Solving the logically constrained multiple choice multiple knapsack problem: A real case

In questa tesi viene analizzato un problema di ottimizzazione proposto da alcuni esercizi commerciali che hanno la necessita` di selezionare e disporre i propri ar- ticoli in negozio. Il problema nasce dall’esigenza di massimizzare il profitto com- plessivo atteso dei prodotti in esposizione, trovando per ognuno una locazione sugli scaffali. I prodotti sono suddivisi in dipartimenti, dai quali solo un ele- mento deve essere selezionato ed esposto. In oltre si prevede la possibilita` di esprimere vincoli sulla locazione e compatibilita` dei prodotti. Il problema risul- tante `e una generalizzazione dei gia` noti Multiple-Choice Knapsack Problem e Multiple Knapsack Problem. Dopo una ricerca esaustiva in letteratura si `e ev- into che questo problema non `e ancora stato studiato. Si `e quindi provveduto a formalizzare il problema mediante un modello di programmazione lineare intera. Si propone un algoritmo esatto per la risoluzione del problema basato su column generation e branch and price. Sono stati formulati quattro modelli differenti per la risoluzione del pricing problem su cui si basa il column generation, per individuare quale sia il piu` efficiente. Tre dei quattro modelli proposti hanno performance comparabili, mentre l’ultimo si `e rivelato piu` inefficiente. Dai risul- tati ottenuti si evince che il metodo risolutivo proposto `e adatto a istanze di dimensione medio-bassa.

Energy consumption of parallel algorithms for solving linear systems on HPC architecture

Modern High-Performance Computing HPC systems are gradually increasing in size and complexity due to the correspondent demand of larger simulations requiring more complicated tasks and higher accuracy. However, as side effects of the Dennard’s scaling approaching its ultimate power limit, the efficiency of software plays also an important role in increasing the overall performance of a computation. Tools to measure application performance in these increasingly complex environments provide insights into the intricate ways in which software and hardware interact. The monitoring of the power consumption in order to save energy is possible through processors interfaces like Intel Running Average Power Limit RAPL. Given the low level of these interfaces, they are often paired with an application-level tool like Performance Application Programming Interface PAPI. Since several problems in many heterogeneous fields can be represented as a complex linear system, an optimized and scalable linear system solver algorithm can decrease significantly the time spent to compute its resolution. One of the most widely used algorithms deployed for the resolution of large simulation is the Gaussian Elimination, which has its most popular implementation for HPC systems in the Scalable Linear Algebra PACKage ScaLAPACK library. However, another relevant algorithm, which is increasing in popularity in the academic field, is the Inhibition Method. This thesis compares the energy consumption of the Inhibition Method and Gaussian Elimination from ScaLAPACK to profile their execution during the resolution of linear systems above the HPC architecture offered by CINECA. Moreover, it also collates the energy and power values for different ranks, nodes, and sockets configurations. The monitoring tools employed to track the energy consumption of these algorithms are PAPI and RAPL, that will be integrated with the parallel execution of the algorithms managed with the Message Passing Interface MPI.