Sequential Linear Integer Programming for Integer Optimal Control with Total Variation Regularization

We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty allows us to prove the existence of minimizers of the integer optimal control problem. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example

On Integer Optimal Control with Total Variation Regularization on Multi-dimensional Domains

We consider optimal control problems with integer-valued controls and a total variation regularization penalty in the objective on domains of dimension two or higher. The penalty yields that the feasible set is sequentially closed in the weak-$^*$ and closed in the strict topology in the space of functions of bounded variation. In turn, we derive first-order optimality conditions of the optimal control problem as well as trust-region subproblems with partially linearized model functions using local variations of the level sets of the feasible control functions. We also prove that a recently proposed function space trust-region algorithm -- sequential linear integer programming -- produces sequences of iterates whose limits are first-order optimal points.Comment: 25 page

Approximationseigenschaften von Sum-Up Rounding

Optimization problems that involve discrete variables are exposed to the conflict between being a powerful modeling tool and often being hard to solve. Infinite-dimensional processes, as e.g. described by differential equations, underlying the optimization may lead to the need to solve for distributed discrete control variables. This work analyzes approximation arguments that replace the need for solving the optimization problem by the need for first solving a relaxation and second computing appropriate roundings to regain discrete controls. We provide sufficient conditions on rounding algorithms and their grid refinement strategies that allow to prove approximation of the relaxed controls by the discrete controls in weaker topologies, a feature due to the infinite-dimensional vantage point. If the control-to-state mapping of the underlying process exhibits suitable compactness properties, state vector approximation follows in the norm topology as well as, under additional assumptions, optimality principles of the computed discrete controls. The conditions are verified for representatives of the family of Sum-Up Rounding algorithms. We apply the arguments on different classes of mixed-integer optimization problems that are constrained by partial differential equations. Specifically, we consider discrete control inputs, which are distributed in the time domain, for evolution equations that are governed by a differential operator that generates a strongly continuous semigroup, discrete control inputs, which are distributed in multi-dimensional spatial domains, for elliptic boundary value problems and discrete control inputs, which are distributed in space-time cylinders, for evolution equations that are governed by differential operators such that the corresponding Cauchy problem satisfies maximal parabolic regularity. Furthermore, we apply the arguments outside the scope of partial differential equations to a signal reconstruction problem. Computational results illustrate the findings.Optimierungsprobleme mit diskreten Variablen befinden sich im Spannungsfeld zwischen hoher Modellierungsmächtigkeit und oft schwerer Lösbarkeit. Zur Optimierung unendlichdimensionaler Prozesse, z.B. beschrieben mit Hilfe von Differentialgleichungen, kann die Lösung nach verteilten diskreten Kontrollvariablen erforderlich sein. Diese Arbeit untersucht Approximationsargumente, mit deren Hilfe die Notwendigkeit einer Lösung des Optimierungsproblems durch die Notwendigkeit zuerst eine Relaxierung zu lösen und anschließend eine passende Rundung zu berechnen, um wieder diskrete Kontrollvariablen zu erhalten, ersetzt wird. Wir geben hinreichende Bedingungen an Rundungsalgorithmen und ihre Gitterverfeinerungsstrategien an, um eine Approximation der relaxierten Kontrollvariablen mit den diskreten Kontrollvariablen in schwächeren Topologien zu erhalten, was aus der unendlichdimensionalen Betrachtung des Problems folgt. Falls der Steuerungs-Zustands-Operator des zugrundeliegenden Prozesses passende Kompaktheitseigenschaften aufweist, folgen die Approximation der Zustandsvektoren in der Normtopologie und, unter zusätzlichen Bedingungen, Optimalitätsprinzipien für die berechneten diskreten Kontrollvariablen. Die Bedingungen werden für Repräsentanten der Familie von Sum-Up Rounding Algorithmen nachgewiesen. Wir wenden die Argumente auf verschiedene Klassen von gemischt-ganzzahligen Optimierungsproblemen, die von partiellen Differentialgleichungen beschränkt werden, an. Insbesondere betrachten wir diskrete, in der Zeit verteilte, Steuerungen in Evolutionsgleichungen mit Differentialoperatoren, die stark stetige Halbgruppen erzeugen; diskrete, mehrdimensional im Ort verteilte, Steuerungen in elliptischen Randwertproblemen und diskrete, in Ort und Zeit verteilte, Steuerungen in Evolutionsgleichungen mit Differentialoperatoren, deren zugehörige Cauchyprobleme maximale parabolische Regularität aufweisen. Des Weiteren wenden wir die Argumente außerhalb des Kontexts partieller Differentialgleichungen auf ein Signalrekonstruktionsproblem an. Numerische Beispiele illustrieren die gezeigten Resultate

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more