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Variational Analysis of Composite Models with Applications to Continuous Optimization
TL;DR: In this article, a comprehensive study of composite models in variational analysis and optimization is presented, with the main attention paid to the new and rather large class of fully subamenable compositions, and the underlying theme of the study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones.
Abstract: The paper is devoted to a comprehensive study of composite models in variational analysis and optimization the importance of which for numerous theoretical, algorithmic, and applied issues of operations research is difficult to overstate. The underlying theme of our study is a systematical replacement of conventional metric regularity and related requirements by much weaker metric subregulatity ones that lead us to significantly stronger and completely new results of first-order and second-order variational analysis and optimization. In this way we develop extended calculus rules for first-order and second-order generalized differential constructions with paying the main attention in second-order variational theory to the new and rather large class of fully subamenable compositions. Applications to optimization include deriving enhanced no-gap second order optimality conditions in constrained composite models, complete characterizations of the uniqueness of Lagrange multipliers and strong metric subregularity of KKT systems in parametric optimization, etc.
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Abstract: The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in variational analysis for more than two decades while being largely underinvestigated. We discover here that parabolic regularity is the key to derive new calculus rules and computation formulas for major second-order generalized differential constructions of variational analysis in connection with some properties of sets that go back to classical differential geometry and geometric measure theory. The established results of second-order variational analysis and generalized differentiation, being married to the developed calculus of parabolic regularity, allow us to obtain novel applications to both qualitative and quantitative/numerical aspects of constrained optimization including second-order optimality conditions, augmented Lagrangians, etc. under weak constraint qualifications.
25 citations
TL;DR: Two versions of the generalized Newton method are developed to compute not merely arbitrary local minimizers of nonsmooth optimization problems but just those, which possess an important stability property known as tilt stability, which are based on graphical derivatives of the latter.
Abstract: This paper aims at developing two versions of the generalized Newton method to compute local minimizers for nonsmooth problems of unconstrained and constrained optimization that satisfy an importan...
16 citations
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TL;DR: In this article, a Newton-type algorithm is proposed to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions, which can be efficiently computed for broad classes of extended real-valued functions.
Abstract: This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order subdifferential of such functions that enjoys extensive calculus rules and can be efficiently computed for broad classes of extended-real-valued functions. Based on this and on metric regularity and subregularity properties of subgradient mappings, we establish verifiable conditions ensuring well-posedness of the proposed algorithm and its local superlinear convergence. The obtained results are also new for the class of equations defined by continuously differentiable functions with Lipschitzian derivatives ($\mathcal{C}^{1,1}$ functions), which is the underlying case of our consideration. The developed algorithm for prox-regular functions is formulated in terms of proximal mappings related to and reduces to Moreau envelopes. Besides numerous illustrative examples and comparison with known algorithms for $\mathcal{C}^{1,1}$ functions and generalized equations, the paper presents applications of the proposed algorithm to the practically important class of Lasso problems arising in statistics and machine learning.
13 citations
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TL;DR: This paper addresses problems of second-order cone programming important in optimization theory and applications by formulate the corresponding version ofsecond-order sufficiency and use it to establish the uniform second- order growth condition for the augmented Lagrangian.
Abstract: This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal-dual convergence of this method.
11 citations
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TL;DR: In this paper, the second-order sufficient condition for local optimality has been shown to justify linear convergence of the primal-dual sequence generated by the augmented Lagrangian method for piecewise linear-quadratic composite optimization problems.
Abstract: Second-order sufficient conditions for local optimality have been playing an important role in local convergence analysis of optimization algorithms. In this paper, we demonstrate that this condition alone suffices to justify the linear convergence of the primal-dual sequence, generated by the augmented Lagrangian method for piecewise linear-quadratic composite optimization problems, even when the Lagrange multiplier in this class of problems is not unique. Furthermore, we establish the equivalence between the second-order sufficient condition and the quadratic growth condition of the augmented Lagrangian problem for this class of composite optimization problems.
9 citations
References
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Book•
11 May 2000
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: Basic notation.- Introduction.- Background material.- Optimality conditions.- Basic perturbation theory.- Second order analysis of the optimal value and optimal solutions.- Optimal Control.- References.
2,067 citations
TL;DR: A very broad and flexible framework is investigated which allows a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence in convex feasibility problems.
Abstract: Due to their extraordinary utility and broad applicability in many areas of classical mathematics and modern physical sciences (most notably, computerized tomography), algorithms for solving convex feasibility problems continue to receive great attention. To unify, generalize, and review some of these algorithms, a very broad and flexible framework is investigated. Several crucial new concepts which allow a systematic discussion of questions on behaviour in general Hilbert spaces and on the quality of convergence are brought out. Numerous examples are given.
1,742 citations
01 Jan 2006
1,528 citations
TL;DR: In this paper, it is shown that if (1) is consistent, one can infer that there is a solution x0 of (1)-close to x. The purpose of this report is to justify and formulate precisely this assertion.
Abstract: (briefly, -4x^6), one arrives at a vector JC that "almost" satisfies (1). It is almost obvious geometrically that, if (1) is consistent, one can infer that there is a solution x0 of (1) "close" to x. The purpose of this report is to justify and formulate precisely this assertion. We shall use fairly general definitions of functions that measure the size of vectors, since it may be possible to obtain better estimates of the constant c (whose important role is described in the statement of the theorem) for some measuring functions than for others. We shall make a few remarks on the estimation of c after completing the proof of the main theorem.
732 citations
Book•
01 Jan 2005
TL;DR: In this article, the authors present a set of principles for using variational techniques in nonlinear functional analysis in the presence of symmetry and symmetry in the context of convex analysis.
Abstract: and Notation.- Variational Principles.- Variational Techniques in Subdifferential Theory.- Variational Techniques in Convex Analysis.- Variational Techniques and Multifunctions.- Variational Principles in Nonlinear Functional Analysis.- Variational Techniques In the Presence of Symmetry.
551 citations