Numerical results are also reported.

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Section: Discussionmentioning

confidence: 99%

Piecewise-quadratic Approximations in Convex Numerical Optimization

Astorino¹,

Frangioni²,

Gaudioso³

et al. 2011

SIAM J. Optim.

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“…The next point in the bundle is then selected in a way that both under-and overestimates predict a significant improvement. This has recently been proposed by [20].…”

Section: Dealing With Nonconvex Objective Functionsmentioning

confidence: 95%

Nonsmooth cryptanalysis, with an application to the stream cipher MICKEY

Tischhauser

2011

Journal of Mathematical Cryptology

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Abstract.A new approach to the cryptanalysis of symmetric algorithms based on nonsmooth optimisation is presented. We develop this technique as a novel way of dealing with nonlinearity over F 2 by modeling the equations corresponding to the algorithm as a continuous optimisation problem that avoids terms of higher degree. The resulting problems are not continuously differentiable, but can be approached with techniques from nonsmooth analysis. Applied to the stream cipher MICKEY, which is part of the eSTREAM final portfolio, this method can solve instances corresponding to the full cipher, although with time complexity greater than brute force. Finally, we compare this approach to classical pseudo-Boolean programming.

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“…bundle-type methods are developed for convex and nonconvex optimization problems [2][3][4][5][6][7][8][9][10]. The algorithms based on smoothing techniques are presented in [11,12].…”

Section: Introductionmentioning

confidence: 99%

Combination of steepest descent and BFGS methods for nonconvex nonsmooth optimization

Yousefpour

2015

Numer Algor

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In this paper, a method is developed for solving nonsmooth nonconvex minimization problems. This method extends the classical BFGS framework. First, we generalize the Wolfe conditions for locally Lipschitz functions and prove that this generalization is well defined. Then, a line search algorithm is presented to find a step length satisfying the generalized Wolfe conditions. Next, the Goldstein ε-subgradient is approximated by an iterative method and a descent direction is computed using a positive definite matrix. This matrix is updated using the BFGS method. Finally, a minimization algorithm based on the BFGS method is described. The algorithm is implemented in MATLAB and numerical results using it are reported.

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Piecewise linear approximations in nonconvex nonsmooth optimization

Cited by 12 publications

References 30 publications

Piecewise-quadratic Approximations in Convex Numerical Optimization

Piecewise-quadratic Approximations in Convex Numerical Optimization

Nonsmooth cryptanalysis, with an application to the stream cipher MICKEY

Combination of steepest descent and BFGS methods for nonconvex nonsmooth optimization

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