Quadratic nonseparable resource allocation problems with generalized bound constraints

We study a 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.

[1]  Gerard J. M. Smit,et al.  DEMKit: a Decentralized Energy Management Simulation and Demonstration Toolkit , 2019, 2019 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe).

[2]  Patrick Jaillet,et al.  Separable Convex Optimization with Nested Lower and Upper Constraints , 2017, INFORMS Journal on Optimization.

[3]  Gerwin Hoogsteen,et al.  Charging electric vehicles, baking pizzas, and melting a fuse in Lochem , 2017 .

[4]  Gerard J. M. Smit,et al.  Multi-commodity support in profile steering , 2017, 2017 IEEE PES Innovative Smart Grid Technologies Conference Europe (ISGT-Europe).

[5]  Andrei Alexandrescu,et al.  Fast Deterministic Selection , 2016, SEA.

[6]  K. Sathish Kumar,et al.  A survey on residential Demand Side Management architecture, approaches, optimization models and methods , 2016 .

[7]  Martijn H. H. Schoot Uiterkamp,et al.  Robust planning of electric vehicle charging , 2016 .

[8]  Jana Vogel,et al.  Elements Of Power System Analysis , 2016 .

[9]  Gerard J. M. Smit,et al.  Demand side management using profile steering , 2015, 2015 IEEE Eindhoven PowerTech.

[10]  Hamidreza Zareipour,et al.  Home energy management systems: A review of modelling and complexity , 2015 .

[11]  Johan Driesen,et al.  Load Balancing With EV Chargers and PV Inverters in Unbalanced Distribution Grids , 2015, IEEE Transactions on Sustainable Energy.

[12]  Hao Xing,et al.  Decentralized Optimal Demand-Side Management for PHEV Charging in a Smart Grid , 2015, IEEE Transactions on Smart Grid.

[13]  Michael Patriksson,et al.  Algorithms for the continuous nonlinear resource allocation problem - New implementations and numerical studies , 2015, Eur. J. Oper. Res..

[14]  Pierluigi Siano,et al.  Demand response and smart grids—A survey , 2014 .

[15]  Ufuk Topcu,et al.  Optimal decentralized protocol for electric vehicle charging , 2011, IEEE Transactions on Power Systems.

[16]  László A. Végh.  Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives , 2011, STOC '12.

[17]  Marina Thottan,et al.  Differentiated services QoS in smart grid communication networks , 2011, Bell Labs Technical Journal.

[18]  Nobuyuki Tsuchimura,et al.  M-Convex Function Minimization by Continuous Relaxation Approach: Proximity Theorem and Algorithm , 2011, SIAM J. Optim..

[19]  Michael Patriksson,et al.  A survey on the continuous nonlinear resource allocation problem , 2008, Eur. J. Oper. Res..

[20]  Krzysztof C. Kiwiel,et al.  Breakpoint searching algorithms for the continuous quadratic knapsack problem , 2007, Math. Program..

[21]  Stephen P. Boyd,et al.  Portfolio optimization with linear and fixed transaction costs , 2007, Ann. Oper. Res..

[22]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[23]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[24]  D. Harville.  Matrix Algebra From a Statistician's Perspective , 1998 .

[25]  Bala Shetty,et al.  Quadratic resource allocation with generalized upper bounds , 1997, Oper. Res. Lett..

[26]  Dorit S. Hochbaum,et al.  About strongly polynomial time algorithms for quadratic optimization over submodular constraints , 1995, Math. Program..

[27]  Dorit S. Hochbaum,et al.  Lower and Upper Bounds for the Allocation Problem and Other Nonlinear Optimization Problems , 1994, Math. Oper. Res..

[28]  Dorit S. Hochbaum,et al.  Strongly Polynomial Algorithms for the Quadratic Transportation Problem with a Fixed Number of Sources , 1994, Math. Oper. Res..

[29]  Arie Tamir,et al.  A strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two-terminal series—parallel networks , 1993, Math. Program..

[30]  Toshihide Ibaraki,et al.  Resource allocation problems - algorithmic approaches , 1988, MIT Press series in the foundations of computing.

[31]  Jeffery L. Kennington,et al.  A polynomially bounded algorithm for a singly constrained quadratic program , 1980, Math. Program..

[32]  E. Spedicato.  A bound to the condition number of canonical rank-two corrections and applications to the variable metric method , 1975 .

[33]  M. V. Wilkes,et al.  The Art of Computer Programming, Volume 3, Sorting and Searching , 1974 .

[34]  Manuel Blum,et al.  Time Bounds for Selection , 1973, J. Comput. Syst. Sci..

[35]  Lalitha Sanathanan,et al.  On an Allocation Problem with Multistage Constraints , 1971, Oper. Res..

[36]  M. Bartlett.  An Inverse Matrix Adjustment Arising in Discriminant Analysis , 1951 .