5 research outputs found
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
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
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 ModellierungsmÀchtigkeit 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 anschlieĂend 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 schwĂ€cheren 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 zusĂ€tzlichen Bedingungen, OptimalitĂ€tsprinzipien fĂŒr die berechneten diskreten Kontrollvariablen. Die Bedingungen werden fĂŒr ReprĂ€sentanten der Familie von Sum-Up Rounding Algorithmen nachgewiesen.
Wir wenden die Argumente auf verschiedene Klassen von gemischt-ganzzahligen Optimierungsproblemen, die von partiellen Differentialgleichungen beschrĂ€nkt 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 RegularitĂ€t aufweisen. Des Weiteren wenden wir die Argumente auĂerhalb des Kontexts partieller Differentialgleichungen auf ein Signalrekonstruktionsproblem an. Numerische Beispiele illustrieren die gezeigten Resultate