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The University of Southampton
CORMSIS Centre for Operational Research, Management Sciences and Information Systems

Mathematical programs with switching constraints: Applications, analysis, and algorithms -- talk by Patrick Mehlitz Event

Time:
14:00 - 16:00
Date:
29 November 2018
Venue:
Building 54, Room 8031, University of Southampton, Highfield Campus, SO17 1BJ

For more information regarding this event, please email Konstantinos Katsikopoulos at K.Katsikopoulos@southampton.ac.uk .

Event details

A mathematical program with switching constraints (MPSC) is a nonlinear optimization problem comprising constraints of the form $G(x)H(x)=0$ where $G,H\colon\mathbb R^n\to\mathbb R$ are given functions. Switching structures appear frequently in the context of optimal control where they are used to model situations where at most one (of several) controls is allowed to be nonzero at any available time instance. Moreover, switching constraints can be used to model so-called semi-continuity conditions on variables arising in portfolio optimization. Finally, logical constraints involving the OR-operator can be reformulated without the aid of binary variables (so-called \emph{big-M-method}) by postulating switching requirements.

It is easily seen that MPSCs are closely related to so-called mathematical programs with complementarity constraints (MPCCs). Indeed, any MPSC is an MPCC (and vice versa) but the associated transformations go along with the loss of structural information.

Thus, a separate investigation of MPSCs which respects the inherent structural properties of this problem class is necessary.

In this talk, we first address some applications of switching-constrained optimization in more detail. Noting that standard constraint qualifications from nonlinear programming are likely to be violated at several feasible points of an MPSC, its Karush-Kuhn-Tucker conditions do not yield a reliable criterion to characterize local minimizers. Thus, weaker stationarity notions and associated constraint qualifications will be introduced to overcome this inherent lack of regularity.

Clearly, switching-constrained optimization problems often possess (almost) disconnected feasible sets which makes their numerical treatment challenging.

In order to deal with this difficulty, a relaxation approach is suggested and its convergence properties are studied. Results of computational experiments addressing real-world applications will be presented.

This talk is based on joint ongoing work with Christian Kanzow and Daniel Steck.

Speaker information

Patrick Mehlitz,Brandenburgische Technische Universitat Cottbus-Senftenberg.,Patrick studied business mathematics at the Technische Universit\"at \emph{Bergakademie} Freiberg (Germany) from October 2008 to September 2013. Afterwards, he stayed in Freiberg and achieved his PhD in July 2017 under supervision of Stephan Dempe. Recently, Patrick moved to the Brandenburgische Technische Universit\"at Cottbus-Senftenberg (Germany) where he got position as a postdoctoral researcher in the group of Gerd Wachsmuth. His research mainly focuses on complementarity-type and bilevel optimization problems in Banach spaces and associated applications from optimal control of ordinary or partial differential equations.

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