Conic Programming in GAMS

Transcription

1 Conc Programmng n GAMS Armn Pruessner, Mchael Busseck, Steven Drkse, Ale Meeraus GAMS Development Corporaton INFORMS 003, Atlanta October 19-

2 Drecton What ths talk s about Overvew: the class of conc programs Conc constrants and eamples usng GAMS Solvers n GAMS and numercal results CONE World - a forum for conc programmng What ths talk s not about: Conc programmng algorthms Detaled applcatons

3 Overvew What are conc programs? Generalzed lnear programs wth the addton of nonlnear conve cones Class ncludes, for eample, Lnear program (LP) (Conve) Quadratc program (QP) (Conve) Quadratcally constraned QP (QCQP) Recently much actvty n ths area! 3

4 Areas and Applcatons Conc programmng used for: Engneerng Truss topology desgn FIR flter desgn Fnance Portfolo optmzaton Statstcs and Numercal Lnear Algebra Robust lnear programmng Norm mnmzaton problems 7th DIMACS Implementaton Challenge on SDP and Second Order Cone Programmng (SOCP) 4

5 Modelng and Solvng Less General Model Generalty More General LP (Conve) QP (Conve) QCQP SOCP SDP Less Dffcult Soluton Dffculty More Dffcult 5

6 Cone Programs General form of conc program mn f T s. t. A b C, [, ] l u where C s a second order cone (dm=k): C k 1:( k 1) = : 1:( k 1) k k 6

7 Eample (quadratc cone) Quadractc cone C sometmes also called Lorentz cone (or ce cream cone) Trval Quadratc Cone: z + y 7

8 Second Order Cone Programs mn f T s. t. C + d a T + b, = 1,..., N R n f, a R n C R d R ( k 1) n k 1 b R Equvalent to conc program Lnear constrants: cone dmenson k=1 Cone constrants: change of varables (vector) T y = C + d, z = a + b 8

9 Types of Cones Quadratc Cone Rotated Quadratc Cone j j k k, k j Sometmes preferable for modelng quadratc nequaltes 9

10 Rotated Quadratc Cone Show equvalence to quadratc cone: y z Rotated quadratc cone y z y + + y z ( ) + y ( y) Quadratc cone 10

11 11 General Transformaton If where C r rotated quadratc cone C q quadratc cone A = L M O M M L L q r C A C

12 Conc Constrants n GAMS In the GAMS modelng language: Conc constrants denoted by =C= Conc programs result n lnear programs n GAMS 1

13 Quadratc Cone n GAMS Set s /s1,s,,sn/; Set t(s) / s,,sn/; Varable (s); Conc LP formulaton ( s1 ) =C= sum(t(s), (s)); Equvalent NLP formulaton ( s1 ) =G= sqrt[sum(t(s),sqr((s)))]; Note: Summaton on rght hand sde 13

14 Rotated Quadratc Cone Set s /s1,s,s3,,sn/; Set t(s) / s3,,sn/; Varable (s); Conc LP formulaton ( s1 )+( s ) =C= sum(t(s), (s)); Equvalent NLP formulaton *( s1 )*( s )=G= sum[t(s),sqr((s))]; Note: Summaton on rght hand sde 14

15 Eample: Comple L1 Norm Comple L1 Norm Mnmzaton Mnmze A-b 1 where A,,b are comple valued We can wrte ths as mn Re( A) Im( A) Im( A) Re( ) Re( A) Im( ) Re( b) Im( b) 15

16 16 Eample (Contnued) Comple L1 Norm Mnmzaton (quadratc cone) ) ( ) ( ) ( ) Im( ) Re( ) Im( ) Re( ) Re( ) Im( ) Im( ) Re( s.t. ) ( mn z z t b b A A A A () z () z t m re m re + =

17 GAMS Model (Quadratc Cone) Objectve.. obj =E= sum(, t()); reseq_re().. res_re() =E= sum(j, A_re(,j)*_re(j)) - sum(j,a_m(,j)*_m(j)) - b_re(); reseq_m().. res_m() =E= coneeq().. t() =C= res_re() + res_m(); Model conemodel /objectve, reseq_re, reseq_m, coneeq/; Solve conemodel usng lp mnmzng obj; 17

18 Portfolo Optmzaton mn s. t. j, j' j j p j σ j j j, j' = 1, j j' r = α D mn 0 Objectve s to mnmze varance (rsk), subject to an epected return σ j,j = covarance =1 / (numdays-1) j = % of nvestment n stock j p j = prce change (return) for stock j r mn = mnmum epected return D j,d = Devaton per day d of stock j wrt to mean return 18

19 Portfolo Optmzaton (Cont.) Can rewrte: mn α D By ntroducng ntermedate varables p,q, and w: mnmze α r subject to w( d) = j D( j, d) ( j) q = 1 qr d w(d) 19

20 GAMS Model (Rotated Cone) Objectve.. obj =E= a*(*r); Budget.. sum(j, (j)) =E= 1; Return.. Sum(j, p(j)*(j)) =G= rmn; Wcone(days).. w(d) = sum(j, D(j,d)*(j); cone_eq1.. q =E= 1; cone_eq.. q + r =C= sum(d, w(d)); Model conemodel / all /; Solve conemodel usng lp mnmzng obj; 0

21 GAMS/MOSEK Solvng Conc Models n GAMS: Newest addton s MOSEK LP (smple or nteror pont) MIP (branch and bound) Conc Programs (conc nteror pont): Conve NLP Solver CONOPT also accepts conc constrants 1

22 Numercal Eamples DIMACS Challenge Models (SDP and SOCP) Chose subset of models from DIMACS (only SOCP models) SOCP models: Sum of norms Antenna array weght desgn Schedulng problems MOSEK solves all SOCP models and s the most effcent

23 Numercal Eamples DIMACS Benchmarks by Hans Mttelmann Solvers: MOSEK.5.1 (et MPS), LOQO 6.03, SDPT3 3.01, SeDuM 1.05R4 18 SOCP problems In SeDuM (MATLAB) format MOSEK: etended QPS format (based on MPS) 3

24 4 DIMACS Results (H. Mttelmann) Sched_50_50_org Sched_100_50_org Sched_100_100_org Sched_00_100_org Sched_50_50_org Sched_100_50_org Sched_100_100_org Sched_00_100_org MM Qssp30 Qssp60 Qssp MM Nql30 Nql60 Nql Nb Nb_L1 Nb_L Nb_L_Bessel SeDuM SDPT3 MOSEK LOQO Problem MM: memory problems

25 DIMACS Results (Cont.) Use Performance Profles (Dolan and Moré, 00) to vsualze results: Cumulatve dstrbuton functon for a performance metrc Performance metrc: rato τ of current solver tme over best tme of all solvers for success Intutvely: probablty of success f gven τ tmes fastest tme (τ=rato) 5

26 Profles (Data: H. Mttelmann) Fgure 1 6

27 CONE World An onlne forum for dscusson and nformaton on cone programmng CONELb lbrary of models GAMS cone format NLP formulaton Conc programmng solvers Lnks and lsts 7

28 ConeLb Conc Models n GAMS 8

29 ConeLb Conc Models n GAMS 9

30 Cone Versus NLP Formulaton Modelng conc constrants can be tedous NLP form of quadratc cone s more natural Rotated quadratc cone for QP s cumbersome BUT Potentally sgnfcant computatonal advantages Compare cone vs. NLP formulaton on CONELb (currently 8 models) NLP formulaton: substtute nto conc constrant (converted usng CONVERT utlty n GAMS) In practce smarter NLP formulatons may est dependng on model 30

31 Cone Vs. NLP (Effcency) Percent Of Models Solved Fgure Performance Profle MOSEK NLP-1 NLP- NLP Tme Factor 31

32 Reproducblty of Results Reproducblty of results s key to valdty n scentfc dscplnes (but sometmes neglected) Open data source (models, solvers, solver optons) Should be easly reproducble As part of the GAMS World, we provde downloadable scrpts to reproduce results Currently only for NLP/Cone models (Global World) See 3

33 Reproducblty of Results 33

34 Reproducblty of Results 34

35 Conclusons Addton of conc programmng capablty wthn GAMS MOSEK as state-of-the-art conc programmng solver Although modelng stll cumbersome, sgnfcant computatonal advantages usng cones CONELb: growng collecton of conc programmng models Presentaton wll be avalable under 35

36 References Lobo, M.S., Vandenberghe, L., Boyd, S. and Lebret, H. Applcatons of Second Order Cone Programmng, Lnear Algebra and ts Applcatons, 84:193-8, November MOSEK Optmzaton Tools Help Desk, Verson, onlne at H.D. Mttelmann, An ndependent benchmarkng of SDP and SOCP solvers, Math Program., Seres B, 00 (appeared electroncally). E. D. Dolan and J. J. More, Benchmarkng optmzaton software wth performance profles, Math. Programmng, 91 (), 01-13,