Test Problems for Large Scale Nonsmooth Minimization

Transcription

1 Reports of the Department of Mathematcal Informaton Technology Seres B. Scentfc Computng No. B. 4/007 Test Problems for Large Scale Nonsmooth Mnmzaton Napsu Karmtsa Unversty of Jyväskylä Department of Mathematcal Informaton Technology P.O. Box 35(Agora) FI Unversty of Jyväskylä FINLAND fax

2 Copyrght c 007 Napsu Karmtsa and Unversty of Jyväskylä ISBN ISSN X

3 Test Problems for Large-Scale NonsmoothMnmzaton NapsuKarmtsa Abstract Many practcal optmzaton problems nvolve nonsmooth(that s, not necessarly dfferentable) functons of hundreds or thousands of varables wth varous constrants. However, there exst only few large-scale academc test problems for nonsmooth case and there s no establshed practce for testng solvers for large-scale nonsmooth optmzaton. For ths reason, we now collect the nonsmooth test problems used n our prevous numercal experments and also gve some new problems. Namely, we gve problems for unconstraned, bound constraned, and nequalty constraned nonsmooth mnmzaton. 1 Introducton Many practcal optmzaton problems nvolve nonsmooth functons wth large amounts of varables(see, e.g.,[1,, 14]). However, there s no establshed practce for testng solvers for large-scale nonsmooth optmzaton and only few largescale nonsmooth academc test problems exst. In ths paper, we gve a collecton of problems for large-scale nonsmooth mnmzaton. The general formula for these problems s wrtten by { mnmze f(x) (1) subjectto x G, wheretheobjectvefuncton f : R n RssupposedtobelocallyLpschtzcontnuousonthefeasbleregon G R n andthenumberofvarables nssupposedtobe large. Note that no dfferentablty or convexty assumptons are made. We shall descrbe three groups of nonsmooth test problems: unconstraned(g = R n n(1),seesecton),boundconstraned(g = {x R n x l x x u forall = 1,..., n}n(1),seesecton3),andnequaltyconstraned(g = {x R n g j (x) 0forall j = 1,...,p}n(1),seeSecton4). TheworkwasfnancallysupportedbyUnverstyofJyväskylä. DepartmentofMathematcalInformatonTechnology,POBox35(Agora),FI-40014Unversty of Jyväskylä, Fnland, hamas@mt.jyu.f 1

4 Unconstranedproblems. Inthssectonwepresent10nonsmoothunconstraned(G = R n n(1))mnmzaton problems frst ntroduced n[7]. The problems have been constructed ether by channg and extendng small exstng nonsmooth problems or by nonsmoothng largesmoothproblems(thats,forexample,byreplacngtheterm x by x ). All these problems can be formulated wth any number of varables. We frst gve the formulatonoftheobjectvefuncton fandthestartngpont x 1 = ( 1,...,x(1) n ) T foreachproblem.then,wecollectsomedetalsoftheproblemsaswellasthereferences to the orgnal(small-scale) problems n Table Generalzaton of MAXQ f(x) = max 1 n x. =, for = 1,...,n/and =, for = n/ + 1,...,n... Generalzaton of MXHILB f(x) = max 1 n n j=1 x j +j 1. = 1.0, forall = 1,..., n..3. Chaned LQ =1 max { x x +1, x x +1 + (x + x +1 1) }. = 0.5, forall = 1,..., n..4.chanedcb3i =1 max { x 4 + x +1, ( x ) + ( x +1 ), e x +x +1 }. =.0, forall = 1,..., n..5. Chaned CB3 II f(x) = max { n 1 ( =1 x 4 + x+1), n 1 =1 (( x ) + ( x +1 ) ), n 1 =1 (e x +x +1 ) }. =.0, forall = 1,..., n.

5 .6. Number of actve faces f(x) = max 1 n {g ( n =1 x ),g(x ) }, where g(y) = ln ( y + 1). = 1.0, forall = 1,..., n..7. Nonsmooth generalzaton of Brown functon f(x) = ( n 1 =1 x x x +1 +1) x. = 1.0, when mod (, ) = 0 and = 1.0, when mod (, ) = 1, = 1,..., n..8. Chaned Mffln ( =1 x + ( x + x +1 1) x + x +1 1 ). = 1.0, forall = 1,..., n..9. Chaned crescent I f(x) = max { n 1 ( =1 x + (x +1 1) + x +1 1 ) (, x (x +1 1) + x )}. n 1 =1 =.0, when mod (, ) = 0 and = 1.5, when mod (, ) = 1, = 1,..., n..10. Chaned crescent II =1 max { x + (x +1 1) + x +1 1, x (x +1 1) + x }. =.0, when mod (, ) = 0 and = 1.5, when mod (, ) = 1, = 1,..., n. Thedetalsoftheproblems.1.10aregvennTable1,where pdenotesthe problemnumber, f(x )sthemnmumvalueoftheobjectvefuncton,andthe symbols (nonconvex) and + (convex) denote the convexty of the problems. Inaddton,thereferencestotheorgnalproblemsneachcasearegvennTable1. 3

6 Table 1: Unconstraned problems. p f(x ) Convex Orgnalproblem Ref MAXQ, n = 0 [15] MXHILB, n = 50 [10].3 (n 1) 1/ + LQ, n = [16].4 (n 1) + CB3, n = [3].5 (n 1) + CB3, n = [3] Number of actve faces [5] Generalzaton of Brown functon [4].8 vares Mffln, n = [6] Crescent, n = [9] Crescent, n = [9] * f(x ) for n = 50, f(x ) for n = 00,and f(x ) for n = Bound constraned problems. Inthssectonwedescrbe10nonsmoothboundconstranedproblems(G = {x R n x l x x u forall = 1,...,n}n(1)). Boundconstranedproblemsare easlyconstructedfromtheproblemsgvennsecton(orn[7])bynclosngthe addtonal bounds x x x forallodd. Here x denotesthesolutonpontfortheunconstranedproblem. Ifthestartngpont x 1 = ( 1,..., n ) T gvennsectonsnotfeasble,we smply project t to the feasble regon(f a strctly feasble startng pont s needed an addtonal safeguard of may be added). The convexty of the bound constraned problems s the same as that of unconstraned problems(see Table 1) Bound constraned generalzaton of MAXQ f(x) = max 1 n x. 0.1 x 1.1 when mod (, ) = 0, = 1,...,n. = 1.1, for = 1,...,n/,when mod (, ) = 0, =, for = 1,...,n/,when mod (, ) = 1, = 0.1, for = n/ + 1,...,n,when mod (, ) = 0, and =, for = n/ + 1,...,n,when mod (, ) = 1. 4

7 3.. Bound constraned generalzaton of MXHILB f(x) = max 1 n n j=1 x j +j x 1.1 when mod (, ) = 0, = 1,...,n.. = 1.0, forall = 1,..., n Bound constraned chaned LQ =1 max { x x +1, x x +1 + (x + x +1 1) } x when mod (, ) = 0, = 1,...,n. = , when mod (, ) = 0 and = 0.5, when mod (, ) = 1, = 1,...,n Bound constraned chaned CB3 I =1 max { x 4 + x +1, ( x ) + ( x +1 ), e x +x +1 }. 1.1 x.1 when mod (, ) = 0, = 1,...,n. =.0, forall = 1,..., n Bound constraned chaned CB3 II f(x) = max { n 1 ( =1 x 4 + x+1), n 1 =1 (( x ) + ( x +1 ) ), n 1 =1 (e x +x +1 ) }. 1.1 x.1 when mod (, ) = 0, = 1,...,n. =.0, forall = 1,..., n Bound constraned number of actve faces f(x) = max 1 n {g ( n =1 x ),g(x ) }, where g(y) = ln ( y + 1). 0.1 x 1.1 when mod (, ) = 0, = 1,...,n. = 1.0, forall = 1,..., n. 5

8 3.7. Bound constraned nonsmooth generalzaton of Brown functon f(x) = ( n 1 =1 x x x +1 +1) x. 0.1 x 1.1 when mod (, ) = 0, = 1,...,n. = 1.0, when mod (, ) = 0 and = 1.0, when mod (, ) = 1, = 1,..., n Bound constraned chaned Mffln ( =1 x + ( x + x +1 1) x + x +1 1 ) x 1.68, x when mod (, ) = 0, = 3,...,n 1, 0.1 x n 1.1. = 0.68, = , when mod (, ) = 0, ( > ), and = 1.0, when mod (, ) = 1, = 1,...,n Bound constraned chaned crescent I f(x) = max { n 1 ( =1 x + (x +1 1) + x +1 1 ) (, x (x +1 1) + x )}. n 1 =1 0.1 x 1.1 when mod (, ) = 0, = 1,...,n. = 1.1, when mod (, ) = 0 and = 1.5, when mod (, ) = 1, = 1,..., n Bound constraned chaned crescent II =1 max { x + (x +1 1) + x +1 1, x (x +1 1) + x }. 0.1 x 1.1 when mod (, ) = 0, = 1,...,n. = 1.1, when mod (, ) = 0 and = 1.5, when mod (, ) = 1, = 1,..., n. 6

9 4 Inequalty constraned problems. Fnally, we descrbe eght nonlnear or nonsmooth nequalty constrants(or constrant combnatons). Some of them(constrants ) have been ntally gven n[8]. TheconstrantscanbecombnedwththeproblemsgvennSectonto obtan80nequaltyconstranedproblems(g = {x R n g j (x) 0forall j = 1,..., p}n(1)). Theconstrantsareselectedsuchthattheorgnalunconstraned mnma of problems n Secton are not feasble. Note that, due to nonconvexty of the constrants, all the nequalty constraned problems formed ths way are nonconvex. Thestartngponts x 1 = ( 1,..., n ) T fornequaltyconstranedproblemsare chosen to be strctly feasble. In what follows, the startng ponts for the problems wth constrants are the same as those for problems wthout constrants(see Secton ) unless stated otherwse Modfcaton of Broyden trdagonal constrant I (for orgnal Broyden trdagonal constrant, see, e.g.,[1]) g j (x) = (3.0.0x j+1 )x j+1 x j.0x j , j [1, n ], forproblems.1,.,.6,.7,.9,and.10nsectonand g j (x) = (3.0.0x j+1 )x j+1 x j.0x j+ +.5, j [1, n ], forproblems.3,.4,.5,and.8nsecton. =.0, = 1,..., j +, forproblems.3and.8nsecton, = 1.0, = 1,..., j +, forproblems.9and.10nsecton,and = 1.0, j + and mod(, ) = 0, forproblem.7nsecton. 4.. Modfcaton of Broyden trdagonal constrant II g 1 (x) = n =1 ((3.0.0x +1)x +1 x.0x ), forproblems.1,.,.6,.7,.9,and.10nsectonand g 1 (x) = n =1 ((3.0.0x +1)x +1 x.0x + +.5), forproblems.3,.4,.5,and.8nsecton. =.0, = 1,..., n, forproblems.3and.8nsecton. 7

10 4.3. Modfcaton of MAD1 I (for orgnal problem, see, e.g.,[13]) g 1 (x) = max {x 1 + x + x 1x 1.0, sn x 1, cosx }, g (x) = x 1 x = 0.5 and = 1.1 forallproblemsnsecton Modfcaton of MAD1 II g 1 (x) = x 1 + x + x 1x 1.0,. g (x) = sn x 1, g 3 (x) = cosx, g 4 (x) = x 1 x = 0.5 and = 1.1 forallproblemsnsecton Smple modfcaton of MAD1 I g 1 (x) = n 1 ( =1 x + x +1 + x x +1.0x.0x ), forproblems.1,.,.6,.7,.9,and.10nsectonand g 1 (x) = n 1 ( =1 x + x +1 + x x ), forproblems.3,.4,.5,and.8nsecton. = 0.5, = 1,..., n, forproblems.1,.,.6,.7,.9,and.10 nsectonand = 0.0, = 1,..., n, forproblems.4,.5,and.8nsecton Smple modfcaton of MAD1 II g j (x) = x j + x j+1 + x jx j+1.0x j.0x j , j [1, n 1], forproblems.1,.,.6,.7,.9,and.10nsectonand g j (x) = x j + x j+1 + x jx j+1 1.0, j [1, n 1], forproblems.3,.4,.5,and.8nsecton. 8

11 = 0.5, = 1,..., j + 1, forproblems.1,.,.6,.7,.9,and.10 nsectonand = 0.0, = 1,..., j + 1, forproblems.4,.5,and.8nsecton Modfcaton of P0 from UFO collecton I (for orgnal problem, see, e.g.,[11]) g j (x) = ( x j+1 )x j+1 x j.0x j , j [1, n ], =.0, = 1,..., j +, forproblems.,.3,.6,.7,.9,and.10 nsectonand =.0, = 1,..., j +, forproblem.8nsecton Modfcaton of P0 from UFO collecton II g 1 (x) = n =1 (( x +1)x +1 x.0x ). =.0, = 1,..., n, forproblems.,.3,.6,.7,and.8 nsecton Acknowledgements The author would lke to thank Prof. Marko M. Mäkelä(Unversty of Turku, Fnland) for contnung support. References [1] BELIAKOV, G., MONSALVE TOBON, J. E., AND BAGIROV, A. M. Parallelzaton of the dscrete gradent method of non-smooth optmzaton and ts applcatons. In Computatonal Scence ICCS 003, Sloot et. al., Ed., Lecture Notes n Computer Scence. Sprnger Berln, Hedelberg, 003, pp [] BEN-TAL, A., AND NEMIROVSKI, A. Non-Eucldean restrcted memory level method for large-scale convex optmzaton. Mathematcal Programmng 10, 3 (005), [3] CHARALAMBOUS, C., ANDCONN, A.R. Aneffcentmethodtosolvethe mnmax problem drectly. SIAM Journal on Numercal Analyss 15, 1(1978), [4] CONN,A.R.,GOULD,N.I.M., ANDTOINT,P.L. Testngaclassofmethods for solvng mnmzaton problems wth smple bounds on the varables. Mathematcs of Computaton 50, 18(1988),

12 [5] GROTHEY, A. Decomposton Methods for Nonlnear Nonconvex Optmzaton Problems. PhD thess, Unversty of Ednburgh, 001. [6] GUPTA, N. A Hgher than Frst Order Algorthm for Nonsmooth Constraned Optmzaton. PhD thess, Washngton State Unversty, [7] HAARALA, M., MIETTINEN, K., AND MÄKELÄ, M. M. New lmted memory bundle method for large-scale nonsmooth optmzaton. Optmzaton Methods and Software 19, 6(004), [8] KARMITSA,N.,MÄKELÄ,M.M.,ANDALI,M.M. Lmtedmemorybundle algorthm for nequalty constraned nondfferentable optmzaton. Reports of the Department of Mathematcal Informaton Technology, Seres B. Scentfc Computng, B. 3/007 Unversty of Jyväskylä, Jyväskylä, 007. [9] KIWIEL, K. C. Methods of Descent for Nondfferentable Optmzaton. Lecture Notes n Mathematcs Sprnger-Verlag, Berln, [10] KIWIEL, K. C. An ellpsod trust regon bundle method for nonsmooth convex optmzaton. SIAM Journal on Control and Optmzaton 7, 4(1989), [11] LUKŠAN,L.,T UMA,M.,ŠIŠKA,M.,VLČEK,J.,ANDRAMEŠOVÁ,N.UFO00. Interactve system for unversal functonal optmzaton. Techncal Report 883, Insttute of Computer Scence, Academy of Scences of the Czech Republc, Prague, 00. [1] LUKŠAN, L., AND VLČEK, J. Sparse and partally separable test problems for unconstraned and equalty constraned optmzaton. Techncal Report 767, Insttute of Computer Scence, Academy of Scences of the Czech Republc, Prague, [13] LUKŠAN, L., AND VLČEK, J. Test problems for nonsmooth unconstraned and lnearly constraned optmzaton. Techncal Report 798, Insttute of Computer Scence, Academy of Scences of the Czech Republc, Prague, 000. [14] MAJAVA, K., HAARALA, N., AND KÄRKKÄINEN, T. Solvng varatonal mage denosng problems usng lmted memory bundle method. In Proceedngs of The nd Internatonal Conference on Scentfc Computng and Partal Dfferental Equatons and The Frst East Asa SIAM Symposum, Hongkong, December 1-16, 005.(to appear, 006), L. Wenbn, N. Mchael, and S. Zhong-C, Eds. [15] SCHRAMM, H. Ene Kombnaton von Bundle- und Trust-Regon-Verfahren zur Lösung nchtdfferenzerbarer Optmerungsprobleme. PhD thess, Bayreuther Mathematsche Schrften, No. 30, Unverstät Bayreuth, [16] WOMERSLEY, R. S. Numercal Methods for Structured Problems n Nonsmooth Optmzaton. PhD thess, Department of Mathematcs, Unversty of Dundee, 1981.