On a reduction for a class of resource allocation problems

In the resource allocation problem (RAP), the goal is to divide a given amount of resource over a set of activities while minimizing the cost of this allocation and possibly satisfying constraints on allocations to subsets of the activities. Most solution approaches for the RAP and its extensions allow each activity to have its own cost function. However, in many applications, often the structure of the objective function is the same for each activity and the difference between the cost functions lies in different parameter choices such as, e.g., the multiplicative factors. In this article, we introduce a new class of objective functions that captures the majority of the objectives occurring in studied applications. These objectives are characterized by a shared structure of the cost function depending on two input parameters. We show that, given the two input parameters, there exists a solution to the RAP that is optimal for any choice of the shared structure. As a consequence, this problem reduces to the quadratic RAP, making available the vast amount of solution approaches and algorithms for the latter problem. We show the impact of our reduction result on several applications and, in particular, we improve the best known worst-case complexity bound of two important problems in vessel routing and processor scheduling from $O(n^2)$ to $O(n \log n)$

A fast algorithm for quadratic resource allocation problems with nested constraints

We study the quadratic resource allocation problem and its variant with lower and upper constraints on nested sums of variables. This problem occurs in many applications, in particular battery scheduling within decentralized energy management (DEM) for smart grids. We present an algorithm for this problem that runs in $O(n \log n)$ time and, in contrast to existing algorithms for this problem, achieves this time complexity using relatively simple and easy-to-implement subroutines and data structures. This makes our algorithm very attractive for real-life adaptation and implementation. Numerical comparisons of our algorithm with a subroutine for battery scheduling within an existing tool for DEM research indicates that our algorithm significantly reduces the overall execution time of the DEM system, especially when the battery is expected to be completely full or empty multiple times in the optimal schedule. Moreover, computational experiments with synthetic data show that our algorithm outperforms the currently most efficient algorithm by more than one order of magnitude. In particular, our algorithm is able to solves all considered instances with up to one million variables in less than 17 seconds on a personal computer

A Class of Convex Quadratic Nonseparable Resource Allocation Problems with Generalized Bound Constraints

We study a convex quadratic nonseparable resource allocation problem that arises in the area of decentralized energy management (DEM), where unbalance in electricity networks has to be minimized. In this problem, the given resource is allocated over a set of activities that is divided into subsets, and a cost is assigned to the overall allocated amount of resources to activities within the same subset. We derive two efficient algorithms with $O(n \log  n)$ worst-case time complexity to solve this problem. For the special case where all subsets have the same size, one of these algorithms even runs in linear time given the subset size. Both algorithms are inspired by well-studied breakpoint search methods for separable convex resource allocation problems. Numerical evaluations on both real and synthetic data confirm the theoretical efficiency of both algorithms and demonstrate their suitability for integration in DEM systems

ACCELERATING DECENTRALIZED ENERGY TRANSITIONS: A SOCIO-TECHNICAL PERSPECTIVE

Whereas past transitions were often long multi-decadal affairs, the current energy transition requires a much shorter time horizon. Reducing carbon emissions to avoid the worst impacts of climate change is essential. Socially and technologically driven pressures are creating opportunities to observe accelerated social-technical change in action. By observing ongoing accelerated transitions, the goal of this dissertation is to further the understanding of the mechanisms of these transitions. This dissertation asks two questions: (1) In the context of accelerated social and technical change, is society or technology the driver? And (2) how can an understanding of this dynamic be used to further accelerate social and technical change? To explore these research questions, this dissertation focuses on a case study of a particular accelerated transition that is currently unfolding—decentralized energy. To operationalize answering the addressing questions, comparative research alongside an in-depth case study analysis was conducted. The dissertation is divided into five manuscript chapters. The first manuscript, Chapter Two, begins with an overall discussion on decentralized energy: its opportunities, challenges, and justice considerations. The next manuscript, Chapter Three, compares the governance dimensions of decentralized energy transitions in three medium-sized northern cities. Using the same three case studies, Chapter Four compares the case studies using energy futures analysis. The remaining two manuscripts, Chapter Five and Chapter Six focus on a single case study of solar energy in Saskatchewan. In Chapter Five, the paper explores the idea of effective public engagement that considers how energy justice issues can be used to drive DE transitions. Chapter Six builds from the previous chapter and argues for practical suggestions to accelerate DE transitions based on observations from the public engagement activities and a discussion on decision-making. This dissertation concludes with three insights that synthesize the aggregated findings. (1) There are unintended consequences to accelerated energy transitions. Energy justice can be used as a framework to unearth tensions and potentially attempt to predict where unintended consequences may appear. (2) A transformed role of the state is needed to facilitate acceleration, one that employs a more interactive form of governance and public policy. (3) Further research that uses a comparative approach with a focus on governance dimensions can lead to more useful insights to understand accelerated transitions

Fast exact algorithms for optimization problems in resource allocation and switched linear systems

University of Minnesota Ph.D. dissertation.June 2019. Major: Industrial Engineering. Advisor: Qie He. 1 computer file (PDF); x, 138 pages.Discrete optimization is a branch of mathematical optimization where some of the decision variables are restricted to real values in a discrete set. The use of discrete decision variables greatly expands the scope and capacity of mathematical optimization models. In the era of big data, efficiency and scalability are increasingly important in evaluating the performance of an algorithm. However, discrete optimization problems usually are challenging to solve. In this thesis, we develop new fast exact algorithms for discrete optimization problems arising in the field of resource allocation and switched linear systems. The first problem is the discrete resource allocation problem with nested bound constraints. It is a fundamental problem with a wide variety of applications in search theory, economics, inventory systems, etc. Given $B$ units of resource and $n$ activities, each of which associated with a convex allocation cost $f_i(\cdot)$, we aim to find an allocation of resources to the $n$ activities, denoted by \bm{x} \in \Ze^n, to minimize the total allocation cost $\sum\limits_{i = 1}^{n} f_i(x_i)$ subject to the total amount of resource constraint as well as lower and upper bound constraints on total resource allocated to subsets of activities. We develop a $\Theta(n^2\log\frac{B}{n})$-time algorithm for it. It is an infeasibility-guided divide-and-conquer algorithm and the worst-case complexity is usually not achieved. Numerical experiments demonstrate that our algorithm significantly outperforms a state-of-the-art optimization solver and the performance of our algorithm is competitive compared to the algorithm with the best worst-case complexity for this problem in the literature. The second problem is the minimum convex cost network flow problem on the dynamic lot size network. In the dynamic lot size network, there are one source node and $n$ sink nodes with demand $d_i, i = 1, \dots, n$. Let $B = \sum_{i=0}^{n}d_i$ be the total demand. We aim to find a flow $\bm{x}$ to minimize the total arc cost and satisfy all the flow balance and capacity constraints. Many optimization models in the literature can be seen as special cases of this problem, including dynamic lot-sizing problem and speed optimization. It is also a generalization of the first problem. We develop the Scaled Flow-improving Algorithm. For the continuous problem, our algorithm finds a solution that is at most $\epsilon$ away from an optimal solution in terms of the infinity norm in $O(n^2\log{\frac{B}{n\epsilon}})$ time. For the integer problem, our algorithm terminates in $O(n^2\log\frac{B}{n})$ time. Our algorithm has the best worst-case complexity in the literature. In particular, it solves the discrete resource allocation problem with nested bound constraints in $O(n\log{n}\log\frac{B}{n})$ time and it also achieves the best worst-case complexity for that problem. We conduct extensive numerical experiments on instances with a variety of convex objectives. The numerical result demonstrates the efficiency of our algorithm in solving large-sized instances. The last problem is the optimal control problem in switched linear systems. We consider the following dynamical system that consists of several linear subsystems: $K$ matrices, each chosen from the given set of matrices, to maximize a convex function over the product of the $K$ matrices and the given vector.This simple problem has many applications in operations research and control, yet a moderate-sized instance is challenging to solve to optimality for state-of-the-art optimization software. We prove the problem is NP-hard. We propose a simple exact algorithm for this problem. Our algorithm runs in polynomial time when the given set of matrices has the oligo-vertex property, a concept we introduce for a set of matrices. We derive several easy-to-verify sufficient conditions for a set of matrices to have the oligo-vertex property. In particular, we show that a pair of binary matrices has the oligo-vertex property. Numerical results demonstrate the clear advantage of our algorithm in solving large-sized instances of the problem over one state-of-the-art global solver. We also pose several open questions on the oligo-vertex property and discuss its potential connection with the finiteness property of a set of matrices, which may be of independent interest

Resource allocation problems in decentralized energy management

Changes in our electricity supply chain are causing a paradigm shift from centralized control towards decentralized energy management. Within the framework of decentralized energy management, devices that offer flexibility in their load profile play an important role. These devices schedule their flexible load profile based on steering signals received from centralized controllers. The problem of finding optimal device schedules based on the received steering signals falls into the framework of resource allocation problems. We study an extension of the traditional problems studied within resource allocation and prove that a divide-and-conquer strategy gives an optimal solution for the considered extension. This leads to an efficient recursive algorithm, with quadratic complexity in the practically relevant case of quadratic objective functions. Furthermore, we study discrete variants of two problems common in decentralized energy management. We show that these problems are NP-hard and formulate natural relaxations of both considered discrete problems that we solve efficiently. Finally, we show that the solutions to the natural relaxations closely resemble solutions to the original, hard problems