Auxiliary problem principle extended to variational inequalities

1988 ◽  
Vol 59 (2) ◽  
pp. 325-333 ◽  
Author(s):  
G. Cohen
Keyword(s):  
Variational Inequalities ◽  
Auxiliary Problem ◽  
Auxiliary Problem Principle

scholarly journals A Predictor-Corrector Method for Solving Equilibrium Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Zong-Ke Bao ◽  
Ming Huang ◽  
Xi-Qiang Xia
Keyword(s):  
Variational Inequalities ◽  
Convergence Analysis ◽  
Equilibrium Problems ◽  
Auxiliary Problem ◽  
The Other ◽  
Stationary Points ◽  
Stopping Criterion ◽  
Auxiliary Problem Principle ◽  
One Step ◽  
Predictor Corrector

We suggest and analyze a predictor-corrector method for solving nonsmooth convex equilibrium problems based on the auxiliary problem principle. In the main algorithm each stage of computation requires two proximal steps. One step serves to predict the next point; the other helps to correct the new prediction. At the same time, we present convergence analysis under perfect foresight and imperfect one. In particular, we introduce a stopping criterion which gives rise toΔ-stationary points. Moreover, we apply this algorithm for solving the particular case: variational inequalities.

scholarly journals Decentralized Voltage Optimization Based on the Auxiliary Problem Principle in Distribution Networks with DERs

2021 ◽  
Vol 11 (10) ◽  
pp. 4509
Author(s):  
Anna Rita Di Fazio ◽  
Chiara Risi ◽  
Mario Russo ◽  
Michele De Santis
Keyword(s):  
Distribution Systems ◽  
Auxiliary Problem ◽  
Optimization Procedure ◽  
Distribution Networks ◽  
Small Dimension ◽  
Quadratic Program ◽  
Voltage Profile ◽  
Distributed Energy ◽  
Auxiliary Problem Principle ◽  
Coordination Process

This paper addresses the problem of optimizing the voltage profile of radially-operated distribution systems by acting on the active and reactive powers provided by distributed energy resources (DERs). A novel voltage optimization procedure is proposed by adopting a decentralized control strategy. To this aim, a centralized voltage optimization problem (VOP), minimizing the distance of all the nodal voltages from their reference values, is firstly formulated as a strictly-convex quadratic program. Then, the centralized VOP is rewritten by partitioning the network into voltage control zones (VCZs) with pilot nodes. To overcome the lack of strictly convexity determined by the reduction to the pilot nodes, the dual centralized VOP working on the augmented Lagrangian function is reformulated and iteratively solved by the method of multipliers. Finally, a fully-distributed VOP solution is obtained by applying a distributed algorithm based on the auxiliary problem principle, which allows for solving in each VCZ a quadratic programming problem of small dimension and to drive the VCZ solutions toward the overall optimum by an iterative coordination process that requires to exchange among the VCZs only scalar values. The effectiveness and feasibility of the proposed method have been demonstrated via numerical tests on the IEEE 123-bus system.

Sign in / Sign up

Export Citation Format

Share Document