Variational Analysis of Composite Models with Applications to Continuous Optimization

Mohammadi, Ashkan; Mordukhovich, Boris S.; Sarabi, M. Ebrahim

doi:10.48550/arxiv.1905.08837

Cited by 13 publications

(47 citation statements)

References 28 publications

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“…As promised, now we establish such an estimate for the general case of continuously prox-regular functions. The following proposition and its proof extend those given in [23,Theorem 4.1]. The obtained result shows that subproblem (7.3) always admit an optimal solution under the tilt stability assumption.…”

Section: Applications To Constrained Optimizationsupporting

confidence: 77%

“…This class strictly includes all the C 2 -cone reducible sets in the sense of [2,Definition 3.135] and encompasses convex polyhedra, the second-order cone, the cone of symmetric and positive semidefinite matrices, etc. Furthermore, parabolic regularity, combined with related developments of [23], occurs to be very instrumental in the study and calculations of second subderivatives and twice epi-differentiability of functions while being employed in our numerical applications given below.…”

Section: Applications To Constrained Optimizationmentioning

confidence: 99%

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Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

Mordukhovich¹,

Ebrahim²

2020

Preprint

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This paper aims at developing two versions of the generalized Newton method to compute not merely arbitrary local minimizers of nonsmooth optimization problems but just those, which possess an important stability property known as tilt stability. We start with unconstrained minimization of continuously differentiable cost functions having Lipschitzian gradients and suggest two second-order algorithms of the Newton type: one involving coderivatives of Lipschitzian gradient mappings, and the other based on graphical derivatives of the latter. Then we proceed with the propagation of these algorithms to minimization of extended-real-valued prox-regular functions, while covering in this way problems of constrained optimization, by using Moreau envelops. Employing advanced techniques of second-order variational analysis and characterizations of tilt stability allows us to establish the solvability of subproblems in both algorithms and to prove the Q-superlinear convergence of their iterations.

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Section: Applications To Constrained Optimizationsupporting

confidence: 77%

Section: Applications To Constrained Optimizationmentioning

confidence: 99%

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

Mordukhovich¹,

Ebrahim²

2020

Preprint

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“…Twice epi-differentiability has been recognized as an important property in second-order variational analysis with numerous applications to optimization; see the aforemention monograph by Rockafellar and Wets and the recent papers [38,39,40] developing a systematic approach to verify epidifferentiability via parabolic regularity, which is a major second-order property of sets and extendedreal-valued functions.…”

Section: Example 52 (Convex Clustering Problems)mentioning

confidence: 99%

Generalized Damped Newton Algorithms in Nonsmooth Optimization via Second-Order Subdifferentials

Khanh¹,

Mordukhovich²,

Phat³

et al. 2021

Preprint

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The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we design such a globally convergent algorithm to solve a class of nonsmooth convex composite problems with extended-real-valued cost functions, which typically arise in machine learning and statistics. Finally, the obtained algorithmic developments and justifications are applied to solving a major class of Lasso problems with detailed numerical implementations. We present the results of numerical experiments and compare the performance of our main algorithm applied to Lasso problems with those achieved by other first-order and second-order methods.

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“…One of the important novel features of this approach is the possibility to obtain some quantitative estimates for adjoint functions as discussed in Remark 5.2. We also plan to implement this approach to deriving second-order optimality conditions for variational problems by extending to infinite dimensions and further developing the recent results of second-order variational analysis achieved in [10,11].…”

Section: Conclusion {Concl}mentioning

confidence: 99%

Optimality Conditions for Variational Problems in Incomplete Functional Spaces

Mohammadi¹,

Mordukhovich²

2021

Preprint

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This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach consists of reducing a variational problem to a (nondynamic) problem of constrained optimization in a normed space and then applying the results recently obtained for the latter class by using generalized differentiation. In this way we derive necessary optimality conditions for nonconvex problems of the calculus of variations with velocity constraints under the weakest metric subregularity-type constraint qualification. The developed approach leads us to a short and simple proof of first-order necessary optimality conditions for such and related problems in broad spaces of functions including those of class C k as k ≥ 1.

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Variational Analysis of Composite Models with Applications to Continuous Optimization

Cited by 13 publications

References 28 publications

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

Generalized Newton Algorithms for Tilt-Stable Minimizers in Nonsmooth Optimization

Generalized Damped Newton Algorithms in Nonsmooth Optimization via Second-Order Subdifferentials

Optimality Conditions for Variational Problems in Incomplete Functional Spaces

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