A strongly convergent proximal bundle method for convex minimization in Hilbert spaces

Ackooij, Wim van; Bello-Cruz, Yunier; Oliveira, Welington de

doi:10.1080/02331934.2015.1004549

Cited by 10 publications

(11 citation statements)

References 31 publications

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“…A proximal bundle algorithm can stop with a satisfactory solution

{\overset{g}{true}}^{l}

when both e k and ∥ q k ∥ are small (van Ackooij, Cruz, & de Oliveira, 2016), where

e^{k} = {normalℒ}_{frakturB}^{k} false(g^{k + 1} false) + < q^{k}, {\overset{g}{true}}^{l} - g^{k + 1} > - ℒ false({\overset{true^}{g}}^{l} false) = δ_{k} - ∥ {\overset{g}{true}}^{l} - g^{k + 1} ∥^{2} / t_{k}

is the aggregate linearization error and

q^{k} = \frac{g^{k + 1} - {\overset{g}{true}}^{l}}{t_{k}}

. We define the optimality gap as (best upper bound‐best lower bound)/best upper bound, that is,

false(UB - ℒ false({\overset{true^}{g}}^{l + 1} false) false) / UB

.…”

Section: The Proximal Bundle Methodsmentioning

confidence: 99%

Two‐agent scheduling with linear resource‐dependent processing times

Wang

Qiu

et al. 2020

Naval Research Logistics

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This paper considers a two‐agent scheduling problem with linear resource‐dependent processing times, in which each agent has a set of jobs that compete with that of the other agent for the use of a common processing machine, and each agent aims to minimize the weighted number of its tardy jobs. To meet the due date requirements of the jobs of the two agents, additional amounts of a common resource, which may be in discrete or continuous quantities, can be allocated to the processing of the jobs to compress their processing durations. The actual processing time of a job is a linear function of the amount of the resource allocated to it. The objective is to determine the optimal job sequence and resource allocation strategy so as to minimize the weighted number of tardy jobs of one agent, while keeping the weighted number of tardy jobs of the other agent, and the total resource consumption cost within their respective predetermined limits. It is shown that the problem is NP‐hard in the ordinary sense, and there does not exist a polynomial‐time approximation algorithm with performance ratio unless P=NP; however it admits a relaxed fully polynomial time approximation scheme. A proximal bundle algorithm based on Lagrangian relaxation is also presented to solve the problem approximately. To speed up convergence and produce sharp bounds, enhancement strategies including the design of a Tabu search algorithm and integration of a Lagrangian recovery heuristic into the algorithm are devised. Extensive numerical studies are conducted to assess the effectiveness and efficiency of the proposed algorithms.

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“…A proximal bundle algorithm can stop with a satisfactory solution

{\overset{g}{true}}^{l}

when both e k and ∥ q k ∥ are small (van Ackooij, Cruz, & de Oliveira, 2016), where

e^{k} = {normalℒ}_{frakturB}^{k} false(g^{k + 1} false) + < q^{k}, {\overset{g}{true}}^{l} - g^{k + 1} > - ℒ false({\overset{true^}{g}}^{l} false) = δ_{k} - ∥ {\overset{g}{true}}^{l} - g^{k + 1} ∥^{2} / t_{k}

is the aggregate linearization error and

q^{k} = \frac{g^{k + 1} - {\overset{g}{true}}^{l}}{t_{k}}

. We define the optimality gap as (best upper bound‐best lower bound)/best upper bound, that is,

false(UB - ℒ false({\overset{true^}{g}}^{l + 1} false) false) / UB

.…”

Section: The Proximal Bundle Methodsmentioning

confidence: 99%

Two‐agent scheduling with linear resource‐dependent processing times

Wang

Qiu

et al. 2020

Naval Research Logistics

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“…As already mentioned, in this paper we are interested in weak and strong convergence of projected gradient methods applied to convex programs as (1). To the best of our knowledge, weak convergence of the projected gradient method has only been shown under the assumption of Lipschitz continuity of ∇f or using exogenous stepsize, like Strategy (d) above.…”

Section: Projected Gradient Methods Initializationmentioning

confidence: 99%

“…We now recall some necessary and sufficient optimality conditions for problem (1), whose proof can be found in [7,Prop. 17.4].…”

Section: Preliminariesmentioning

confidence: 99%

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On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

Bello-Cruz

Oliveira

2016

Numerical Functional Analysis and Optimization

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This work focuses on convergence analysis of the projected gradient method for solving constrained convex minimization problems in Hilbert spaces. We show that the sequence of points generated by the method employing the Armijo line search converges weakly to a solution of the considered convex optimization problem. Weak convergence is established by assuming convexity and Gateaux differentiability of the objective function, whose Gateaux derivative is supposed to be uniformly continuous on bounded sets. Furthermore, we propose some modifications in the classical projected gradient method in order to obtain strong convergence. The new variant has the following desirable properties: the sequence of generated points is entirely contained in a ball with diameter equal to the distance between the initial point and the solution set, and the whole sequence converges strongly to the solution of the problem that lies closest to the initial iterate. Convergence analysis of both methods is presented without Lipschitz continuity assumption.

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“…There is a lot of prior work related to the method proposed in this paper. The work on variable metric bundle methods [9,18,25,36,37,[40][41][42]45,49,50,54,61,63,66,70] and (inexact) proximal Newton-type methods [8,11,12,27,38,39,43,44,51,59,60,62,71] are probably the most closely related, though our method differs in key ways arising from our assumed access methods. Proximal Newton methods are similar in philosophy to our approach, as both types of methods really shine when the structured part of the objective can be minimized efficiently.…”

Section: Related Workmentioning

confidence: 99%

Minimizing Oracle-Structured Composite Functions

Shen

Ali

Boyd

2021

Preprint

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We consider the problem of minimizing a composite convex function with two different access methods: an oracle, for which we can evaluate the value and gradient, and a structured function, which we access only by solving a convex optimization problem. We are motivated by two associated technological developments. For the oracle, systems like PyTorch or TensorFlow can automatically and efficiently compute gradients, given a computation graph description. For the structured function, systems like CVXPY accept a high level domain specific language description of the problem, and automatically translate it to a standard form for efficient solution. We develop a method that makes minimal assumptions about the two functions, does not require the tuning of algorithm parameters, and works well in practice across a variety of problems. Our algorithm combines a number of well-known ideas, including a low-rank quasi-Newton approximation of curvature, piecewise affine lower bounds from bundle-type methods, and two types of damping to ensure stability. We illustrate the method on stochastic optimization, utility maximization, and risk-averse programming problems.

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A strongly convergent proximal bundle method for convex minimization in Hilbert spaces

Cited by 10 publications

References 31 publications

Two‐agent scheduling with linear resource‐dependent processing times

Two‐agent scheduling with linear resource‐dependent processing times

On Weak and Strong Convergence of the Projected Gradient Method for Convex Optimization in Real Hilbert Spaces

Minimizing Oracle-Structured Composite Functions

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