Double Bundle Method for finding Clarke Stationary Points in Nonsmooth DC Programming

Joki, Kaisa; Bagirov, Adil M.; Karmitsa, Napsu; Mäkelä, Marko M.; Taheri, Sona

doi:10.1137/16m1115733

Cited by 50 publications

(50 citation statements)

References 30 publications

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“…The bundle philosophy has been extensively used in handling DC optimization, introducing the cutting plane model for f (1) and/or f (2) . Some recent references are Astorino and Miglionico (2016), de Oliveira (2019), de Oliveira (2020), Gaudioso et al (2018b), Gaudioso et al (2020a), Gaudioso et al (2020b), Joki et al (2018).…”

Section: Nonconvex Nso and DC Programmingmentioning

confidence: 99%

Essentials of numerical nonsmooth optimization

2022

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Approximately sixty years ago two seminal findings, the cutting plane and the subgradient methods, radically changed the landscape of mathematical programming. They provided, for the first time, the practical chance to optimize real functions of several variables characterized by kinks, namely by discontinuities in their derivatives. Convex functions, for which a superb body of theoretical research was growing in parallel, naturally became the main application field of choice. The aim of the paper is to give a concise survey of the key ideas underlying successive development of the area, which took the name of numerical nonsmooth optimization. The focus will be, in particular, on the research mainstreams generated under the impulse of the two initial discoveries.

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Section: Nonconvex Nso and DC Programmingmentioning

confidence: 99%

Essentials of numerical nonsmooth optimization

2022

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“…However, one major drawback of a critical point is that it does not need to be a local optimum or even a saddle point. In the worst case, the algorithm may stop at a point, where the original DC function f is differentiable and the opposite of the gradient of f constructs a descent direction decreasing the value of f [65].…”

Section: Single-objective DC Optimization Problemmentioning

confidence: 99%

Bundle Methods for Nonsmooth DC Optimization

Joki

Bagirov

2020

Numerical Nonsmooth Optimization

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Due to the complexity of many practical applications, we encounter optimization problems with nonsmooth functions, that is, functions which are not continuously differentiable everywhere. Classical gradient-based methods are not applicable to solve such problems, since they may fail in the nonsmooth setting. Therefore, it is imperative to develop numerical methods specifically designed for nonsmooth optimization. To date, bundle methods are considered to be the most efficient and reliable general purpose solvers for this type of problems.

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“…In the last years, nonsmooth-tailored DC programming has experienced a lot of attention as it has a lot of practical applications (see [28,42]). In fact, several nonsmooth DC algorithms have been developed ( [30,[43][44][45][46][47]).…”

Section: Solving Dc-mil Using a Nonsmooth DC Algorithmmentioning

confidence: 99%

SVM-Based Multiple Instance Classification via DC Optimization

et al. 2019

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A multiple instance learning problem consists of categorizing objects, each represented as a set (bag) of points. Unlike the supervised classification paradigm, where each point of the training set is labeled, the labels are only associated with bags, while the labels of the points inside the bags are unknown. We focus on the binary classification case, where the objective is to discriminate between positive and negative bags using a separating surface. Adopting a support vector machine setting at the training level, the problem of minimizing the classification-error function can be formulated as a nonconvex nonsmooth unconstrained program. We propose a difference-of-convex (DC) decomposition of the nonconvex function, which we face using an appropriate nonsmooth DC algorithm. Some of the numerical results on benchmark data sets are reported.

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Double Bundle Method for finding Clarke Stationary Points in Nonsmooth DC Programming

Cited by 50 publications

References 30 publications

Essentials of numerical nonsmooth optimization

Essentials of numerical nonsmooth optimization

Bundle Methods for Nonsmooth DC Optimization

SVM-Based Multiple Instance Classification via DC Optimization

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