Stochastic Programming handling CVAR in objective and constraint
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1 Sochasc Programmng handlng CVAR n obecve and consran Leondas Sakalaskas VU Inse of Mahemacs and Informacs Lhana ICSP XIII Jly Bergamo Ialy
2 Olne Inrodcon Lagrangan & KKT condons Mone-Carlo samplng Sochasc Varable Merc Dscsson
3 Inrodcon Real lfe decsons are ofen made nder ncerany and rsk. De o coherency properes Condonal Vale A Rsk CVAR becomes poplar for modellng rsk n bankng and fnance Arner Ph. e al Pflg 2.
4 Inrodcon Operaonally convenen defnon of CVAR s gven by Rockafelar and Uryasev 22: CVaR mn E[ f ] f - loss fncon - random varables - confdence probably.
5 Inrodcon Rockafellar & Uryasev 2 sggesed approach for lnear opmaon of CVAR see also Prékopa 23 Kall & Mayer 25 ec. However aaned reslng LP problem may be very large and so o solve may reqre hge compng resorces on he oher hand s no clear how o solve he problem when some loss fncons are nonlnear.
6 ITRODUCTIO rher he Rsk Averson problem s consdered n a framework of nonlnear sochasc programmng handlng CVAR on fnancal nsrmens n obecve and consrans. Usally he level of rsk-averson s epressed as weghed sm wh weghs on epeced loss fncon and rsk measre.
7 Lagrangan & KKT Le s consder he rsk-averson level opmaon problem wh mlple consrans: f CVaR CVaR[ ] 2... m E f mn Assme random loss fncons be Lpshan he probablsc measre be absolely connos and defned by densy p..
8 Lagrangan & KKT condons The problem can be rewren as eqvalen blevel sochasc program: mn mn. m d p d p d p R R R l l l
9 Lagrangan & KKT condons Le s nrodce he eended Lagrangan whch s an epecaon of random Lagrangan: m R R d p d p L n l m l
10 Lagrangan & KKT condons Snce ncerany s descrbed by absolely connos measre he epecaons n formlas above are smoohly dfferenable fncons wh respec o and see Shapro&Rbnsen 983 Barke&Sakalaskas 27 ec.
11 Lagrangan & KKT condons Therefore sbgradens of loss fncons are avalable de o Lpsh propery. Of corse hese sbgradens can be aken as sochasc gradens: Ths he gradens of loss fncons can be epressed as epecaons oo:. G f EG...m
12 Lagrangan & KKT condons KKT condons are as follows: f * * CVaR[ m * CVAR[ ] * * * ] 2... m Pr... m.
13 Mone-Carlo samplng Snce epecaons and her dervaves consdered are comple mlvarae negrals he random samplng s a nversal echnqe o solve he problem consdered.
14 Mone-Carlo samplng Samplng for Sochasc Programmng: Samplng Average Appromaon LP Decomposon Benders Uryasev Sen ec Ineror-Pon mehod Brge Seqenal samplng Sochasc Appromaon Gpal orkn Kshner ec Seres of Mone-Carlo esmaors Homem d Mello&Shapro Sakalaskas
15 Mone-Carlo samplng Assme he Mone-Carlo samples are avalable n for any :... 2 are vecors dencally dsrbed wh respec o densy. p
16 Mone-Carlo samplng The correspondng Mone-Carlo esmaors are comped as well: ~ f ~ f Pr.... m
17 The samplng varances become avalable oo: Mone-Carlo samplng ~ Pr ~ 2 2 f D ~ ~ 2 2 f D m
18 The Mone-Carlo esmaors of obecve and consran fncons can be comped oo: f oherwse Mone-Carlo samplng Pr ~ # G G g ~ # G g # G G # G
19 Le s nrodce sochasc graden of Lagrangan and conseqenly he Mone-Carlo esmaor of Lagrangan graden: Mone-Carlo samplng m G G G Q # # m Q g g q ~ ~
20 Aferwards he samplng covarance mar s easy o compe: Mone-Carlo samplng T q Q q Q A
21 Sochasc Varable mehod The Mone-Carlo esmaors consdered can be appled o fnd he solon of Rsk-Averson problem by sochasc graden search e he Sochasc Varable Merc Uryasev mehod s developed sng he merc ndced by samplng covarance mar.
22 Sochasc Varable Merc mehod: Sochasc Varable mehod q A ] ~ ~ ma[ D f Pr m...
23 Sochasc Varable mehod Mone-Carlo esmaors appled are random n general. The followng rle s proposed o reglae he sample se whch ensres convergence wh lnear rae and enables s o perform compaon n a raonal way Sakalaskas 22: 2 n T q A q
24 Sochasc Varable mehod Termnaon of he algorhm cold be performed n sascal way by esng sascal hypohess of eqaly of Lagrangan graden o ero; 2 valdy of consrans; 3 confdence nervals of obecve and consran fncons.
25 Ths he algorhm s ermnaed whenever: 2 3 Sochasc Varable mehod 2 n q A q T ~ ~ D f m D... / 2 v Pr Pr Pr
26 Comper smlaon The approach consdered has been esed by Mone-Carlo mehod rals solvng he es problem wh pecewse lnear loss fncons n=2-5 m=. : l l c a... m ma c U e d l a U g b In
27 Comper Smlaon n=2 n=5 n= n=2 n= reqency of ermnaon
28 Dsscsson &Conclsons Obecve fncon : n=2 Obecve fncon : n=
29 Dsscsson &Conclsons Pr n=2 Pr
30 Comper Smlaon Mone-Carlo sample se: n= n=2 n=2
31 Dscssson & conclsons The sochasc erave mehod has been developed o solve he nonlnear rsk averson problems by a fne seqenal Mone-Carlo search. The mehod s gronded by he probablsc ermnaon procedre and he rle for erave reglaon of se of Mone-Carlo samples as well as akng no accon he sochasc model rsk.
32 Dscssson & conclsons The reglaon of sample se when hs se s aken nversely proporonal o he sqare of he Machalanobs norm of he graden of he Mone-Carlo esmae allows s o solve rsk averson problems raonally from he compaonal vewpon and ensres he convergence a.s. a a lnear rae.
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