Piecewise-quadratic Approximations in Convex Numerical Optimization

Astorino, Annabella; Frangioni, Antonio; Gaudioso, Manlio; Gorgone, Enrico

doi:10.1137/100817930

Cited by 29 publications

(34 citation statements)

References 29 publications

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“…The dual for (4) is also very similar in structure to the duals of (2) or (3). Thus, the subproblem (4) is no harder (or at least, cannot be much harder) to solve than (2) or (3). Moreover, it turns out that the (unique) solution to problem (4) is also a solution to at least one of the problems (2) or (3); see Lemma 1 below.…”

Section: Introductionmentioning

confidence: 98%

“…As is well known, for the same modelf k and the same stability centerx k , one can find the proximal and level parameters τ k and ℓ k such that the two versions above generate the same next iterate x k+1 (i.e., for some choice of parameters, the solution of the subproblems (2) and (3) is the same). In this (formal, theoretical) sense the two approaches can be considered equivalent.…”

Section: Introductionmentioning

confidence: 99%

“…We immediately comment that (4) can be reformulated as a quadratic program (if X is polyhedral), just like (2) or (3), with just one extra scalar bound constraint compared to (2), or one extra scalar variable and scalar bound constraint compared to (3); see (8) below. The dual for (4) is also very similar in structure to the duals of (2) or (3).…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, the only other bundle-type method which employs some kind of double stabilization is [2], where the proximal and trust-region features are present for piecewise quadratic models of f . However, the motivation for this and the resulting algorithm are rather different from ours.…”

Section: Introductionmentioning

confidence: 99%

“…However, the motivation for this and the resulting algorithm are rather different from ours. For example, the subproblems in [2] are that of minimizing a quadratic function subject to quadratic constraints.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

A doubly stabilized bundle method for nonsmooth convex optimization

Oliveira

Solodov

2015

Math. Program.

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We propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations. Most bundle algorithms use a cutting-plane model of the objective function to formulate a subproblem whose solution gives the next iterate. Proximal bundle methods employ the model in the objective function of the subproblem, while level methods put the model in the subproblem's constraints. The proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints. One advantage when compared to the proximal approach is that the level set constraint provides a certain Lagrange multiplier, which is used to update the proximal parameter in a novel manner. We also show that in the case of inexact function and subgradient evaluations, no additional procedure needs to be performed by our variant to deal with inexactness (as opposed to the proximal bundle methods that require special modifications). Numerical experiments on almost one thousand instances of different types of problems are presented. Our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions, and compares favorably to both of them.Keywords nonsmooth optimization · proximal bundle method · level bundle method · inexact oracle.Mathematics Subject Classification (2010) 90C25 · 90C30 · 65K05.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A doubly stabilized bundle method for nonsmooth convex optimization

Oliveira

Solodov

2015

Math. Program.

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Łukasiewicz Logic and Artificial Neural Networks

Nola

Vitale

2020

Studies in Computational Intelligence

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Standard Bundle Methods: Untrusted Models and Duality

Frangioni

2020

Numerical Nonsmooth Optimization

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We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods". In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function.

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Piecewise-quadratic Approximations in Convex Numerical Optimization

Cited by 29 publications

References 29 publications

A doubly stabilized bundle method for nonsmooth convex optimization

A doubly stabilized bundle method for nonsmooth convex optimization

Łukasiewicz Logic and Artificial Neural Networks

Standard Bundle Methods: Untrusted Models and Duality

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