Abstract
This document proposes a Newton step graph-based model for Quadratic separable Problems (QSP). The Newton step is well suited for this kind of problems, but when the problem size grows the matrix-based QSP model will grow in a non linear manner. Furthermore, handling the constraints becomes the main problem as we have to select the right constraints in the different solution steps. When this happens, the sparse matrix representation is the path to follow, but very little has been made in order to fully explode the sparsity structure. Indeed, the Hessian matrix for the QSP model has a very particular structure which can be exploited by using the graph underlying the problem, this is the approach taken in this document. To this end a graph is built derived from the components involved in the Newton step which describes the solution for the QSP problem. Based on this graph, the gradient can be evaluated directly based on the graph topology, as it will be shown, the information needed for such evaluation is embedded within the graph. These links eventually will guide the solution process in this approach. A deeper analysis of these links is done which leads to its complete understanding. It will be seen that the main effect of the link weakening operation is to allow the computation of the exact gradient. However, the solution will be reinforced by taking into account the second order information provided by the linking structure. Furthermore, the link weakening is used to separate coupled problems which in turn leads to a decentralized scheme. Finally, several decentralization approaches for the Newton step graph-based model for QSP are proposed.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- \( N \) :
-
Number of decision variables
- \( L \) :
-
Number of equality constraints
- M :
-
Number of inequality constraints
- \( z_{i} \) :
-
Decision variable i
- \( \left\lceil {z_{i} } \right\rceil \) :
-
Upper limit value of variable \( z_{i} \)
- \( \left\lfloor {z_{i} } \right\rfloor \) :
-
Lower limit value of variable \( z_{i} \)
- \( \overline{{z_{i} }} \) :
-
Slack variable for \( z_{i} \) upper bound
- \( \underline{{z_{i} }} \) :
-
Slack variable for \( z_{i} \) lower bound
- \( \varDelta z_{i} \) :
-
Variable \( x_{i} \) increment
- \( f(z) \) :
-
Objective function
- \( g_{l} (z) \) :
-
Equality constraint \( l \)
- \( h_{m} (z) \) :
-
Inequality constraint \( m \)
- \( a_{li} \) :
-
\( i \)th coefficient in equality constraint \( l \)
- \( b_{mi} \) :
-
\( i \)th coefficient in inequality constraint m
- \( r_{l} \) :
-
RHS of equality constraint l
- \( s_{m} \) :
-
RHS of inequality constraint m
- \( \lambda_{l} \) :
-
Dual variable for equality constraint l
- \( \mu_{m} \) :
-
Dual variable for inequality constraint m
- \( \overline{\rho }_{i} \) :
-
Dual variable for \( z_{i} \) upper bound
- \( \underline{\rho }_{i} \) :
-
Dual variable for \( z_{i} \) lower bound
- \( \wp (S) \) :
-
Power set of set S
- \( \varGamma_{i} \) :
-
Set of variables connected to \( z_{i} \)
References
A.G. Bakirtzis, P.N. Biskas, A decentralized solution of the dc-opf of interconnected power systems. IEEE Trans. Power Syst. 18(3), 1007–1013 (2003)
A. Bakirtzis, P.N. Biskas, N. Macheras, N. Pasialis, A decentralized implementation of dc optimal power flow on a network of computers. IEEE Trans. Power Syst. 20(1), 25–33 (2000)
P. Biskas, A. Bakirtzis, Decentralised opf of large multiarea power systems. IEE Proc. Gener. Transm. Distrib. 153(1), 99–105 (2006)
A.J. Conejo, J.A. Aguado, Multi-area coordinated decentralized dc optimal power flow. IEEE Trans. Power Syst. 13(4), 1272–1278 (1998)
G. Cohen, Optimization by decomposition and coordination: a unified approach. IEEE Trans. Autom. Control 2(2), 222–232 (1978)
J. Cerda, D. De Roure, A graph-based decomposition method for quadratic separable programs, in SIAM Conference on Optimization, (Society for Industria and Applied mathematics, SIAM, Ed. Boston, Massachusetts, USA) 10–13 May 2008
J. Cerda, A. Avalos, A graph model proposal for convex non linear separable problems with linear constraints, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering and Computer Science, WCECS 2013, San Francisco, CA, USA, pp. 129–133, 23–25 Oct 2012
J. Cerda, D. De Roure, A graph-based economical dispatch model, in Proceedings of WORLDCOMP 2010: Foundation of Computer Sciences, ed. by H.R. Arabnia, G.A. Gravvanis, A.M.G. Solo (CSREA Press, Las Vegas, Nevada, USA) pp. 132–138, 6–9 April 2010
P. Cira, J. Cerda, J.J. Flores, A graph-based economical dispatch model, in Procedia Technology: Proceedings of The 2012 Iberoamerican Conference on Electronics Engineering and Computer Science, vol. 3 (Elsevier, Guadalajara, Mexico) pp. 304–315, 23–25 Oct 2012
J. Cerda, M. Graff, A graph-based decentralization proposal for convex non linear separable problems with linear constraints, in Lecture Notes in Engineering and Computer Science: Proceedings of The World Congress on Engineering and Computer Science, WCECS 2013, San Francisco, CA, USA, pp. 41–45, 23–25 Oct 2012
N. Jennings, M. Wooldridge, Intelligent agents: theory and practice. Knowl. Eng. Rev. 10(2), 115–152 (1995)
J. Cerda, D. De Roure, E. Gerding, An agent-based electrical power market simulator, in AAMAS ‘08: Proceedings of the 7th International Joint Conference on Autonomous Agents and Multi-agent Systems. Richland, SC: International Foundation for Autonomous Agents and Multiagent Systems, 2008, pp. 1655–1656
J. Cerda, D. De Roure, An agent-based decentralisation approach for the electricity power market, in Proceedings of the 2010 IEEE Electronics, Robotics and Automotive Mechanics Conference, ed. by P. Kellenberger (IEEE, Cuernavaca Morelos, México), pp. 666–671, Sept 2010
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Cerda, J., Avalos, A., Graff, M. (2014). A Graph-Based Model for Quadratic Separable Programs and Its Decentralization. In: Kim, H., Ao, SI., Amouzegar, M. (eds) Transactions on Engineering Technologies. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9115-1_18
Download citation
DOI: https://doi.org/10.1007/978-94-017-9115-1_18
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-017-9114-4
Online ISBN: 978-94-017-9115-1
eBook Packages: EngineeringEngineering (R0)