Transactions on Engineering Technologies pp 229-245 | Cite as

# A Graph-Based Model for Quadratic Separable Programs and Its Decentralization

## 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.

### Keywords

DC-OPF Decentralization Graphs NLP Optimization QSP### List of Symbols

- \( N \)
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 \( z_{i} \)

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