Convex Analysis and Minimization Algorithms I

This paper presents an accelerated variant of the hybrid proximal extragradient (HPE) method for convex optimization, referred to as the accelerated HPE (A-HPE) framework. Iterationcomplexity results are established for the A-HPE framework, as well as a special version of it, where a large stepsize condition is imposed. Two specific implementations of the A-HPE framework are described in the context of a structured convex optimization problem whose objective function consists of the sum of a smooth convex function and an extended real-valued non-smooth convex function. In the first implementation, a generalization of a variant of Nesterov's method is obtained for the case where the smooth component of the objective function has Lipschitz continuous gradient. In the second implementation, an accelerated Newton proximal extragradient (A-NPE) method is obtained for the case where the smooth component of the objective function has Lipschitz continuous Hessian. It is shown that the A-NPE method has a O(1/k 7/2 ) convergence rate, which improves upon the O(1/k 3 ) convergence rate bound for another accelerated Newton-type method presented by Nesterov. Finally, while Nesterov's method is based on exact solutions of subproblems with cubic regularization terms, the A-NPE method is based on inexact solutions of subproblems with quadratic regularization terms, and hence is potentially more tractable from a computational point of view.

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“…Statement a) is proved in [2, Proposition 3] and b) is a classical result which can be found, for example, in Proposition 4.2.2 of [5].…”

Section: Basic Definition and Notationmentioning

confidence: 99%

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

Monteiro¹,

Svaiter²

2013

SIAM J. Optim.

108

159

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“…Readers not familiar with this field may use [17] (especially its Chap. C) for an elementary introduction, while [16,22] are more complete.…”

Section: Introducing Cut-generating Functionsmentioning

confidence: 99%

Cut-Generating Functions and S-Free Sets

et al. 2015

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We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To this aim, we introduce a concept: cut-generating functions (cgf) and we develop a formal theory for them, largely based on convex analysis. They are intimately related to S-free sets and we study this relation, disclosing several definitions for minimal cgf's and maximal S-free sets. Our work unifies and puts in perspective a number of existing works on S-free sets; in particular, we show how cgf's recover the celebrated Gomory cuts.

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“…The bound z 0 (R) is easily computed after computing the highest distance (following · a − ) from each a ∈ A to the rectangle R, and the projection (following · a + ) of such points onto R. Since the function · −a a − is convex, it attains its maximum on R at some vertex of R. Hence finding max x∈R x − a a − reduces to evaluating the four vertices of R. On the other hand, although calculating min x∈R x − a a + amounts in general to solve a convex programme, [12], a straightforward procedure has been suggested for the case in which · a + is an l p norm, see [8,17] for details.…”

Section: Remarkmentioning

confidence: 99%

“…• Let (x * , t * ) be optimal for (12). Then x * is optimal for (11), since for anŷ x feasible for (11), definingt = ( x − a a − ) a∈A , one obtains…”

Section: Remarkmentioning

confidence: 99%

See 1 more Smart Citation

Semi-obnoxious location models: A global optimization approach

Morales

Carrizosa

Conde

1997

European Journal of Operational Research

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In the last decades there has been an increasing interest in environmental topics. This interest has been reflected in modeling the location of obnoxious facilities, as shown by the important number of papers published in this field. However, a very common drawback of the existing literature is that, as soon as environmental aspects are taken into account, economical considerations (e.g. transportation costs) are ignored, leading to models with dubious practical interest. In this paper we take into account both the environmental impact and the transportation costs caused by the location of an obnoxious facility, and propose as solution method of the well-known BSSS, with a new bounding scheme which exploits the structure of the problem.

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Convex Analysis and Minimization Algorithms I

Cited by 1,373 publications

References 0 publications

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

An Accelerated Hybrid Proximal Extragradient Method for Convex Optimization and Its Implications to Second-Order Methods

Cut-Generating Functions and S-Free Sets

Semi-obnoxious location models: A global optimization approach

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