A major part of J. Outrata’s mathematical work has been devoted to the investigation and application of the calmness property for multifunctions. This concept has turned out to be crucial in particular as a constraint qualification in order to derive M-stationarity conditions for MPECs etc. J. Outrata has made many significant contributions to the analysis of calmness, among them the highly important discovery that it may replace the much stronger Aubin property in the derivation of a crucial preimage formula for normal cones. The talk aims at an illustrative introduction of the calmness concept, at a reminiscence about the roots of common work with J. Outrata as well as a presentation of several more recent results on calmness issues. Among those figure a comparison of calmness between the canonical and enhanced perturbation mappings for generalized equations and estimations of the calmness modulus in linear and nonlinear programming.
A multivalued mapping between finite dimensional spaces is called polyhedral (Robinson 1982) if its graph is the union of finitely many polyhedral convex sets. For example, the optimal solution set mapping of a "canonically perturbed" linear or quadratic program is polyhedral.
Similarly, the solution multifunction of an affine variational inequality under suitable parametrization has this property. Polyhedral multifunctions are of particular interest in the stability analysis of the classes of problems just mentioned and are also used in the study of linear inequalities, piecewise linear equations, complementarity problems, disjunctive optimization and many other subjects. The pioneering work in the 1970ies of F. Nozicka (a founder of the series of "Paraopt" conferences) on parametric linear optimization is one of the fundamentals for this theory. In our talk, we present characterizations and applications of several types of Lipschitz properties (e.g., upper Lipschitz behavior, metric regularity, strong regularity) for optimization and variational problems involving a polyhedral structure. We cover both classical and more recent developments on this topic. In particular, we show how to apply results from variational analysis to the classes of problems under consideration. This talk is co-authored by Bernd Kummer, Humboldt University Berlin.
Diethard Klatte is a retired Professor for Mathematics at the University of Zurich. He received his PhD in 1977 and his Habilitation in 1984 at Humboldt University Berlin in the Department of Mathematics. His areas of expertise are mathematical programming and set-valued and variational analysis, with a focus to parametric and sensitivity analysis of nonlinear optimization, complementarity and equilibrium problems. He is co-author of two monographs on variational analysis and parametric optimization, author/co-author of more than 60 original publications in refereed journals and of a huge number of talks at international conferences, and he co-edited three volumes of conference proceedings. He is very active in the scientific community, in particular, has been for many years an editorial board member of the Journal of Convex Analysis and served in the past as associate editor of SIAM Journal on Optimization, Operations Research Letters and Optimization, and has been in the program committees of many international conferences in his field.
The presentation starts with a brief introduction on topology optimization as a mathematical tool for optimal design of mechanical components. Although now routinely used in the industry, software for topology optimization suffers from limitations, in particular when used for complex three-dimensional structures. Several ways will be presented on how to substantially improve efficiency of topology optimization software using modern tools of numerical linear algebra and numerical optimization. These are based on domain decomposition and multigrid techniques and, for the more involved problems, on decomposition of large-scale matrix inequalities using recent results of graph theory.
MK's is a professor in mathematical optimization at the University of Birmingham, UK, and a researcher at UTIA. He studied applied mathematics at MFF UK and received his PhD at the AVCR. Before joining the University of Birmingham in 2007, he was working on research projects at the Universities in Bayreuth and Erlangen, Germany. Before joining the academic sphere, he worked for several years in the industry.
His research interests include nonlinear and semidefinite optimization, optimization of elastic structures, and optimization with equilibrium constraints.
He is a (co-)author of a monograph (with JVO and Jochem Zowe) and 50 journal articles on various aspects of mathematical optimization and optimization of mechanical structures. He developed or co-developed several computer programs for nonlinear and semidefinite optimization (PENNON, PENSDP) and for optimization of elastic structures. He was a long-term visitor at the Institute of Mathematics and its Applications, University of Minnesota (2003), the Technical University of Denmark (2007) and the Institute for Pure and Applied Mathematics, UCLA (2010).
I am going to demonstrate how (nonlinear) metric subregularity of set-valued mappings can be treated in the framework of the theory of linear error bounds of real-valued functions. For this purpose, the machinery of error bounds has been extended to functions defined on the product of two (metric or normed) spaces. A general classification scheme of necessary and sufficient conditions for the local error bound property/metric subregularity will be presented. Several kinds of primal space and subdifferential slopes for real-valued functions and set-valued mappings will be discussed.
Alexander Kruger received his PhD in mathematics from the Belarusian State University in 1981. Prior to moving to Australia in 2003, he had worked at several universities and research institutions in Minsk, Belarus and also spent 4 months as a Fulbright Scholar at Indiana University South Bend, USA. He has published about 85 papers in the fields of generalized differentiation, optimisation theory and variational analysis. Currently Alexander is Research Director of Centre for Informatics and Applied Optimization (CIAO) at Federation University Australia.
We introduce the notions of critical and noncritical multipliers for parametric variational systems, which include those arising in problems of nonlinear programming, composite and minimax constrained optimization, conic programming, etc. These notions extend the corresponding ones developed by Izmailov and Solodov for the classical KKT system. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. Based on advanced tools and results of second-order variational analysis and generalized differentiation, we obtain efficient characterizations of critical and noncritical multipliers in terms of the given data of the problems under consideration. Furthermore, it is shown that critical multipliers can be ruled out by full stability of local minimizers in various classes of composite optimization and conic programming problems. We also establish connections between noncriticality of multipliers and appropriate notions of calmness, which strongly relate to some recent developments by Ji\v r\'i Outrata and his collaborators.
Boris Mordukhovich is Distinguished University Professor of Mathematics at Wayne State University. He has around 400 publications including several monographs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is in the list of Highly Cited Researchers in Mathematics.