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![Figure 2.1: Plot of p 1 (x 2 , 2x 2 ; λ), for λ ∈ [0, 1].](profile/Jennifer-Johnstone/publication/220531232/figure/fig4/AS:669056160002051@1536526827525/Plot-of-p-1-x-2-2x-2-l-for-l-0-1_Q320.jpg)
![Figure 3.1: Plot of p 1 (x 2 , 2x 2 + 1; λ), for λ ∈ [0, 1], done in...](profile/Jennifer-Johnstone/publication/220531232/figure/fig5/AS:669056160002052@1536526827559/Plot-of-p-1-x-2-2x-2-1-l-for-l-0-1-done-in-Scilab-using-the-original_Q320.jpg)
Abstract The proximal average operator is recognized for its ability to transform two convex functions into another convex function. However, we prove with examples that the proximal average operator does have limitations, with respect to convexity. We also look at the importance of ‚ 2 [0;1] and describe an idea of how to plot the proximal average...
Citations
... However, results concerning the convergence of the entire iterative solution procedures are often lacking. See [19,20,89,96,109,111,112,114,123,146,158,210] for more information on the Moreau-Yosida regularization and related analytical and theoretical advancement, as well as practical applications and the associated analysis. ...
Nichtglatte Optimierungsprobleme in reflexiven Banachräumen treten in vielen Anwendungen auf. Häufig wird angenommen, dass alle vorkommenden Nichtdifferenzierbarkeiten durch Lipschitz-stetige Operatoren wie abs, min und max gegeben sind. Bei solchen Problemen kann es sich zum Beispiel um optimale Steuerungsprobleme mit möglicherweise nicht glatten Zielfunktionen handeln, welche durch partielle Differentialgleichungen (PDG) eingeschränkt sind, die ebenfalls nicht glatte Terme enthalten können. Eine effiziente und robuste Lösung erfordert eine Kombination numerischer Simulationen und spezifischer Optimierungsalgorithmen. Lokal Lipschitz-stetige, nichtglatte Nemytzkii-Operatoren, welche direkt in der Problemformulierung auftreten, spielen eine wesentliche Rolle in der Untersuchung der zugrundeliegenden Optimierungsprobleme. In dieser Dissertation werden zwei spezifische Methoden und Algorithmen zur Lösung solcher nichtglatter Optimierungsprobleme in reflexiven Banachräumen vorgestellt und diskutiert. Als erste Lösungsmethode wird in dieser Dissertation die Minimierung von nichtglatten Operatoren in reflexiven Banachräumen mittels sukzessiver quadratischer Überschätzung vorgestellt, SALMIN. Ein neuartiger Optimierungsansatz für Optimierungsprobleme mit nichtglatten elliptischen PDG-Beschränkungen, welcher auf expliziter Strukturausnutzung beruht, stellt die zweite Lösungsmethode dar, SCALi. Das zentrale Merkmal dieser Methoden ist ein geeigneter Umgang mit Nichtglattheiten. Besonderes Augenmerk liegt dabei auf der zugrundeliegenden nichtglatten Struktur des Problems und der effektiven Ausnutzung dieser, um das Optimierungsproblem auf angemessene und effiziente Weise zu lösen.
... Given proper convex functions f 0 and f 1 , their proposed solution, the proximal average, used Fenchel conjugates to define a parameterized function P A(x, λ) such that P A is epi-continuous with respect to λ, and P A(x, 0) = f 0 (x), P A(x, 1) = f 1 (x) for all x. The proximal average has been studied extensively since its original conception, and many favourable properties and applications of this approach have arisen [1,3,4,6,7,8,10,9,11,14,15,16,18]. For example, the minimizers of the proximal average function change continuously with respect to λ [10]. ...
The NC-proximal average is a parametrized function used to continuously transform one proper, lsc, prox-bounded function into another. Until now, it has been defined for two functions. The purpose of this article is to redefine it so that any finite number of functions may be used. The layout generally follows that of [11], extending those results to the more general case and in some instances giving alternate proofs by using techniques developed after the publication of that paper. We conclude with an example examining the discontinuity of the minimizers of the NC-proximal average.
... A parametrized function of the proxparameter r, the Moreau envelope is defined as the infimal convolution of f with the scaled norm-squared function r 2 · −x 2 . It is largely used in matters of minimization of convex functions [1,2,5,6,15,17,19,20,31,33,34], and more recently it has found a place in non-convex optimization as well [4,10,11,12,13,14,16,25,26]. ...
... For over the last years, an enormous amount of effort by some authors has been devoted to understand the extension of various operator concepts when the involved operators are replaced by convex functionals. For instance, some operator means, the operator inverse A −→ A −1 , the operator logarithm A −→ log A, the power operator A −→ A m (−1 < m < 1), the (Tsallis) operator entropy and the shorted operator have been extended from the case that the variables are positive linear operators to the case that the variables are convex functionals, see [1,2,3,4,6] and the related references cited therein. The above extensions were investigated in the sense that, if A −→ φ(A) is a map between positive operators then its extension f −→ Φ(f ) for convex functionals satisfies ...
The operator-product ABA appears in many mathematical contexts, such as in algebraic Riccati equation, in operator entropy and in operator-mean theory. The purpose of the present paper is to investigate a reasonable analogue of ABA when the positive linear operators A and B are convex functionals. As consequence, the square of a convex functional extending A(2) is provided as well.
... A parametrized function of the proxparameter r, the Moreau envelope is defined as the infimal convolution of f with the scaled norm-squared function r 2 · −x 2 . It is largely used in matters of minimization of convex functions [1,2,5,6,15,17,19,20,31,33,34], and more recently it has found a place in non-convex optimization as well [4,10,11,12,13,14,16,25,26]. ...
Introduced in the 1960s, the Moreau envelope has grown to become a key tool in nonsmooth analysis and optimization. Essentially an infimal convolution with a parametrized norm squared, the Moreau envelope is used in many applications and optimization algorithms. An important aspect in applying the Moreau envelope to nonconvex functions is determining if the function is prox-bounded, that is, if there exists a point x and a parameter r such that the Moreau envelope is finite. The infimum of all such r is called the threshold of proxboundedness (prox-threshold) of the function f: In this paper, we seek to understand the proxthresholds
of piecewise linear-quadratic (PLQ) functions. (A PLQ function is a function whose domain is a union of finitely many polyhedral sets, and that is linear or quadratic on each piece.) The main result provides a computational technique for determining the prox-threshold for a PLQ function, and further analyzes the behavior of the Moreau envelope of the function using the prox-threshold. We provide several examples to illustrate the techniques and challenges.
The $\epsilon$-subdifferential of convex univariate piecewise linear-quadratic functions can be computed in linear worst-case time complexity as the level-set of a convex function. Using dichotomic search, we show how the computation can be performed in logarithmic worst-case time. Furthermore, a new algorithm to compute the entire graph of the $\epsilon$-subdifferential in linear time is presented. Both algorithms are not limited to convex PLQ functions but are also applicable to any convex piecewise-defined function with little restrictions.
We present several techniques that take advantage of convexity and use optimal computational geometry algorithms to build fast (log-linear or linear) time algorithms in computational convex analysis. The techniques that have strong potential to be reused include: monotonicity of the argmax and injecting convexity to use that monotonicity, Lipschitzness of the argmin, exploiting various formulas in convex analysis, using a graph data structure to vectorize computation, and building a parametrization of the graph. We also point out the potential for parallelization. The techniques can be used as a check list on open problems to find an efficient algorithm. Finally, we list several currently open questions in computational convex analysis with links to computational geometry.
The notion of the geometric mean of two positive reals is extended by Ando
(1978) to the case of positive semidefinite matrices and . Moreover, an interesting
generalization of the geometric mean of and to convex functions
was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex
analysis. The present work aims at providing a further development of the geometric
mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic
self-dual operator for convex functions named “the geometric mean of parameterized
arithmetic and harmonic means of convex functions” is proposed, and its essential
properties are investigated.
We present a new algorithm to compute the Legendre–Fenchel conjugate of convex piecewise linear-quadratic (PLQ) bivariate functions. The algorithm stores a function using a (primal) planar arrangement. It then computes the (dual) arrangement associated with the conjugate by looping through vertices, edges, and faces in the primal arrangement and building associated dual vertices, edges, and faces. Using optimal computational geometry data structures, the algorithm has a linear time worst-case complexity. We present the algorithm, and illustrate it with numerical examples. We proceed to build a toolbox for convex bivariate PLQ functions by implementing the addition, and scalar multiplication operations. Finally, we compose these operators to compute classical convex analysis operators such as the Moreau envelope, and the proximal average.
The NC-proximal average is a parametrized function used to continuously transform
one proper, lsc, prox-bounded function into another. Until now it has been defined for
two functions. The purpose of this article is to redefine it so that any finite number of
functions may be used. The layout generally follows that of [Hare 2009], extending
those results to the more general case and in some instances giving alternate proofs
by using techniques developed after the publication of that paper. We conclude with an
example examining the discontinuity of the minimizers of the NC-proximal average.
