Computing Proximal Points of Convex Functions with Inexact Subgradients

Hare, Warren; Planiden, Chayne

doi:10.1007/s11228-016-0394-3

Cited by 7 publications

(10 citation statements)

References 41 publications

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“…In this section, we examine the convergence of the DFO VU-algorithm, starting with the V-step. By [26,Corollary 4.6], if the V-step never terminates, then lim j→∞ z j+1 − z j = 0.…”

Section: Convergencementioning

confidence: 99%

“…Then [26,Theorem 4.9] states that if f is locally K-Lipschitz (which a finite-max function is), then…”

Section: Convergencementioning

confidence: 99%

“…We now consider how to make implementable the conceptual VU-algorithm in a derivative-free setting as provided by Assumptions 1.4 through 1.6. In order to prove convergence, we make use of the results of [21] and [26]. We use the techniques in [21] to approximate a subgradient, the VU-decomposition, the Ugradient and the U-Hessian for the function f at a point x.…”

Section: End Algorithm 3 Defining Inexact Subgradients and Related Ap...mentioning

confidence: 99%

“…However, in [21] it was shown that the VUdecomposition, U-gradient, and U -Hessian can be approximated numerically with controlled precision for finite-max functions. Moreover, in [26] a derivative-free algorithm for computing proximal points of convex functions that only requires approximate subgradients was developed. Finally, in [24] it was shown how to approximate subgradients for convex finite-max functions using only function values.…”

Section: Introductionmentioning

confidence: 99%

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A derivative-free 𝒱𝒰-algorithm for convex finite-max problems

Hare

Planiden

Sagastizábal

2019

Optimization Methods and Software

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The VU-algorithm is a superlinearly convergent method for minimizing nonsmooth, convex functions. At each iteration, the algorithm works with a certain V-space and its orthogonal U -space, such that the nonsmoothness of the objective function is concentrated on its projection onto the V-space, and on the U -space the projection is smooth. This structure allows for an alternation between a Newtonlike step where the function is smooth, and a proximal-point step that is used to find iterates with promising VU-decompositions. We establish a derivative-free variant of the VU-algorithm for convex finite-max objective functions. We show global convergence and provide numerical results from a proof-of-concept implementation, which demonstrates the feasibility and practical value of the approach. We also carry out some tests using nonconvex functions and discuss the results.

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“…In this section, we examine the convergence of the DFO VU-algorithm, starting with the V-step. By [26,Corollary 4.6], if the V-step never terminates, then lim j→∞ z j+1 − z j = 0.…”

Section: Convergencementioning

confidence: 99%

“…Then [26,Theorem 4.9] states that if f is locally K-Lipschitz (which a finite-max function is), then…”

Section: Convergencementioning

confidence: 99%

Section: End Algorithm 3 Defining Inexact Subgradients and Related Ap...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A derivative-free 𝒱𝒰-algorithm for convex finite-max problems

Hare

Planiden

Sagastizábal

2019

Optimization Methods and Software

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“…The set of all solution points to (1.1) is known as the proximal mapping of f , denoted by Prox f . The proximal mapping is a key component of many optimization algorithms, such as the proximal point method and its variants [5,8,13,16,20,21,37]. Because of the above and other nice features, the Moreau envelope and proximal mapping have been thoroughly researched and applied to many situations in the convex [10,17,24,29,33,35] and nonconvex [22,25,26,28,30,34,39] settings.…”

Section: Introductionmentioning

confidence: 99%

Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness

Planiden¹

2019

Preprint

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This work advances knowledge of the threshold of prox-boundedness of a function; an important concern in the use of proximal point optimization algorithms and in determining the existence of the Moreau envelope of the function. In finite dimensions, we study general prox-bounded functions and then focus on some useful classes such as piecewise functions and Lipschitz continuous functions. The thresholds are explicitly determined when possible and bounds are established otherwise. Some calculus rules are constructed; we consider functions with known thresholds and find the thresholds of their sum and composition.

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Linear Convergence of the Derivative-Free Proximal Bundle Method on Convex Nonsmooth Functions, with Application to the Derivative-Free $\mathcal{VU}$-Algorithm

Planiden,

Rajapaksha

2024

Set-Valued Var. Anal

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Proximal bundle methods are a class of optimisation algorithms that leverage the proximal operator to address nonsmoothness in the objective function efficiently. This study focuses on a derivative-free (DFO) proximal bundle method and one of its applications called the DFO $\mathcal{VU}$ VU -algorithm. These algorithms incorporate approximate proximal points as subprocedures in order to optimise convex nonsmooth functions based on approximated subdifferential information. Interestingly, the classical $\mathcal{VU}$ VU -algorithm, which operates on true subgradient values, achieves superlinear convergence. At each iteration, the algorithm divides the whole space into two: the smooth $\mathcal{U}$ U -space and the nonsmooth $\mathcal{V}$ V -space. It takes a Newton-like step on the $\mathcal{U}$ U -space and a proximal-point step on the $\mathcal{V}$ V -space, enabling it to handle both smooth and nonsmooth parts effectively and converge faster. In this work, we reveal the worst possible convergence rate for the DFO $\mathcal{VU}$ VU -method by showing the linear convergence of the DFO proximal bundle method. This will be done by presenting a suitable framework and using the subdifferential-based error bound on the distance to critical points.

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Computing Proximal Points of Convex Functions with Inexact Subgradients

Cited by 7 publications

References 41 publications

A derivative-free 𝒱𝒰-algorithm for convex finite-max problems

A derivative-free 𝒱𝒰-algorithm for convex finite-max problems

Conditions for the existence, identification and calculus rules of the threshold of prox-boundedness

Linear Convergence of the Derivative-Free Proximal Bundle Method on Convex Nonsmooth Functions, with Application to the Derivative-Free $\mathcal{VU}$-Algorithm

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