Dynamic Resource Allocation

Computer systems are subject to continuously increasing performance demands. However, energy consumption has become a critical issue, both for high-end large-scale parallel systems [12], as well as for portable devices [34]. In other words, more work needs to be done in less time, preferably with the same or smaller energy budget. Future performance and efficiency goals of computer systems can only be reached with large-scale, heterogeneous architectures [6]. Due to their distributed nature, control software is required to coordinate the parallel execution of applications on such platforms. Abstraction, arbitration and multi-objective optimization are only a subset of the tasks this software has to fulfill [6, 31]. The essential problem in all this is the allocation of platform resources to satisfy the needs of an application.\ud \ud This work considers the dynamic resource allocation problem, also known as the run-time mapping problem. This problem consists of task assignment to (processing) elements and communication routing through the interconnect between the elements. In mathematical terms, the combined problem is defined as the multi-resource quadratic assignment and routing problem (MRQARP). An integer linear programming formulation is provided, as well as complexity proofs on the N P-hardness of the problem.\ud \ud This work builds upon state-of-the-art work of Yagiura et al. [39, 40, 42] on metaheuristics for various generalizations of the generalized assignment problem. Specifically, we focus on the guided local search (GLS) approach for the multi-resource quadratic assignment problem (MRQAP). The quadratic assignment problem defines a cost relation between tasks and between elements. We generalize the multi-resource quadratic assignment problem with the addition of a capacitated interconnect and a communication topology between tasks. Numerical experiments show that the performance of the approach is comparable with commercial solvers. The footprint, the time versus quality trade-off and available metadata make guided local search a suitable candidate for run-time mapping

Matheuristics: using mathematics for heuristic design

Matheuristics are heuristic algorithms based on mathematical tools such as the ones provided by mathematical programming, that are structurally general enough to be applied to different problems with little adaptations to their abstract structure. The result can be metaheuristic hybrids having components derived from the mathematical model of the problems of interest, but the mathematical techniques themselves can define general heuristic solution frameworks. In this paper, we focus our attention on mathematical programming and its contributions to developing effective heuristics. We briefly describe the mathematical tools available and then some matheuristic approaches, reporting some representative examples from the literature. We also take the opportunity to provide some ideas for possible future development

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

Low-rank semidefinite programming for the MAX2SAT problem

This paper proposes a new algorithm for solving MAX2SAT problems based on combining search methods with semidefinite programming approaches. Semidefinite programming techniques are well-known as a theoretical tool for approximating maximum satisfiability problems, but their application has traditionally been very limited by their speed and randomized nature. Our approach overcomes this difficult by using a recent approach to low-rank semidefinite programming, specialized to work in an incremental fashion suitable for use in an exact search algorithm. The method can be used both within complete or incomplete solver, and we demonstrate on a variety of problems from recent competitions. Our experiments show that the approach is faster (sometimes by orders of magnitude) than existing state-of-the-art complete and incomplete solvers, representing a substantial advance in search methods specialized for MAX2SAT problems.Comment: Accepted at AAAI'19. The code can be found at https://github.com/locuslab/mixsa

A Feasible Lagrangian Approach with Application to the Generalized Assignment Problem

Lagrangian relaxation is a widely used decomposition approach to solve difficult optimization problems that exhibit special structure. It provides a lower bound on the optimal objective of a minimization problem. On the other hand, an upper bound and quality feasible solutions may be obtained by perturbing solutions of the subproblem. In this thesis, we enhance the Lagrangian approach by using information at the subproblem to push for feasibility to the original problem. We exploit the idea that if the solution for the subproblem is pushed towards feasibility to the original problem, it may lead to improved lower bounds as well as good feasible solutions. Our proposed strategy is to solve the subproblem repeatedly at each iteration of the Lagrangian procedure and strengthen it with valid inequalities. As cuts are added to the subproblem, it inevitably becomes harder to solve. We propose to solve it under a time limit and adjust the Lagrangian bound accordingly. Two variants of the approach are explored that we call a Modified Lagrangian approach and a Feasible Lagrangian approach. We use the Generalized Assignment Problem for testing. We develop two methodologies based on minimal covering inequalities. The first solves the subproblem repeatedly for a given number of iterations and generates minimal cover inequalities that are either discarded or passed on to subsequent Lagrangian iterations. The second starts with initial multipliers and repeatedly solves the subproblem until a feasible solution is attained. At that point, the regular Lagrangian approach is used to find a lower bound. We test on GAP instances from the literature and compare the lower bound to the Lagrangian bound and the feasible solution to the best known solution in the literature. The results demonstrate that the proposed feasible Lagrangian approach leads to improved lower bounds and good quality feasible solutions