On the Minimization Over Sparse Symmetric Sets: Projections, Optimality Conditions, and Algorithms

Beck, Amir; Hallak, Nadav

doi:10.1287/moor.2015.0722

Cited by 51 publications

(61 citation statements)

References 24 publications

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“…(i) As already mentioned the u-step, defined through (6.2), reduces to the computation of the proximal mapping of the function h. Thus, this step can be efficiently computed when the proximal map of h is accessible, i.e., via an and explicit formula or via simple computations, see for instance, [26,13,6] for interesting examples.…”

Section: Two Fundamental Instances Of a ρ And The Corresponding Albummentioning

confidence: 99%

Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence

2018

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We introduce a novel approach addressing global analysis of a difficult class of nonconvexnonsmooth optimization problems within the important framework of Lagrangian-based methods. This genuine nonlinear class captures many problems in modern disparate fields of applications. It features complex geometries, qualification conditions, and other regularity properties do not hold everywhere. To address these issues we work along several research lines to develop an original general Lagrangian methodology which can deal, all at once, with the above obstacles. A first innovative feature of our approach is to introduce the concept of Lagrangian sequences for a broad class of algorithms. Central to this methodology is the idea of turning an arbitrary descent method into a multiplier method. Secondly, we provide these methods with a transitional regime allowing us to identify in finitely many steps a zone where we can tune the step-sizes of the algorithm for the final converging regime. Then, despite the min-max nature of Lagrangian methods, using an original Lyapunov method we prove that each bounded sequence generated by the resulting monitoring schemes are globally convergent to a critical point for some fundamental Lagrangian-based methods in the broad semialgebraic setting, which to the best of our knowledge, are the first of this kind.

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Section: Two Fundamental Instances Of a ρ And The Corresponding Albummentioning

confidence: 99%

Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence

2018

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“…(7)] (with parameter β = 0.2), and the nonmonotone accelerated FBS (denoted AFBS) proposed in[26, Alg. 2] for fully nonconvex problems.…”

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confidence: 99%

Forward-Backward Envelope for the Sum of Two Nonconvex Functions: Further Properties and Nonmonotone Linesearch Algorithms

Themelis¹,

Stella²,

Patrinos³

2018

SIAM J. Optim.

145

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We propose ZeroFPR, a nonmonotone linesearch algorithm for minimizing the sum of two nonconvex functions, one of which is smooth and the other possibly nonsmooth. ZeroFPR is the first algorithm that, despite being fit for fully nonconvex problems and requiring only the black-box oracle of forward-backward splitting (FBS) -namely evaluations of the gradient of the smooth term and of the proximity operator of the nonsmooth one -achieves superlinear convergence rates under mild assumptions at the limit point when the linesearch directions satisfy a Dennis-Moré condition, and we show that this is the case for quasi-Newton directions. Our approach is based on the forward-backward envelope (FBE), an exact and strictly continuous penalty function for the original cost. Extending previous results we show that, despite being nonsmooth for fully nonconvex problems, the FBE still enjoys favorable first-and second-order properties which are key for the convergence results of ZeroFPR. Our theoretical results are backed up by promising numerical simulations. On large-scale problems, by computing linesearch directions using limited-memory quasi-Newton updates our algorithm greatly outperforms FBS and its accelerated variant (AFBS).

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“…Existing algorithms for handling sparsity constrained optimization problems in a direct manner are mainly focused on the problems with symmetric sets as feasible regions which admit efficient computation for the corresponding projection operators [14][15][16]. However, such projected gradient type methods can not be extended to a general SLP problem, since the corresponding feasible region lacks the symmetry and the underlying projection is generally difficult to compute.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian duality and saddle points for sparse linear programming

Zhao

Luo

et al. 2019

Sci. China Math.

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A sparse linear programming (SLP) problem is a linear programming problem equipped with a sparsity (or cardinality) constraint, which is nonconvex and discontinuous theoretically and generally NP-hard computationally due to the combinatorial property involved. By rewriting the sparsity constraint into a disjunctive form, we present an explicit formula of its Lagrangian dual in terms of an unconstrained piecewise-linear convex programming problem which admits a strong duality. A semi-proximal alternating direction method of multipliers (sPADMM) is then proposed to solve this dual problem by taking advantage of the efficient computation of the proximal mapping of the vector Ky-Fan norm function. Based on the optimal solution of the dual problem, we design a dualprimal algorithm for pursuing a global solution of the original SLP problem. Numerical results illustrate that our proposed algorithm is promising especially for large-scale problems.

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On the Minimization Over Sparse Symmetric Sets: Projections, Optimality Conditions, and Algorithms

Cited by 51 publications

References 24 publications

Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence

Nonconvex Lagrangian-Based Optimization: Monitoring Schemes and Global Convergence

Forward-Backward Envelope for the Sum of Two Nonconvex Functions: Further Properties and Nonmonotone Linesearch Algorithms

Lagrangian duality and saddle points for sparse linear programming

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