Mathematics

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(nlogn) 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.

Optimal experimental design (OED) is a principled framework for maximizing information gained from limited data in inverse problems. Unfortunately, conventional methods for OED are prohibitive when applied to expensive models with high-dimensional parameters, as we target here. We develop a fast and scalable computational framework for goal-oriented OED of large-scale Bayesian linear inverse problems that finds sensor locations to maximize the expected information gain (EIG) for a predicted quantity of interest. By employing low-rank approximations of appropriate operators, an online-offline decomposition, and a new swapping greedy algorithm, we are able to maximize EIG at a cost measured in model solutions that is independent of the problem dimensions. We demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm for a contaminant transport inverse problem with infinite-dimensional parameter field.

In this article, we study algorithms for nonnegative matrix factorization (NMF) in various applications involving streaming data. Utilizing the continual nature of the data, we develop a fast two-stage algorithm for highly efficient and accurate NMF. In the first stage, an alternating non-negative least squares (ANLS) framework is used, in combination with active set method with warm-start strategy for the solution of subproblems. In the second stage, an interior point method is adopted to accelerate the local convergence. The convergence of the proposed algorithm is proved. The new algorithm is compared with some existing algorithms in benchmark tests using both real-world data and synthetic data. The results demonstrate the advantage of our algorithm in finding high-precision solutions.

A numerical method is developed to solve linear semi-infinite programming problem (LSIP) in which the iterates produced by the algorithm are feasible for the original problem. This is achieved by constructing a sequence of standard linear programming problems with respect to the successive discretization of the index set such that the approximate regions are included in the original feasible region. The convergence of the approximate solutions to the solution of the original problem is proved and the associated optimal objective function values of the approximate problems are monotonically decreasing and converge to the optimal value of LSIP. An adaptive refinement procedure is designed to discretize the index set and update the constraints for the approximate problem. Numerical experiments demonstrate the performance of the proposed algorithm.

This paper introduces a general framework for solving constrained convex quaternion optimization problems in the quaternion domain. To soundly derive these new results, the proposed approach leverages the recently developed generalized HR -calculus together with the equivalence between the original quaternion optimization problem and its augmented real-domain counterpart. This new framework simultaneously provides rigorous theoretical foundations as well as elegant, compact quaternion-domain formulations for optimization problems in quaternion variables. Our contributions are threefold: (i) we introduce the general form for convex constrained optimization problems in quaternion variables, (ii) we extend fundamental notions of convex optimization to the quaternion case, namely Lagrangian duality and optimality conditions, (iii) we develop the quaternion alternating direction method of multipliers (Q-ADMM) as a general purpose quaternion optimization algorithm. The relevance of the proposed methodology is demonstrated by solving two typical examples of constrained convex quaternion optimization problems arising in signal processing. Our results open new avenues in the design, analysis and efficient implementation of quaternion-domain optimization procedures.

We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its efficient solution.

Considering the sequential clearing of energy and reserves in Europe, enabling inter-area reserve exchange requires optimally allocating inter-area transmission capacities between these two markets. To achieve this, we provide a market-based allocation framework and derive payments with desirable properties. The proposed min-max least core selecting payments achieve individual rationality, budget balance, and approximate incentive compatibility and coalitional stability. The results extend the works on private discrete items to a network of continuous public choices.

Given a graph G=(V,E) , the b -coloring problem consists in attributing a color to every vertex in V such that adjacent vertices receive different colors, every color has a b -vertex, and the number of colors is maximized. A b -vertex is a vertex adjacent to vertices colored with all used colors but its own. The b -coloring problem is known to be NP-Hard and its optimal solution determines the b -chromatic number of G , denoted ? b (G) . This paper presents an integer programming formulation and a very effective multi-greedy randomized heuristic which can be used in a multi-start metaheuristic. In addition, a matheuristic approach is proposed combining the multi-start multi-greedy randomized metaheuristic with a MIP (mixed integer programming) based local search procedure using the integer programming formulation. Computational experiments establish the proposed multi-start metaheuristic as very effective in generating high quality solutions, along with the matheuristic approach successfully improving several of those results. Moreover, the computational results show that the multi-start metaheuristic outperforms a state-of-the-art hybrid evolutionary metaheuristic for a subset of the large instances which were previously considered in the literature. An additional contribution of this work is the proposal of a benchmark instance set, which consists of newly generated instances as well as others available in the literature for classical graph problems, with the aim of standardizing computational comparisons of approaches for the b -coloring problem in future works.

Knowledge spillovers occur when a firm researches a new technology and that technology is adapted or adopted by another firm, resulting in a social value of the technology that is larger than the initially predicted private value. As a result, firms systematically under--invest in research compared with the socially optimal investment strategy. Understanding the level of under--investment, as well as policies to correct it, is an area of active economic research. In this paper, we develop a new model of spillovers, taking inspiration from the available microeconomic data. We prove existence and uniqueness of solutions to the model, and we conduct some initial simulations to understand how indirect spillovers contribute to the productivity of a sector.

Mathematical models of street traffic allowing assessment of the importance of their individual segments for the functionality of the street system is considering. Based on methods of cooperative games and the reliability theory the suitable measure is constructed. The main goal is to analyze methods for assessing the importance (rank) of road fragments, including their functions. A relevance of these elements for effective accessibility for the entire system will be considered.