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Efficient high-precision matrix algebra on parallel architectures for nonlinear combinatorial optimization

MATHEMATICAL PROGRAMMING COMPUTATION 2010年第2期2卷 103-124页

作者： Gunnels, John Lee, Jon Margulies, Susan IBM TJ Watson Res Ctr Yorktown Hts NY 10598 USA Rice Univ Dept Computat & Appl Math Houston TX USA

We provide a first demonstration of the idea that matrix-based algorithms for nonlinear combinatorial optimization problems can be efficiently implemented. Such algorithms were mainly conceived by theoretical computer scientists for proving efficiency. We are able to demonstrate the practicality of our approach by developing an implementation on a massively parallel architecture, and exploiting scalable and efficient parallel implementations of algorithms for ultra high-precision linear algebra. Additionally, we have delineated and implemented the necessary algorithmic and coding changes required in order to address problems several orders of magnitude larger, dealing with the limits of scalability from memory footprint, computational efficiency, reliability, and interconnect perspectives.

关键词： nonlinear combinatorial optimization Matroid optimization High-performance computing High-precision linear algebra

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Asset-Task Assignment Algorithms in the Presence of Execution Uncertainty

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COMPUTER JOURNAL 2011年第9期54卷 1514-1525页

作者： Timotheou, Stelios Univ London Imperial Coll Sci Technol & Med Dept Elect & Elect Engn Intelligent Syst & Networks Grp London SW7 2BT England

We investigate the assignment of assets to tasks where each asset can potentially execute any of the tasks, but assets execute tasks with a probabilistic outcome of success. There is a cost associated with each possible assignment of an asset to a task, and if a task is not executed there is also a cost associated with the non-execution of the task. As we proposed in [Gelenbe, E., Timotheou, S., and Nicholson, D. (2010). Fast distributed near optimum assignment of assets to tasks. Comput. J., doi:10.1093/comjnl/bxq010], we formulate the allocation of assets to tasks in order to minimize the overall expected cost, as a nonlinear combinatorial optimization problem. We propose the use of network flow algorithms which are based on solving a sequence of minimum cost flow problems on appropriately constructed networks with estimated arc costs. We introduce three different schemes for the estimation of the arc costs and we investigate their performance compared with a random neural network algorithm and a greedy algorithm. We also develop an approach for obtaining tight lower bounds to the optimal solution based on a piecewise linear approximation of the considered problem.

关键词： optimum asset assignment execution uncertainty random neural networks nonlinear combinatorial optimization network flow problems piecewise linear approximation lower bounds

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Tridiagonal maximum-entropy sampling and tridiagonal masks

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DISCRETE APPLIED MATHEMATICS 2023年第1期337卷 120-138页

作者： Al-Thani, Hessa Lee, Jon Univ Michigan IOE Dept Ann Arbor MI 48109 USA

The NP-hard maximum-entropy sampling problem (MESP) seeks a maximum (log-) determinant principal submatrix, of a given order, from an input covariance matrix C. We give an efficient dynamic-programming algorithm for MESP when C (or its inverse) is tridiagonal and generalize it to the situation where the support graph of C (or its inverse) is a spider graph with a constant number of legs (and beyond). We give a class of arrowhead covariance matrices C for which a natural greedy algorithm solves MESP. A mask M for MESP is a correlation matrix with which we pre-process C, by taking the Hadamard product M degrees C. Upper bounds on MESP with M degrees C give upper bounds on MESP with C. Most upper-bounding methods are much faster to apply, when the input matrix is tridiagonal, so we consider tridiagonal masks M (which yield tridiagonal M degrees C). We make a detailed analysis of such tridiagonal masks, and develop a combinatorial local-search based upper-bounding method that takes advantage of fast computations on tridiagonal matrices.(c) 2023 Elsevier B.V. All rights reserved.

关键词： nonlinear combinatorial optimization Covariance matrix Differential entropy Maximum-entropy sampling Dynamic programming Local search Spider Arrowhead Tridiagonal Mask Correlation matrix

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Fast Distributed Near-Optimum Assignment of Assets to Tasks

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COMPUTER JOURNAL 2010年第9期53卷 1360-1369页

作者： Gelenbe, Erol Timotheou, Stelios Nicholson, David Univ London Imperial Coll Sci Technol & Med Dept Elect & Elect Engn London SW7 2BT England BAE Syst Bristol BS34 7QW Avon England

We investigate the assignment of assets to tasks where each asset can potentially execute any of the tasks, but assets execute tasks with a probabilistic outcome of success. There is a cost associated with each possible assignment of an asset to a task, and if a task is not executed, there is also a cost associated with the non-execution of the task. Thus, any assignment of assets to tasks will result in an expected overall cost which we wish to minimize. We formulate the allocation of assets to tasks in order to minimize this expected cost, as a nonlinear combinatorial optimization problem. A neural network approach for its approximate solution is proposed based on selecting parameters of a random neural network (RNN), solving the network in equilibrium, and then identifying the assignment by selecting the neurons whose probability of being active is the highest. Evaluations of the proposed approach are conducted by comparison with the optimum (enumerative) solution as well as with a greedy approach over a large number of randomly generated test cases. The evaluation indicates that the proposed RNN-based algorithm is better in terms of performance than the greedy heuristic, consistently achieving on average results within 5% of the cost obtained by the optimal solution for all problem cases considered. The RNN-based approach is fast and is of low polynomial complexity in the size of the problem, while it can be used for decentralized decision making.

关键词： optimum asset assignment distributed decisions nonlinear combinatorial optimization random neural network

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A priority based genetic algorithm for nonlinear transportation costs problems

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COMPUTERS & INDUSTRIAL ENGINEERING 2016年 96卷 86-95页

作者： Tari, Farhad Ghassemi Hashemi, Zahra Sharif Univ Technol Dept Ind Engn Azadi AvePOB 11155-9414 Tehran Iran

In this manuscript, a vehicle allocation problem involving a heterogeneous fleet of vehicles for delivering products from a manufacturing firm to a set of depots is considered. Each depot has a specific order quantity and transportation costs consist of fixed and variable transportation cost. The objective is to assign the proper type and number of vehicle to each depot route to minimize the total transportation costs. It is assumed that the number of chartering vehicle types is limited. It is also assumed that a discount mechanism is applied to the vehicles renting cost. The discount mechanism is applied to the fixed cost, based on the number of vehicles to be rented. A mathematical programming model is proposed which is then converted to a mixed 0-1 integer programming model. Due to the computational complexity of the proposed mathematical model, a priority based genetic algorithm capable of solving the real world size problems was proposed. A computational experiment is conducted through which, the performance of the proposed algorithm is evaluated. The results reveal that the proposed algorithm is capable of providing the astonishing solutions with minimal computational effort, comparing with the CPLEX solutions. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Fixed cost transportation Priority genetic algorithm Discounted transportation costs nonlinear combinatorial optimization

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Network Topology optimization via Deep Reinforcement Learning

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IEEE TRANSACTIONS ON COMMUNICATIONS 2023年第5期71卷 2847-2859页

作者： Li, Zhuoran Wang, Xing Pan, Ling Zhu, Lin Wang, Zhendong Feng, Junlan Deng, Chao Huang, Longbo Tsinghua Univ Inst Interdisciplinary Informat Sci IIIS Beijing 100084 Peoples R China China Mobile Res Inst Beijing 100053 Peoples R China

Topology impacts important network performance metrics, including link utilization, throughput and latency, and is of central importance to network operators. However, due to the combinatorial nature of network topology, it is extremely difficult to obtain an optimal solution, especially since topology planning in networks also often comes with management-specific constraints. As a result, local optimization with hand-tuned heuristic methods from human experts is often adopted in practice. Yet, heuristic methods cannot cover the global topology design space while taking into account constraints, and cannot guarantee to find good solutions. In this paper, we propose a novel deep reinforcement learning (DRL) algorithm for graph searching, called DRL-GS, for network topology optimization. DRL-GS consists of three novel components, including a verifier to validate the correctness of a generated network topology, a graph neural network (GNN) to efficiently approximate topology rating, and a DRL agent to conduct a topology search. DRL-GS can efficiently search over relatively large topology space and output topology with satisfactory performance. We conduct a case study based on a real-world network scenario, and our experimental results demonstrate the superior performance of DRL-GS in terms of both efficiency and performance.

关键词： Network topology Topology optimization Approximation algorithms Graph neural networks Costs Planning nonlinear combinatorial optimization deep reinforcement learning graph neural network

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APPROXIMATE nonlinear optimization OVER WEIGHTED INDEPENDENCE SYSTEMS

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SIAM JOURNAL ON DISCRETE MATHEMATICS 2009年第4期23卷 1667-1681页

作者： Lee, Jon Onn, Shmuel Weismantel, Robert IBM Corp Thomas J Watson Res Ctr Yorktown Hts NY 10598 USA Technion Israel Inst Technol IL-32000 Haifa Israel Otto VonGuericke Univ Magdegurg D-39106 Magdeburg Germany

We consider optimizing a nonlinear objective function over a weighted independence system presented by a linear-optimization oracle. We provide an efficient algorithm that determines an r-best solution for nonlinear functions of the total weight of an independent set, where r depends only on certain Frobenius numbers of the individual weights and is independent of the size of the ground set. In contrast, we show that finding an optimal (0-best) solution requires exponential time.

关键词： nonlinear combinatorial optimization independence system approximation algorithm Frobenius numbers

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Exact mixed-integer quadratic formulation and solution for large-scale thermal unit commitment

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INTERNATIONAL JOURNAL OF LOW-CARBON TECHNOLOGIES 2024年第19期19卷 1003-1012页

作者： Kang, Chuanxiong Wang, Yongwen Wu, Shaofei Ding, Guili Chen, Chen Nanchang Inst Technol Natl & Prov Joint Engn Lab Hydraul Engn Safety & E 289 Tianxiang Rd Nanchang 330099 Peoples R China Nanchang Inst Technol Jiangxi Engn Res Ctr High Power Elect & Grid Smart 289 Tianxiang Rd Nanchang 330099 Peoples R China China Renewable Energy Engn Inst 57 Andingmenwai St Beijing 100120 Peoples R China

Thermal unit commitment (UC) is a nonlinear combinatorial optimization problem that minimizes total operating costs while considering system load balance, on/off restrictions and other constraints. Successfully solving the thermal UC problem contributes to a more reliable power system and reduces thermal costs. This paper presents an exact mixed-integer quadratic programming (EMIQP) method for large-scale thermal UC problems. EMIQP revolutionizes the landscape by seamlessly translating the intricate nonlinear combinatorial optimization problem of UC into an exact mixed-integer quadratic formulation. This approach also elegantly reimagines on/off constraints as mixed-integer linear equations, employing both the sum and respective approaches. Our case studies unequivocally demonstrate the exceptional prowess of the EMIQP method, consistently securing the global optimum. Moreover, the mathematical-based EMIQP method produces identical results at each run, which is extremely important for UC in the real world.

关键词： thermal unit commitment nonlinear combinatorial optimization mixed-integer quadratic programming exact formulation

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A faster algorithm for the resource allocation problem with convex cost functions

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JOURNAL OF DISCRETE ALGORITHMS 2015年 34卷 137-146页

作者： Shi, Cong Zhang, Huanan Qin, Chao Univ Michigan Ind & Operat Engn Ann Arbor MI 48109 USA Northwestern Univ Ind Engn & Management Sci Evanston IL 60208 USA

We revisit the classical resource allocation problem with general convex objective functions, subject to an integer knapsack constraint. This class of problems is fundamental in discrete optimization and arises in a wide variety of applications. In this paper, we propose a novel polynomial-time divide-and-conquer algorithm (called the multi-phase algorithm) and prove that it has a computational complexity of O(n log n log N), which outperforms the best known polynomial-time algorithm with O(n( log N)(2)). (C) 2015 Elsevier B.V. All rights reserved.

关键词： Polynomial-time algorithms Complexity analysis Data structure nonlinear combinatorial optimization

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Two-level reconfiguration algorithm of branch exchange and variable neighbourhood search for active distribution network

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SYSTEMS SCIENCE & CONTROL ENGINEERING 2018年第3期6卷 109-117页

作者： Hao, Qingxiang Gao, Zhengzhong Bai, Xingzhen Cao, Maoyong Shandong Univ Sci & Technol Coll Elect Engn & Automat Qingdao Peoples R China

In this paper, a two-level algorithm is proposed to solve the distribution network reconfiguration with an objective of minimum power loss. In the first level reconfiguration, switches of maximum power loss reduction are disconnected by the branch exchange (BE) algorithm. Based on the results, neighbourhoods of disconnected switches are constructed by the deterministic transform method in the second level. The variable neighbourhood search (VNS) algorithm keeps searching the neighbourhoods to obtain a better solution with a lower power loss. Simulations are carried out on IEEE33 and PG&69 distribution networks to verify the superiority of the proposed algorithm. The obtained results are compared with the other methods available in the paper. It can be concluded that the presented method has both high stability and rapidity.

关键词： Distribution system reconfiguration nonlinear combinatorial optimization branch exchange algorithm variable neighbourhood search algorithm distributed generation

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