Enforcing Termination of Interprocedural Analysis

Interprocedural analysis by means of partial tabulation of summary functions may not terminate when the same procedure is analyzed for infinitely many abstract calling contexts or when the abstract domain has infinite strictly ascending chains. As a remedy, we present a novel local solver for general abstract equation systems, be they monotonic or not, and prove that this solver fails to terminate only when infinitely many variables are encountered. We clarify in which sense the computed results are sound. Moreover, we show that interprocedural analysis performed by this novel local solver, is guaranteed to terminate for all non-recursive programs --- irrespective of whether the complete lattice is infinite or has infinite strictly ascending or descending chains

A New Class of Cycle Inequality for the Time-Dependent Traveling Salesman Problem

The Time-Dependent Traveling Salesman Problem is a generalization of the well-known Traveling Salesman Problem, where the cost for travel between two nodes is dependent on the nodes and their position in the tour. Inequalities for the Asymmetric TSP can be easily extended to the TDTSP, but the added time information can be used to strengthen these inequalities. We look at extending the Lifted Cycle Inequalities, a large family of inequalities for the ATSP. We define a new inequality, the Extended Cycle (X-cycle) Inequality, based on cycles in the graph. We extend the results of Balas and Fischetti for Lifted Cycle Inequalities to define Lifted X-cycle Inequalities. We show that the Lifted X-cycle Inequalities include some inequalities which define facets of the submissive of the TDTS Polytope

Convex Analysis and Optimization with Submodular Functions: a Tutorial

Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of submodular functions is presented, in a self-contained way, with all results shown from first principles. A good knowledge of convex analysis is assumed

Gomory-Chvátal cutting planes and the elementary closure of Polyhedra

The elementary closure P\u27; of a polyhedrom P is the intersection of P with all its Gomory-Chvátal cutting planes. P\u27; is a rational polyhedron provided that P is rational. The Chvátal-Gomory procedure is the iterative application of the elementary closure operation to P. The Chvátal rank is the minimal number of iterations needed to obtain P_I. It is always finite, but already in |R² one can construct polytopes of arbitrary large Chvátal rank. We show that the Chvátal rank of polytopes contained in the n-dimensional 0/1 cube is O(n² log n) and prove the lower bound (1+E) n, for some E> 0. We show that the separation problem for the elementary closure of a rational polyhedron is NP-hard. This solves a problem posed by Schrijver. Last we consider the elementary closure in fixed dimension. the known bounds for the number of inequalities defining P\u27; are exponential, even fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone from this polynomial description in fixed dimension with a maximal degree of violation in a natural sense.Die elemtare Hülle P\u27; eines Polyeders P ist der Durchschnitt von P mit all seinen Gomory-Chvátal Schnittebenen. P\u27; ist ein rationales Polyeder, falls P rational ist. Die Chvátal-Gomory Prozedur ist das wiederholte Bilden der elementaren Hülle, beginnend mit P. Die minimale Anzahl der Iterationen, die bis zum Erhalt der ganzzahligen Hülle P1 von P nötig sind, heißt Chvátal-Rang von P. Der Chvátal-Rang eines rationalen Polyeders ist endlich. Jedoch lassen sich bereits im |R² Beispiele mit beliebig hohem Chvátal-Rang konstruieren. Wir zeigen, dass der Chvátal-Rang eines Polytops im n-dimensionalen 0/1 Würfel durch O(n² log n) beschränkt ist, und beweisen die untere Schranke (1 + epsilon) * n, für ein epsilon > 0. Wir zeigen, dass das Separationsproblem für die elementare Hülle eines rationalen Polyeders NP-hart ist. Dies löst ein von Schrijver formuliertes Problem. Schließlich wenden wir uns der elementaren Hülle rationaler Polyeder in fester Dimension zu. Die bislang bekannten Schranken für die Anzahl der Ungleichungen, die zur Darstellung von P\u27; benötigt werden, sind exponentiell, selbst in fester Dimension. Wir zeigen, dass in fester Dimension P\u27; durch polynomiell viele Ungleichungen beschrieben werden kann. Wir entwerfen außerdem einen, in beliebiger Dimension polynomiellen, Algorithmus, der zu einem spitzen Kegel P eine Schnittebene aus der polynomiellen Darstellung von P\u27; berechnet, die zudem einen maximalen Grad der Verletzung in einem natürlichen Sinne aufweist

Some mathematical problems related to the 2nd order optimal shape of a crystallization interface

We consider the problem to optimize the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimization principle of the free energy, while the temperature is solving the heat equation with a radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallization, the interface is expected to have a global graph representation. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second surface derivatives, and for the surface temperature gradient