## About

42

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Introduction

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July 2014 - present

## Publications

Publications (42)

This paper concerns smoothing by infimal convolution for two large classes of functions: convex, proper and lower semicontinous as well as for (the nonconvex class of) convex-composite functions. The smooth approximations are constructed so that they epi-converge (to the underlying nonsmooth function) and fulfill a desirable property with respect t...

A new class of matrix support functionals is presented which establish a connection
between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo matrix-fractional function, the nuclear norm, and multi-task learning. The support function is based on the graph of the product of a matrix with its tran...

Smoothing methods have become part of the standard tool set for the study and
solution of nondifferentiable and constrained optimization problems as well as
a range of other variational and equilibrium problems. In this note we
synthesize and extend recent results due to Beck and Teboulle on infimal
convolution smoothing for convex functions with t...

Mathematical programs with equilibrium constraints (MPECs) are difficult optimization problems whose feasible sets do not
satisfy most of the standard constraint qualifications. Hence MPECs cause difficulties both from a theoretical and a numerical
point of view. As a consequence, a number of MPEC-tailored solution methods have been suggested durin...

A mathematical program with vanishing constraints (MPVC) is a constrained optimization problem arising in certain engineering applications. The feasible set has a complicated structure so that the most familiar constraint qualifications are usually violated. This, in turn, implies that standard penalty functions are typically non-exact for MPVCs. W...

In this paper we establish necessary and sufficient conditions for the existence of line segments (or flats) in the sphere of the nuclear norm via the notion of simultaneous polarization and a refined expression for the subdifferential of the nuclear norm. This is then leveraged to provide (point-based) necessary and sufficient conditions for uniqu...

This paper provides a variational analysis of the unconstrained formulation of the LASSO problem, ubiquitous in statistical learning, signal processing, and inverse problems. In particular, we establish smoothness results for the optimal value as well as Lipschitz properties of the optimal solution as functions of the right-hand side (or measuremen...

We study the question as to when the closed convex hull of the graph of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions.

This paper is devoted to the study of the metric subregularity constraint qualification for general optimization problems, with the emphasis on the nonconvex setting. We elaborate on notions of directional pseudo-and quasi-normality, recently introduced by Bai et al., which combine the standard approach via pseudo-and quasi-normality with modern to...

We study the question as to when the closed convex hull of a K-convex map equals its K-epigraph. In particular, we shed light onto the smallest cone K such that a given map has convex and closed K-epigraph, respectively. We apply our findings to several examples in matrix space as well as to convex composite functions.

The projection onto the epigraph or a level set of a closed proper convex function can be achieved by finding a root of a scalar equation that involves the proximal operator as a function of the proximal parameter. This paper develops the variational analysis of this scalar equation. The approach is based on a study of the variational-analytic prop...

Image deblurring is a notoriously challenging ill-posed inverse problem. In recent years, a wide variety of approaches have been proposed based upon regularization at the level of the image or on techniques from machine learning. In this article, we adapt the principal of maximum entropy on the mean (MEM) to both deconvolution of general images and...

In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward. The results are established under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity a...

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often...

Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often...

Deep neural networks are vulnerable to adversarial perturbations: small changes in the input easily lead to misclassification. In this work, we propose an attack methodology catered not only for cases where the perturbations are measured by $\ell_p$ norms, but in fact any adversarial dissimilarity metric with a closed proximal form. This includes,...

In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the desired results under a verifiable Slater-type condition, with relaxed monotonicity and without lower semicontinuity...

Barcode encoding schemes impose symbolic constraints which fix certain segments of the image. We present, implement, and assess a method for blind deblurring and denoising based entirely on Kullback-Leibler divergence. The method is designed to incorporate and exploit the full strength of barcode symbologies. Via both standard barcode reading softw...

We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which such functions are closed and proper as well as formulas for the ready computation of both their conjugates a...

This paper is devoted to the study of the metric subregularity constraint qualification (MSCQ) for optimization problems with nonconvex constraints. We propose a unified theory for several prominent sufficient conditions for MSCQ, which is achieved by means of a new constraint qualification that combines the well-established approach via pseudo- an...

The paper studies the partial infimal convolution of the generalized matrix-fractional function with a closed, proper, convex function on the space of real symmetric matrices. Particular attention is given to support functions and convex indicator functions. It is shown that this process yields approximate smoothings of all Ky-Fan norms, weighted n...

Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support function representation for GMF fu...

We consider sums of sequences of extended real-valued functions focusing on
the epigraphical
liminf and limsup inequalities.
Specifically, we note a deficiency in the statement of Theorem 7.46 in
\textit{Variational Analysis},
Springer, Berlin, Heidelberg, 1998.
An elementary counterexample is provided, and a remedy that is well suited to the cla...

A new class of matrix support functionals is pre sented which establish a connection between optimal value functions for quadratic optimization problems, the matrix-fractional function, the pseudo-matrix-fractional function, the nuclear norm, and multitask learning. The support function is based on the graph of the product of a matrix with its tran...

A well-known technique for the solution of quasi-variational inequalities (QVIs) consists in the reformulation of this problem as a constrained or unconstrained optimization problem by means of so-called gap functions. In contrast to standard variational inequalities, however, these gap functions turn out to be nonsmooth in general. Here, it is sho...

We consider a class of generalized Nash equilibrium problems, where both objective functions and constraints are allowed to depend on the decision variables of the other players. It is well known that this problem can be reformulated as a constrained optimization problem via the (regularized) Nikaido–Isoda-function, but this reformulation is usuall...

We consider a class of generalized Nash equilibrium problems (GNEPs) where both the objective functions and the constraints are allowed to depend on the decision variables of the other players. It is well-known that this problem can be reformulated as a constrained optimization problem via the (regularized) Nikaido-Isoda-function, but this reformul...

We point out and discuss two erroneous statements in the paper [T. Hoheisel, C. Kanzow, B.S. Mordukhovich, H. Phan, Generalized Newton’s method based on graphical derivatives, Nonlinear Analysis TMA 75 (2012) 1324–1340] that were brought to our attention by Bernd Kummer. Furthermore, the correction of the corresponding global convergence result is...

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothi...

Chen and Mangasarian (Comput Optim Appl 5:97–138, 1996) developed smoothing approximations to the plus function built on integral-convolution with density functions. X. Chen (Math Program 134:71–99, 2012) has recently picked up this idea constructing a large class of smoothing functions for nonsmooth minimization through composition with smooth map...

The paper considers a numerical approach for the solution of mathematical problems with vanishing constraints (MPVC). It is well known that direct numerical approaches for the treatment of MPVCs typically fail because standard constraint qualifications usually are not satisfied at a local minimizer. This parallels the situation in the related class...

Motivated by a recent method introduced by Kanzow and Schwartz [C. Kanzow and A. Schwartz, A new regularization method for mathematical programs with complementarity constraints with strong convergence properties, Preprint 296, Institute of Mathematics, University of Würzburg, Würzburg, 2010] for mathematical programs with complementarity constrain...

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanc...

A new class of optimization problems name 'mathematical programs with vanishing constraints (MPVCs)' is considered. MPVCs are on the one hand very challenging from a theoretical viewpoint, since standard constraint qualifications such as LICQ, MFCQ, or ACQ are most often violated, and hence, the Karush-Kuhn-Tucker conditions do not provide necessar...

We consider a class of optimization problems that is called a mathematical program with vanishing constraints (MPVC for short). This class has some similarities to mathematical programs with equilibrium constraints (MPECs for short), and typically violates standard constraint qualifications, hence the well-known Karush–Kuhn–Tucker conditions do not...

We consider a special class of optimization problems that we call Mathematical Programs with Vanishing Constraints, MPVC for short, which serves as a unified framework for several applications in structural and topology optimization. Since
an MPVC most often violates stronger standard constraint qualification, first-order necessary optimality condi...

We consider a special class of optimization problems that we call a Mathematical Programme with Vanishing Constraints. It has a number of important applications in structural and topology optimization, but typically does not satisfy standard constraint qualifications like the linear independence and the Mangasarian–Fromovitz constraint qualificatio...

We consider a regularization method for the numerical solution of mathematical programs with complementarity constraints (MPCC) introduced by Gui-Hua Lin and Masao Fukushima. Existing convergence results are improved in the sense that the MPCC-LICQ assumption is replaced by the weaker MPCC-MFCQ. Moreover, some preliminary numerical results are pres...

Mathematical programmes with equilibrium or vanishing constraints MPECs or MPVCs are both known to be difficult optimization problems which typically violate all standard constraint qualifications. A number of methods try to exploit the particular structure of MPECs and MPVCs in order to overcome these difficulties. In a recent paper by Steffensen...

## Projects

Projects (3)

Understanding theoretically and exploiting numerically the maximum entropy method, in particular for image processing.