Integer Convex Minimization in Low Dimensions

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1 DISS. ETH NO Integer Convex Minimization in Low Dimensions A thesis submitted to attain the degree of DOCTOR OF SCIENCES of ETH ZURICH (Dr. sc. ETH Zurich) presented by TIMM OERTEL Diplom-Mathematiker, Otto-von-Guericke Universität Magdeburg born on citizen of Germany accepted on the recommnedation of Prof. Dr. Robert Weismantel, examiner Prof. Dr. Friedrich Eisenbrand, co-examiner 2014

2 Abstract In this dissertation we discuss several approaches to solve integer and mixedinteger convex minimization problems. That is, we try to minimize a convex function over a convex set with the additional constraint that a small number variables must be integral. The thesis consists of four parts. In the first part we apply the Mirror-Descent Method from continuous convex optimization to the mixed-integer setting. The main feature of this method is that the number of iterations is independent of the dimension, however, this method relies on a strong oracle, the so called improvement oracle. We present an efficient realization of such an oracle for the case when only two variables are required to be integral. The second part contains two alternative, short, and geometrically motivated proofs of the well known result that minimizing a convex function over the integral points of a bounded convex set is polynomial in fixed dimension. In particular, we present an oracle-polynomial algorithm that is based on a mixed-integer linear optimization oracle. Then, in the third part, we extend the Method of Centers of Gravity to the integer and mixed-integer setting. The crucial step consists in replacing the points of center of gravity by more general center-points, allowing us to use measures other than the volume. We introduce the concepts of center-points and approximate center-points. For special instances we prove properties of the (approximate) center-points. In the integer setting and when the dimension is fixed, we present an algorithm to compute approximate center-points. Furthermore, we establish optimality certificates for (mixed-) integer minimization problems based on lattice free polyhedra and we present a algorithm based on center-points that terminates with such an optimality certificate. In the last part we consider a special class of, not necessarily convex, optimization problems in variable dimension. We aim to optimize f(w x) over a set P Z n, where f is a non-linear function, P R n is a polyhedron and W Z d n. The dimension n may vary, however, we assume that the dimension d is fixed. We obtain an efficient transformation from the latter class of problems to integer linear problems. The core result is a representation

3 iv ABSTRACT theorem, characterizing the set W (P Z n ), which can be seen as Frobenius type theorem for polyhedra.

4 Zusammenfassung In der vorliegenden Dissertationsschrift betrachten wir ganzzahlige und gemischt-ganzzahlige Minimierungsprobleme. Das sind Probleme, bei denen eine konvexe Zielfunktion und eine konvexe Menge gegeben sind und versucht wird die Zielfunktion über die konvexe Menge zu minimieren, wobei wir fordern, dass eine, in unserem Fall kleine, Anzahl an Variablen ganzzahlig zu sein hat. Wir diskutieren mehrere algorithmische Ansätze um dies zu lösen. Die Arbeit besteht aus vier Teilen. Im ersten Teil betrachten wir einen Algorithmus aus der kontinuierlichen Optimierung, nämlich die Mirrow-Descent Method. Das Besondere dieser Methode ist, dass die Anzahl der Iterationen unabhängig von der Dimension ist. Jedoch beruht dieser Ansatz auf einem starken Orakel, einem sogenannten Verbesserungsorakel. Unser Kernbeitrag in diesem Teil ist eine effiziente Realisierung dieses Orakels für den Fall, dass höchstens zwei Variablen ganzzahlig sind. Im zweiten Teil geben wir zwei alternative, kurze und geometrisch motivierte Beweise, dass das Minimieren einer konvexen Funktion über den ganzzahligen Punkten in einer konvexen Menge in polynomialer Zeit in der Eingabegröße realisierbar ist, vorausgesetzt, dass die Dimension der konvexen Menge fix ist. Speziell zeigen wir einen Orakel-polynomiellen Algorithmus der allein auf einem gemischt-ganzzahligen linearen Optimierungsorakel basiert. Dann, im dritten Teil, betrachten wir die Method of Centers of Gravity für kontinuierliche konvexe Minimierung und adaptieren diese für den ganzzahligen und gemischt-ganzzahligen Fall. Dazu ersetzen wir den Massenmittelpunkt durch allgemeinere Mittelpunkte; dies erlaubt uns dann andere Maße als das Volumen zu nutzen. Als erstes definieren wir Mittelpunkte und approximative Mittelpunkte, dann beweisen wir für spezielle Fälle einige ihrer Eigenschaften und wir zeigen wie approximative Mittelpunkte gefunden werden können. Danach entwickeln wir Optimalitätskriterien die auf gitterpunktfreien Polyedern beruhen. Zum Schluss präsentieren wir noch einen Algorithmus der auf den zuvor genannten Mittelpunkten beruht und der mit den besagten Optimalitätskriterien terminiert. Im letzten Teil untersuchen wir eine spezielle Klasse von, nicht notwendigerweise konvexen, Optimierungsproblemen in variabler Dimension. Wir wollen

5 vi ZUSAMMENFASSUNG f(w x) über P Z n optimieren, wobei f eine nichtlineare Funktion, P R n ein Polyeder und W eine Matrix aus Z d n ist. Dabei nehmen wir an, dass n variabel, d jedoch fix ist. Wir erhalten eine effiziente Transformation die das zuvor genannte Problem auf ganzzahlige lineare Unterprobleme reduziert. Das zentrale Resultat ist hierbei eine kompakte Charakterisierung der Menge W (P Z n ), dieses kann als eine Art von Frobenius-Theorem für Polyeder gesehen werde.

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