A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization

Fuduli, Antonio; Gaudioso, Manlio; Giallombardo, Giovanni

doi:10.1080/10556780410001648112

Cited by 52 publications

(22 citation statements)

References 13 publications

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“…The consequence is that bundle algorithms are necessarily much more complicated in the nonconvex case. Other contributions to nonconvex bundle methods since Kiwiel's book was published in 1985 include [FGG02,Gro02,LSB91,LV98,MN92,OKZ98,SZ92]. Despite this activity in the field, the only publicly available nonconvex bundle software of which we are aware are the Bundle Trust (BT) fortran code dating from 1991 [SZ92] and some more recent fortran codes of [LV98].…”

Section: Introductionmentioning

confidence: 99%

A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

2005

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Let f be a continuous function on R n , and suppose f is continuously differentiable on an open dense subset. Such functions arise in many applications, and very often minimizers are points at which f is not differentiable. Of particular interest is the case where f is not convex, and perhaps not even locally Lipschitz, but whose gradient is easily computed where it is defined. We present a practical, robust algorithm to locally minimize such functions, based on gradient sampling. No subgradient information is required by the algorithm.When f is locally Lipschitz and has bounded level sets, and the sampling radius ǫ is fixed, we show that, with probability one, the algorithm generates a sequence with a cluster point that is Clarke ǫ-stationary. Furthermore, we show that if f has a unique Clarke stationary pointx, then the set of all cluster points generated by the algorithm converges tox as ǫ is reduced to zero.

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Section: Introductionmentioning

confidence: 99%

A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

2005

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“…The present trust region method should be compared to the approach of Fuduli, Gaudioso, and Giallombardo [20,21] for general nonsmooth and nonconvex locally Lipschitz functions, where the authors design a trust region with the help of the first order affine approximations a(y) = g (y − y k+1 ) + f (y k+1 ), g ∈ ∂f (y k+1 ) of the objective f at the trial points y k+1 . As these affine models are not support functions to the objective, the authors classify them according to whether a(x) > f (x) or a(x) ≤ f (x), using this information to devise a trust region around the current x.…”

Section: Elements From Nonsmooth Analysismentioning

confidence: 99%

A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization

Apkarian¹,

Noll²,

Prot³

2008

SIAM J. Optim.

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Abstract. We present a nonsmooth optimization technique for nonconvex maximum eigenvalue functions and for nonsmooth functions which are infinite maxima of eigenvalue functions. We prove global convergence of our method in the sense that for an arbitrary starting point, every accumulation point of the sequence of iterates is critical. The method is tested on several problems in feedback control synthesis. Here our interest is in nonconvex eigenvalue programs, which arise frequently in automatic control applications and especially in controller synthesis. In particular, solving bilinear matrix inequalities (BMIs) is a prominent application, which may be addressed via nonconvex eigenvalue optimization. In [45,46] we have shown how to adapt the approach of [41,48] to handle nonconvex situations. Applications and extensions of these ideas are presented in [4,1,57,10,5,6].

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“…In literature, differently from the convex case, there are few algorithms tackling problem (4), when f is nonconvex. Some references are [10,17,19,24]. In particular we focus on the recent approach reported in [11], which is an extension to nonconvex functions of a classical bundle method.…”

Section: Bundle Methods For Nonsmooth Minimizationmentioning

confidence: 99%

Nonsmooth Optimization Techniques for Semisupervised Classification

Astorino

Fuduli

2007

IEEE Trans. Pattern Anal. Mach. Intell.

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We apply nonsmooth optimization techniques to classification problems, with particular reference to the TSVM (Transductive Support Vector Machine) approach, where the considered decision function is nonconvex and nondifferentiable and then difficult to minimize.We present some numerical results obtained by running the proposed method on some standard test problems drawn from the binary classification literature.

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A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization

Cited by 52 publications

References 13 publications

A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization

A Trust Region Spectral Bundle Method for Nonconvex Eigenvalue Optimization

Nonsmooth Optimization Techniques for Semisupervised Classification

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