On a Solution of Fractional Programs via D.C. Optimization Theory

Transcription

1 On a Soluton of Fractonal Programs va D.C. Optmzaton Theory Tatana V. Gruzdeva Aleander S. Strekalovsky Matrosov Insttute for System Dynamcs and Control Theory of SB RAS Lermontov str., , Irkutsk, Russa gruzdeva@cc.ru, strekal@cc.ru Abstract Ths paper addresses the problems of fractonal programmng whch data s gven by d.c. functons. Such problems are, n general, nonconve (wth numerous local etrema). We develop an effcent global search method for fractonal program based on two dfferent approaches. The frst approach develops the Dnkelbach s dea and uses a soluton of an equaton wth the optmal value of an aulary d.c. optmzaton problem wth a vector parameter. The second one deals wth another aulary problem wth nonconve nequalty constrants. Both aulary problems are d.c. optmzaton problems, whch allows us to apply the Global Optmzaton Theory and develop two correspondng global search algorthms. The algorthms have been substantated and tested on an etended set of sum-of-ratos problems wth up to 200 varables and 200 terms n the sum. Also we propose an algorthm for solvng problems of fractonal programmng, whch combnes the two approaches. 1 Introducton Durng the recent decades the problems wth the functons representable as a dfference of two conve functons (.e. d.c. functons) can be consdered as one of most attractve n nonconve optmzaton [Hrart-Urruty, 1985, Horst&Pardalos, 1995, Horst&Tuy, 1996, Evtushenko& Posypkn, 2013, Khamsov, 1999, Strekalovsky, 2003, Strongn&Sergeyev, 2000, Tuy, 1998]. Actually, any contnuous optmzaton problem can be appromated by a d.c. problem wth any prescrbed accuracy [Hrart-Urruty, 1985, Horst&Tuy, 1996, Tuy, 1998]. In addton, the space of d.c. functons s a lnear space, any polynomal and moreover any twce dfferentable functon belong to the space of d.c. functons [Hrart-Urruty, 1985, Horst&Pardalos, 1995, Horst&Tuy, 1996, Strekalovsky, 2003, Strongn&Sergeyev, 2000, Tuy, 1998]. Besdes, ths space s closed wth respect to all standard operatons used n Analyss and Optmzaton. On account of ths stuaton the general Copyrght c by the paper s authors. Copyng permtted for prvate and academc purposes. In: Yu. G. Evtushenko, M. Yu. Khachay, O. V. Khamsov, Yu. A. Kochetov, V.U. Malkova, M.A. Posypkn (eds.): Proceedngs of the OPTIMA-2017 Conference, Petrovac, Montenegro, 02-Oct-2017, publshed at 246

2 problem of mathematcal optmzaton (P dc ) { f0 () := g 0 () h 0 () mn, S, f j () := g j () h j () 0, j = 1,..., m, where the functons g j ( ), h j ( ), j = 1,..., m,, are conve on IR n, S s a conve set, S IR n, can be consdered as a rather attractve object from theoretcal and practcal ponts of vew. Recently, we have succeeded n developng the Global Optmalty Condtons for Problem (P dc ), whch perfectly fts optmzaton theory and proved to be rather effcent n terms of computatons [Strekalovsky, 2003, Strekalovsky, 2014]. Now we apply ths theory to solvng the followng problem of the fractonal optmzaton [Bugarn et al., 2016, Schable&Sh, 2003] (P f ) f() := m ψ () φ () mn, S, where S IR n s a conve set and ψ : IR n IR, φ : IR n IR, (H 0 ) ψ () > 0, φ () > 0 S, = 1,..., m. It s well-known that the fractonal programmng problem s NP-complete [Freund&Jarre, 2001]. Surveys on methods for solvng ths problem can be found n [Bugarn et al., 2016, Schable&Sh, 2003], but the development of new effcent methods for a fractonal program stll remans an mportant feld of research n mathematcal optmzaton, because the sum-of-ratos programs arse n varous economc applcatons and real-lfe problems [Schable&Sh, 2003]. Here we develop effcent global search method for fractonal program, whch s based on two followng approaches. Generalzng the Dnkelbach s dea, we propose to reduce the fractonal program wth d.c. functons to solvng an equaton wth the vector parameter that satsfes the nonnegatvty assumpton. In ths case, we need to solve the aulary d.c. mnmzaton problems,.e. Problem (P dc ) wth f () 0, I. We also propose reducton of the sum-of-ratos problem to a mnmzaton of a lnear functon on the nonconve feasble set gven by d.c. nequalty constrants,.e. Problem (P dc ) where f 0 () s a lnear functon. Thus, based on the soluton of these two partcular cases of general d.c. optmzaton problem (P dc ) we develop two-method technology for solvng a general fractonal program, based on the Global Search Theory for d.c. optmzaton [Strekalovsky, 2003, Strekalovsky, 2014]. Ths paper s an etenson of our research [Gruzdeva&Strekalovsky, 2017], where the theoretcal framework and some computatonal smulatons were presented. The outlne of the paper s as follows. In Sect. 2, we descrbe two approaches to solvng Problem (P f ) usng aulary d.c. optmzaton problems. In Sect. 3, we show some comparatve computatonal testng of two approaches on fractonal program nstances wth up to 200 varables and 200 terms n the sum generated randomly and elaborate an algorthm for solvng sum-of-ratos problem combnng the two proposed methods. 2 D.C. Programmng Approach to Sum-of-Ratos Problems Instead of consderng a fractonal program drectly, we develop effcent global search methods, whch are based on two approaches. The frst method adopts a reducton of the fractonal mnmzaton problem to the soluton of an equaton wth an optmal value of the d.c. optmzaton problem wth a vector parameter. The second method s based on the reducton of the sum-of-ratos problem to the optmzaton problem wth nonconve constrants. 2.1 Reducton to the D.C. Mnmzaton Problem To solve fractonal program usng d.c. mnmzaton we consder the followng aulary optmzaton problem (P α ) Φ(, α) := m where α = (α 1,..., α m ) IR m s the vector parameter. [ψ () α φ ()] mn, S, 247

3 In ths subsecton we recall some results [Gruzdeva&Strekalovsky, 2016] concernng the relatons between Problems (P f ) and (P α ). Note, that the data of Problem (P f ) must satsfy the nonnegatvty condton,.e. the followng nequaltes hold (H(α)) ψ () α φ () 0 S, = 1,..., m. Let us ntroduce the functon V(α) of the optmal value to Problem (P α ): { m } V(α) := nf{φ(, α) S} = nf [ψ () α φ ()] : S. In addton, suppose that the followng assumptons are fulflled: (H 1 ) (a) V(α) > α K, where K s a conve set from IR m ; (b) α K IR m there ests a soluton z = z(α) to Problem (P α ) Further, suppose that n Problem (P α0 ) the followng equalty takes place: { m } V(α 0 ) : = nf [ψ () α 0 φ ()] : S = 0. (1) In [Gruzdeva&Strekalovsky, 2016] we proved that any soluton z = z(α 0 ) to Problem (P α0 ) s a soluton to Problem (P f ), so that z Sol(P α0 ) Sol(P f ). Therefore n order to verfy the equalty (1), we should be able to fnd a global soluton to Problem (P α ) for every α IR + m. Snce ψ ( ), φ ( ), = 1,..., m, are smply conve or generally d.c. functons t can be readly seen that Problem (P α ) belongs to the class of d.c. mnmzaton. As a consequence, n order to solve Problem (P α ), we can apply the Global Search Theory [Strekalovsky, 2003, Strekalovsky, 2014]. Due to the theoretcal foundaton developed n [Gruzdeva&Strekalovsky, 2016], we are able to avod the consderaton of Problem (P f ), and address the parametrzed problem (P α ) wth α IR + m. Hence, nstead of solvng Problem (P f ), we propose to combne solvng of Problem (P α ) wth a search wth respect to the parameter α IR + m n order to fnd the vector α 0 IR + m such that V(α 0 ) = 0. Denote Φ () := ψ () α kφ (), = 1,..., m. Let [v k, u k ] be a k-segment for varyng α. Let a soluton z(α k ) to Problem (P α k) be gven and we computed V k := V(α k ). Step 1. If V k > 0, then set v k+1 = α k, α k+1 = 1 2 (uk + α k ). Step 2. If V k < 0, then set u k+1 = α k, α k+1 = 1 2 (vk + α k ). α k -bsecton algorthm Step 3. If V k = 0 and mn Φ (z(α k )) < 0, then set α k+1 = ψ (z(α k )) φ (z(α k )) : Φ (z(α k )) < 0, α k+1 = α k : Φ (z(α k )) 0. In addton, set v k+1 = 0, u k+1 = t k+1 α k+1, where t k+1 = α+. ma α Stop: we computed the values α k+1, v k+1 and u k+1. Further, let [0, α + ] be an ntal segment for varyng α. To choose α + we should take nto account that due to (H(α)) and (H 0 ), we have = 1,..., m : α f () ψ () m ϕ () ψ () = f() S, ϕ () so, for eample, α + can be chosen as α + = f ( 0 ), = 1,..., m. 248

4 Algorthm for solvng fractonal problem Step 0. (Intalzaton) k = 0, v k = 0, u k = α +, α k = α+ 2 [vk, u k ]. Step 1. Fnd a soluton z(α k ) to Problem (P α k) usng the global search strategy for d.c. [Strekalovsky, 2003, Strekalovsky, 2014]. mnmzaton Step 2. (Stoppng crteron) If V k := V(α k ) = 0 and mn Φ (z(α k )) 0, then STOP: z(α k ) Sol(P f ). Step 3. Implement α k -bsecton algorthm to fnd new parameters α k+1, v k+1 and u k+1 ; k = k + 1 and go to Step 1. Let us emphasze the fact that the algorthm for solvng Problem (P f ) of fractonal optmzaton conssts of 3 basc stages: the (a) local and (b) global searches n Problem (P α ) wth a fed vector parameter α (Step 1) and (c) the method for fndng the vector parameter α (Step 3) at whch the optmal value of Problem (P α ) s zero,.e. V(α) = Reducton to the Problem wth D.C. Constrants In ths subsecton we reduce the fractonal program to the followng optmzaton problem wth a nonconve feasble set f 0 := m α mn (P), S, (,α) f := ψ () α φ () 0, = 1,..., m. In [Gruzdeva&Strekalovsky, 2016] t was proved that for any soluton (, α ) IR n IR m to the problem (P), the pont wll be a soluton to Problem (P f ). Further we solved problem (P) usng the eact penalzaton approach for d.c. optmzaton developed n [Strekalovsky, 2016, Strekalovsky, 2017]. Therefore, we ntroduce the penalzed problem as follows (P σ ) θ σ () = f 0 () + σ ma{0, f (), I} mn, S. It can be readly seen that the penalzed functon θ σ ( ) s a d.c. functon, because the functons f ( ) = g ( ) h ( ), I {0}, are the same, and we can apply the global search theory to solvng ths class of nonconve problems [Strekalovsky, 2003, Strekalovsky, 2014]. Actually, snce σ > 0, θ σ () = G σ () H σ (), H σ () := h 0 () + σ I h (), { m G σ ():= θ σ ()+H σ () = g 0 ()+σ ma h (); ma [g () + I j I, j h j ()] t s clear that G σ ( ) and H σ ( ) are conve functons. Let the Lagrange multplers, assocated wth the constrants and correspondng to the pont z k, k {1, 2,...}, be denoted by λ := (λ 1,..., λ m ) IR m. Global search scheme Step 1. Usng the local search method from [Strekalovsky&Gruzdeva, 2007, Strekalovsky, 2015], fnd a crtcal pont z k n (P). Step 2. Set σ k := m λ. Choose a number β : nf(g σ, S) β sup(g σ, S). Choose an ntal β 0 = G σ (z k ), ζ k = θ σ (z k ). Step 3. Construct a fnte appromaton R k (β) = {v 1,..., v N k H σ (v ) = β + ζ k, = 1,..., N k, N k = N k (β)} of the level surface {H σ () = β + ζ k } of the functon H σ ( ). Step 4. Fnd a δ k -soluton ū of the followng Lnearzed Problem: (P σ L ) G σ () H σ (v ), mn, S. so that G σ (ū ) H σ (v ), ū δ k nf {G σ() H σ (v ), }. }, 249

5 Step 5. Startng from the pont ū, fnd a KKT-pont u by the local search method from [Strekalovsky&Gruzdeva, 2007, Strekalovsky, 2015]. Step 6. Choose the pont u j : f(u j ) mn{f 0 (u ), = 1,..., N}. Step 7. If f 0 (u j ) < f 0 (z k ), then set z k+1 = u j, k = k + 1 and go to Step 2. Step 8. Otherwse, choose a new value of β and go to Step 3. The pont z resultng from the global search strategy wll be a soluton to the orgnal fracton program [Gruzdeva&Strekalovsky, 2016]. It should be noted that, n contrast to the approach from Subsect. 2.1, α wll be found smultaneously wth the soluton vector, because they are varables, although aulary ones, of the optmzaton problem. 3 Computatonal Smulatons Two approaches from above for solvng the fractonal programs (P f ) va d.c. optmzaton problems were verfed. The algorthm based on the method for solvng fractonal problem from Subsect. 2.1 (F1-algorthm) and the global search scheme from Subsect. 2.2 (F2-algorthm) were coded n C++ language and appled to an etended set of test eamples generated randomly wth up to 200 varables and 200 terms n the sum. All computatonal eperments were performed on the Intel Core K CPU 4.0 GHz. All conve aulary problems (lnearzed problems) on the steps of F1-, F2-algorthms were solved by the software package IBM ILOG CPLEX Some of the computatonal eperments of testng the proposed algorthms on fractonal problems wth quadratc functons n the numerators of ratos s presented here. We generated the problems of the type proposed by [Jong, 2013] as follows: f() := m, A + b, c, mn, Q q, [1, 5] n, (2) where A = U D 0 U T, U = V 1 V 2 V 3, = 1,..., n; V j = I 2 wjwt j w j 2, j = 1, 2, 3; w 1 = + rand(n, 1), w 2 = 2+rand(n, 1), w 3 = 3+rand(n, 1); c = rand(n, 1), b = + rand(n, 1), Q = 1+2 rand(5, n), q = rand(5, 1)[Jong, 2013]. (We denote by rand(k 1, k 2 ) the random matr wth k 1 rows, k 2 columns and the elements generated randomly on [0, 1].) Table 1: Randomly generated problems (2) wth quadratc functons n m f( 0 ) f(z) Tme F1 Tme F Note that Problems (P α ) correspondng to the problems (2) happened to be conve, because the constructed functons ψ =, A + b,, = 1,..., m, are conve. That s why the F1-algorthm s run-tme turned out to be much less than the run-tme of F2-algorthm, n whch t s necessary to solve nonconve problems wth m d.c. constrants. 250

6 The shortcomng of the F2-algorthm s the run-tme whch s spent on solvng problems wth d.c. constrants, but F1-algorthm wastes a lot of run-tme for fndng the vector parameter α at whch the optmal value of Problem (P α ) s zero. The results of computatonal eperments suggest that the solvng of the fracton programs need a combnaton of the two approaches. For eample, we can use the soluton to Problem (P) to search for the parameter α that reduces the optmal value functon of Problem (P α ) to zero. Ths dea could be mplemented by the followng algorthm. Step 0. (Intalzaton) k = 0, v k = 0, u k = α +. Two-component algorthm Step 1. Startng from feasble pont k, mplement the local search method from [Strekalovsky&Gruzdeva, 2007, Strekalovsky, 2015] to fnd a crtcal pont z k = (ẑ(α k ), α k ) n d.c. constrants problem (P). Step 2. (Stoppng crteron) If V k := V(α k ) = 0 and mn Φ (ẑ(α k )) 0, then STOP: ẑ(α k ) Sol(P f ). Step 3. Fnd a soluton z(α k ) to Problem (P α k) usng the global search strategy for d.c. [Strekalovsky, 2003, Strekalovsky, 2014]. Step 4. (Stoppng crteron) If V k := V(α k ) = 0 and mn Φ (z(α k )) 0, then STOP: z(α k ) Sol(P f ). Step 5. Implement α k -bsecton algorthm to fnd new parameters α k+1, v k+1 and u k+1. Set k+1 = (z(α k ), α k+1 ), k = k + 1 and go to Step 1. mnmzaton The two-component algorthm was tested on a lots of test eamples and ts run-tme turned out to be less than the run-tme of F1-algorthm. 4 Conclusons In ths paper, we showed how fractonal programs can be solved by applyng the Global Search Theory of d.c. optmzaton. The methods developed were substantated and tested on an etended set of test eamples generated randomly. In addton, we proposed a two-component technology based on both the global search algorthm for d.c. mnmzaton problem and algorthm for solvng d.c. constrants problem. Acknowledgements Ths work s carred out under fnancal support of Russan Scence Foundaton (project no ). References [Bugarn et al., 2016] Bugarn, F., Henron, D. & Lasserre J.-B. Mnmzng the sum of many ratonal functons. Mathematcal Programmng Computaton, 8, do: /s z [Evtushenko& Posypkn, 2013] Evtushenko, Y. & Posypkn, M. A determnstc approach to global boconstraned optmzaton. Optmzaton Letters, 7(4), do: /s [Freund&Jarre, 2001] Freund, R.W. & Jarre, F. Solvng the sum-of-ratos problem by an nteror-pont method. Journal of Global Optmzaton, 19(1), [Gruzdeva&Strekalovsky, 2016] Gruzdeva, T.V. & Strekalovsky, A.S. An approach to fractonal programmng va D.C. optmzaton. AIP Conference Proceedngs, 1776, do: / [Gruzdeva&Strekalovsky, 2016] Gruzdeva, T.V. & Strekalovsky, A.S. An Approach to Fractonal Programmng va D.C. Constrants Problem: Local Search. In: Yu. Kochetov, et al. (eds.) DOOR Lecture Notes n Computer Scence, 9869 (pp ) Hedelberg: Sprnger. do: / [Gruzdeva&Strekalovsky, 2017] Gruzdeva, T.V. & Strekalovsky, A.S. A D.C. Programmng Approach to Fractonal Problems. In: R. Battt, D. Kvasov, Y. Sergeyev (eds.) Learnng and Intellgent Optmzaton Conference LION11, Nzhny Novgorod, Russa. Lecture Notes n Computer Scence, Sprnger. To appear. 251

7 [Hrart-Urruty, 1985] Hrart-Urruty, J.B. Generalzed Dfferentablty, Dualty and Optmzaton for Problems dealng wth Dfference of Conve fnctons. In: Ponsten J. (ed.) Convety and Dualty n Optmzaton, Lecture Notes n Economcs and Mathematcal Systems, 256 (pp.37-69). Berln: Sprnger Verlag. [Horst&Tuy, 1996] Horst, R. & Tuy, H. Global Optmzaton: Determnstc Approaches. Berln: Sprnger Verlag. [Horst&Pardalos, 1995] Horst, R. & Pardalos P.M. (Eds.) Handbook of Global Optmzaton, Nonconve Optmzaton and ts Applcatons. Dordrecht: Kluwer Academc Publshers. [Jong, 2013] Jong, Y.-Ch. An effcent global optmzaton algorthm for nonlnear sum-of-ratos problems. Optmzaton Onlne HTML/2012/08/3586.html [Khamsov, 1999] Khamsov, O. On optmzaton propertes of functons, wth a concave mnorant. Journal of Global Optmzaton, 14 (1), [Schable&Sh, 2003] Schable, S. & Sh, J. Fractonal programmng: the sum-of-ratos case. Optmzaton Methods and Software, 18, [Strekalovsky, 2003] Strekalovsky, A.S. Elements of nonconve optmzaton [n Russan]. Novosbrsk: Nauka. [Strekalovsky, 2014] Strekalovsky, A.S. On solvng optmzaton problems wth hdden nonconve structures. In T.M. Rassas, C.A. Floudas, & S. Butenko (Eds.), Optmzaton n scence and engneerng (pp ). New York: Sprnger. [Strekalovsky, 2015] Strekalovsky, A.S. On local search n d.c. optmzaton problems. Appled Mathematcs and Computaton, 255, do: [Strekalovsky, 2016] Strekalovsky, A.S. On the mert and penalty functons for the d.c. optmzaton. In: Yu. Kochetov, et al. (eds.) DOOR Lecture Notes n Computer Scence, 9869, (pp ). Hedelberg: Sprnger. do: / [Strekalovsky, 2017] Strekalovsky, A.S. Global Optmalty Condtons n Nonconve Optmzaton. Journal of Optmzaton Theory and Applcatons, 173, do: /s [Strekalovsky&Gruzdeva, 2007] Strekalovsky, A.S. & Gruzdeva, T.V. Local Search n Problems wth Nonconve Constrants. Computatonal Mathematcs and Mathematcal Physcs, 47, do: /s [Strongn&Sergeyev, 2000] Strongn, R.G. & Sergeyev, Ya.D. Global Optmzaton wth Non-conve Constrants: Sequental and Parallel Algorthms. Dordrecht: Kluwer Academc Publshers. [Tuy, 1998] Tuy, H. Conve Analyss and Global Optmzaton. Dordrecht: Kluwer Academc Publshers. 252