We propose a unifying framework based on configuration linear programs and randomized rounding, for different energy optimization problems in the dynamic speedscaling setting. We apply our framework to various scheduling and routing problems in heterogeneous computing and networking environments. We first consider the energy minimization problem of scheduling a set of jobs on a set of parallel speed scalable processors in a fully heterogeneous setting. For both the preemptive-non-migratory and the preemptive-migratory variants, our approach allows us to obtain solutions of almost the same quality as for the homogeneous environment. By exploiting the result for the preemptive-non-migratory variant, we are able to improve the best known approximation ratio for the single processor non-preemptive problem. Furthermore, we show that our approach allows to obtain a constant-factor approximation algorithm for the power-aware preemptive job shop scheduling problem. Finally, we consider the min-power routing problem where we are given a network modeled by an undirected graph and a set of uniform demands that have to be routed on integral routes from their sources to their destinations so that the energy consumption is minimized. We improve the best known approximation ratio for this problem.

Mathematical optimization offers highly-effective tools for finding solutions for problems with well-defined goals, notably scheduling. However, optimization solvers are often unexplainable black boxes whose solutions are inaccessible to users and which users cannot interact with. We define a novel paradigm using argumentation to empower the interaction between optimization solvers and users, supported by tractable explanations which certify or refute solutions. A solution can be from a solver or of interest to a user (in the context of ‘what-if’ scenarios). Specifically, we define argumentative and natural language explanations for why a schedule is (not) feasible, (not) efficient or (not) satisfying fixed user decisions, based on models of the fundamental makespan scheduling problem in terms of abstract argumentation frameworks (AFs). We define three types of AFs, whose stable extensions are in one-to-one correspondence with schedules that are feasible, efficient and satisfying fixed decisions, respectively. We extract the argumentative explanations from these AFs and the natural language explanations from the argumentative ones.

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Decision trees usefully represent sparse, high-dimensional, and noisy data. Having learned a function from these data, we may want to thereafter integrate the function into a larger decision-making problem, for example, for picking the best chemical process catalyst. We study a large-scale, industrially relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pretrained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models or they may wish to optimize a discrete model that accurately represents a data set. We develop several heuristic methods to find feasible solutions and an exact branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on a concrete mixture design instance and a chemical catalysis industrial instance.

We study the problem of scheduling a set of jobs with release dates, deadlines and processing requirements (or works), on parallel speed-scaled processors so as to minimize the total energy consumption. We consider that both preemption and migration of jobs are allowed. An exact polynomial-time algorithm has been proposed for this problem, which is based on the Ellipsoid algorithm. Here, we formulate the problem as a convex program and we propose a simpler polynomial-time combinatorial algorithm which is based on a reduction to the maximum flow problem. Our algorithm runs in O(nf (n)logP ) time, where n is the number of jobs, P is the range of all possible values of processors' speeds divided by the desired accuracy and f (n) is the complexity of computing a maximum flow in a layered graph with O(n) vertices. Independently, Albers et al. [3] proposed an O(n 2 f (n))-time algorithm exploiting the same relation with the maximum flow problem. We extend our algorithm to the multiprocessor speed scaling problem with migration where the objective is the minimization of the makespan under a budget of energy.

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We study the problem of scheduling a set of jobs with release dates, deadlines and processing requirements (or works) on parallel speed scalable processors so as to minimize the total energy consumption. We consider that both preemptions and migrations of jobs are allowed. For this problem, there exists an optimal polynomial-time algorithm which uses as a black box an algorithm for linear programming. Here, we formulate the problem as a convex program and we propose a combinatorial polynomial-time algorithm which is based on finding maximum flows. Our algorithm runs in O(nf (n) log U) time, where n is the number of jobs, U is the range of all possible values of processors' speeds divided by the desired accuracy and f (n) is the time needed for computing a maximum flow in a layered graph with O(n) vertices.