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Kaplan, A.; Tichatschke, R.

doi:10.1023/a:1008321423879

Cited by 79 publications

(9 citation statements)

References 24 publications

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“…In addition, given f ∈ Γ c , we have already seen that the sequence {x k } generated by (14) can be interpreted as a sequence obtained from applying the gradient descent to the θ-envelope e θ λ f . By Corollary 2, we know that ∇e θ λ f is L-Lipschitz which implies that e θ λ f satisfies (15) and thus the following result follows readily. Proposition 6.…”

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confidence: 76%

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A regularization interpretation of the proximal point method for weakly convex functions

Hoheisel¹,

Laborde²,

Oberman³

2020

Journal of Dynamics &Amp; Games

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Empirical evidence and theoretical results suggest that the proximal point method can be computed approximately and still converge faster than the corresponding gradient descent method, in both the stochastic and exact gradient case. In this article we provide a perspective on this result by interpreting the method as gradient descent on a regularized function. This perspective applies in the case of weakly convex functions where proofs of the faster rates are not available. Using this analysis we find the optimal value of the regularization parameter in terms of the weak convexity.2010 Mathematics Subject Classification. Primary: 26B25, 65K10; Secondary: 90C25.

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confidence: 76%

“…Let c > 0 and f ∈ Γ c . In addition, assume that f is a C 1 , coercive bounded from below function satisfying (15). Now, let {x k } be generated by (14).…”

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confidence: 99%

A regularization interpretation of the proximal point method for weakly convex functions

Hoheisel¹,

Laborde²,

Oberman³

2020

Journal of Dynamics &Amp; Games

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“…Thus, the method of proximal gradient [17] can be employed for its minimization. The method is more commonly employed in convex optimization problems, but it can also be employed in non-convex problems [19] as it is the case in this work. Defining the vector φ that contains all the phase variables of our optimization problem, we also define…”

Section: Derivation Of the Proximal Gradient Algorithmmentioning

confidence: 99%

Proximal Algorithms for Discrete-Level Phase-Shifting Mask Design with Application to Optogenetics

et al. 2021

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This work studies the problem of designing computer-generated holograms using phase-shifting masks limited to represent only a small number of discrete phase levels. This problem has various applications, notably in the emerging field of optogenetics and lithography. A novel regularized cost function is proposed for the problem at hand that penalizes the unfeasible phase levels. Since the proposed cost function is non-smooth, we derive proper proximal gradient algorithms for its minimization. Simulation results, considering an optogenetics application, demonstrate that the proposed proximal gradient algorithm yields better performance as compared to other algorithms proposed in the literature.

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“…In [15], a proximal bundle method was presented for an unconstrained non-convex problem, which was subsequently applied to the nonsmooth non-convex constrained optimization problem using the improvement function in [12,13]. In view of the applications in [7,10,9,14,20,26,28], it has been shown that the proximal point method is an efficient tool that can transform certain problems into a sequence of easily solvable sub-problems, and many popular algorithms have been adopted. However, the global optimization issue has seldom been addressed in these studies.…”

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confidence: 99%

“…Based on the early research work on unconstrained non-convex problems, which was proposed by Kaplan and Tichatschke in [14], in this study, the proximal point method is extended to constrained non-convex problems and corresponding new theoretical results are proposed, especially regarding the relationship between the initial point and global optimization. The main contributions of this study are as follows:…”

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confidence: 99%

Global optimization for non-convex programs via convex proximal point method

Zhao

Xing

2023

JIMO

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<p style='text-indent:20px;'>In this study, a convex proximal point algorithm (CPPA) is considered for solving constrained non-convex problems, and new theoretical results are proposed. It is proved that every cluster point of CPPA is a stationary point, and the initial point of CPPA is key to global optimization. Several sufficient conditions for the initial point selection are provided for CPPA to find the global minimum. Motivated by these results, numerical experiments were conducted on non-convex quadratic programming problems with convex quadratic constraints. The performance of CPPAs was compared, with the initial point randomly selected or obtained through the Lagrangian dual problem. The numerical results demonstrate that the quality of the CPPA with the computed Lagrangian dual initial point is much better than that with the random initial point, in terms of the objective function value.</p>

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Untitled

Cited by 79 publications

References 24 publications

A regularization interpretation of the proximal point method for weakly convex functions

A regularization interpretation of the proximal point method for weakly convex functions

Proximal Algorithms for Discrete-Level Phase-Shifting Mask Design with Application to Optogenetics

Global optimization for non-convex programs via convex proximal point method

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