DistOpt and Distributed Optimization Cassino, Italy

Transcription

1 DistOpt: A Ptolemy-based Tool to Model ad Evaluate the Solutios of Optimizatio Problems i Distributed Eviromets Arturo Losi Uiversità degli Studi di Cassio, Italy losi@uicas.it UC Berkeley Ptolemy Miicoferece March 22-23, 2001 Cassio, Italy Rome Cassio Naples Arturo Losi, UC Berkeley, 2 1

2 Cotets Credits Itroductio ad motivatio Optimizatio problem ad Auxiliary Problem Priciple DistOpt structure Usig DistOpt Arturo Losi, UC Berkeley, 3 Credits Publicatios related to DistOpt J. Cotreras, A. Losi, M. Russo, ad F.F. Wu, DistOpt: A Software Framework for Modelig ad Evaluatig Optimizatio Problem Solutios i Distributed Eviromets, Joural of Parallel ad Distributed Computig, Jue 2000, pp J. Cotreras, A. Losi, M. Russo, ad F.F. Wu, Simulatio ad Evaluatio of Optimizatio Problem Solutio i Distributed Eergy Maagemet Systems, accepted for publicatio i IEEE Trasactios o Power Systems. A. Losi, ad M. Russo, A Note o the Applicatio of the Auxiliary Problem Priciple, to be re-submitted (after the first review) to the Joural of Optimizatio Theory ad Applicatios Arturo Losi, UC Berkeley, 4 2

3 Itroductio ad motivatio Coarse-grai algorithms, with focus o the structure of the optimizatio problem (ot o the structure of the techique) Splittig of the optimizatio problem ito subproblems, with grais correspodig to subproblems Decompositio/coordiatio approach Suitable formulatio of the problem Decompositio of the optimizatio problem ito smaller subproblems, iteratively solved Coordiatio of the subproblems to drive their solutios to a solutio of the origial problem Arturo Losi, UC Berkeley, 5 Itroductio ad motivatio Desig variables decompositio/coordiatio method splittig of the problem ito subproblems techique for solvig the subproblems sychroizatio computig capacity of the processors characteristics of the commuicatios (other packages) DistOpt helps modelig ad evaluatig coarsegrai distributed optimizatio algorithms, for wide classes of optimizatio problems ad decompositio/coordiatio methods Arturo Losi, UC Berkeley, 6 3

4 Optimizatio problem mi X x f ( x ) X (feasibility set) = U 1 = U 2 = u1 U 1, s.t. mi y i, j u U i, j = 1, f ( u, u ) 1 - y = 0, j,i, ( j > i ), with u 1 =, u { x,y,,y }, 1 1, 2 1, = { x,y, y }, 1,-1 Arturo Losi, UC Berkeley, 7 Auxiliary Problem Priciple The Auxiliary Problem Priciple (APP - by Guy Cohe, Frace) is a geeral decompositio/ coordiatio theory that: does ot require separability assumptios (such as additive cost or additive costraits) is a geeralizatio of may kow methods ecompasses both oe-level ad two-level methods for the two-level methods, the covergece coditios are quite geeral DistOpt is based o the APP s two-level methods Arturo Losi, UC Berkeley, 8 4

5 Auxiliary Problem Priciple u1 U 1, s.t. mi y i, j u U i, j = 1, f ( u, u ) 1 - y j,i = 0,,( j> i ), with u 1 =,u { x1,y1, 2,,y1,}, = { x,y, y }, 1,-1 k k k ( ui ) + ef u ( u,...,u ) K ( u ) i 1 - ui i k k k p + c( y -y ),y mi ui ui U i + e i = 1,..., p k + 1 i, j iterative scheme (slightly modified) K i = p k i, j j= 1 j i + r i, j k + 1 k + 1 ( y - y ) i, j i, j j,i, j,i i, j,u for j = 1,...,, j i i + k + 1 Arturo Losi, UC Berkeley, 9 Auxiliary Problem Priciple Global covergece coditios Covex (ad differetiable) objective fuctio, f closed covex feasibility sets, U i (i=1,,) covex couplig costraits (affie i the equality case) strogly covex core fuctio, K defied bouds o the parameters c, r, e I practical o-covex cases, with APP iterative scheme optimality ecessary coditios are geerally met i the limit, if covergece results. Arturo Losi, UC Berkeley, 10 5

6 DistOpt structure Modular, flexible, extedible ad coducive to cooperatio OOP Ptolemy (classic) Discrete-Evet (DE) domai Three levels of abstractio defiitio of the problem ad its splittig ito subproblems formulatio of the trasformed problem ad of the subproblems, ad set-up of the two-level algorithm solutio of the optimizatio subproblems Arturo Losi, UC Berkeley, 11 DistOpt structure First level: Defiitio of the problem ad its splittig descriptio of the optimizatio problem (objective fuctio ad costraits), ad iformatio o the splittig ito subproblems - provided by the user o costraits o the codig of the problem (data structure ad fuctios iteral to the applicatio) formal defiitio of the methods defied data structure for the data exchage - through a file Arturo Losi, UC Berkeley, 12 6

7 DistOpt structure Secod level: formulatio of trasformed problem, subproblems, two-level decompositio/ coordiatio algorithm problem trasformatio, with variable duplicatio subproblem formulatio ad coordiatio set-up of APP parameters (core fuctio K, parameters c, e, r) choice of sychroous/asychroous executio of subproblems Arturo Losi, UC Berkeley, 13 DistOpt structure Secod level: algorithm buildig palette 2000 Academic Press - Joural of Parallel ad Distributed Computig Arturo Losi, UC Berkeley, 14 7

8 DistOpt structure Secod level: block SUB_3 Arturo Losi, UC Berkeley, 15 DistOpt structure Third level: solutio of the subproblems choice of the solver for each subproblem 2000 Academic Press - Joural of Parallel ad Distributed Computig Arturo Losi, UC Berkeley, 16 8

9 Usig DistOpt A DistOpt uiverse Arturo Losi, UC Berkeley, 17 Usig DistOpt Set-up of the computatio Arturo Losi, UC Berkeley, 18 9

10 Usig DistOpt Set-up of the computatio Arturo Losi, UC Berkeley, 19 Usig DistOpt Choice of the core fuctio K 2000 Academic Press - Joural of Parallel ad Distributed Computig Arturo Losi, UC Berkeley, 20 10

11 Usig DistOpt Choice of covergece parameters 2000 Academic Press - Joural of Parallel ad Distributed Computig Arturo Losi, UC Berkeley, 21 Usig DistOpt Some results sychroous asychroous Arturo Losi, UC Berkeley, 22 11

12 ftp://sistelet.ig.uicas.it/distopt readme.txt userguide.pdf DistOpt.tar.gz Arturo Losi, UC Berkeley, 23 12