HYBRIDISING LOCAL SEARCH WITH BRANCH-AND-BOUND FOR CONSTRAINED PORTFOLIO SELECTION PROBLEMS

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1 HYBRIDISING LOCAL SEARCH WITH BRANCH-AND-BOUND FOR CONSTRAINED PORTFOLIO SELECTION PROBLEMS Fang He 1, 2 and Rong Qu 1 1 The Automated Schedulng, Optmsaton and Plannng (ASAP) Group, School of Computer Scence The Unversty of Nottngham, Nottngham, NG8 1BB, UK 2 Department of Computer Scence, Faculty of Scence and Technology, Unversty of Westmnster, W1B 2HW, UK Emal: hef@westmnster.ac.uk, rxq@cs.nott.ac.uk KEYWORDS Hybrd algorthm; Branch-and-Bound; local search; portfolo selecton problems ABSTRACT In ths paper, we nvestgate a constraned portfolo selecton problem wth cardnalty constrant, mnmum sze and poston constrants, and non-convex transacton cost. A hybrd method named Local Search Branch-and-Bound (LS-B&B) whch ntegrates local search wth B&B s proposed based on the property of the problem,.e. cardnalty constrant. To elmnate the computatonal burden whch s manly due to the cardnalty constrant, the correspondng set of bnary varables s dentfed as core varables. Varable fxng (Bxby, Fenelon et al. 2000) s appled on the core varables, together wth a local search, to generate a sequence of smplfed s. The default B&B search then solves these restrcted and smplfed subproblems optmally due to ther reduced sze comparng to the orgnal one. Due to the nherent smlar structures n the s, the soluton nformaton s reused to evoke the reparng heurstcs and thus accelerate the solvng procedure of the subproblems n B&B. The tght upper bound dentfed at early stage of the search can dscard more subproblems to speed up the LS-B&B search to the optmal soluton to the orgnal problem. Our study s performed on a set of portfolo selecton problems wth non-convex transacton costs and a number of tradng constrants based on the extended mean-varance model. Computatonal experments demonstrate the effectveness of the algorthm by usng less computatonal tme. INTRODUCTION In ths paper, we tackle the sngle-perod portfolo selecton problem (PSP). In the problem concerned, a number of transactons can be carred out to adjust the portfolo durng a gven tradng perod. We take nto account these transacton costs as well as a set of tradng constrants. These nclude the cardnalty constrant (a lmt on the total number of assets held n the portfolo,.e. select k out n (k<n) assets to be held n the portfolo), the mnmum poston sze constrant (bounds on the amount of each asset), the mnmum trade sze constrant (bounds on the amount of transacton occurred on each asset) and transacton costs. The goal of the problem s to mnmze the rsk of the adjusted portfolo and the transacton costs ncurred, whle satsfyng the set of tradng constrants n feasble portfolos. The am of ths paper s to develop a hybrd method to solve the complex PSP effcently. The technques developed here are employed to solve a specfc problem, but t could be appled to other varants of PSP wth cardnalty constrant, and possble other combnatoral problems outsde ths doman. If the transacton cost functon s lnear, then the problem s generally easy to solve. However, a functon whch better reflects realstc transacton costs s usually non-convex (Konno and Wjayanayake 2001). Some research show that realstc transacton costs usually nclude a fxed fee, and thus the cost s relatvely hgher when the amount of transacton s smaller (Konno and Wjayanayake 2001, Konno and Wjayanayake 2002). The transacton cost s thus usually represented by a lnear pecewse concave functon. Ths turns the problem nto a non-convex optmsaton problem, whch s more dffcult to solve. In ths paper, we propose a new hybrd approach whch ntegrates local search wth B&B to solve the nonconvex portfolo selecton problem heurstcally. We conceptually dvde the decson varables nto two parts: the set of core varables whch defnes the cardnalty constrant and the rest of varables. Varable fxng s appled to the core varables. The result of varable fxng has two facets: values (.e. 0, 1) are assgned to the core bnary varables and smplfed s generated. A local search together wth varable fxng are performed on the core varables to generate a sequence of smplfed s. These s are traversed heurstcally to fnd the Proceedngs 30th European Conference on Modellng and Smulaton ECMS Thorsten Claus, Frank Herrmann, Mchael Mantz, Olver Rose (Edtors) ISBN: / ISBN: (CD)

2 promsng s,.e. whose lower bounds are not greater than upper bounds. The promsng sub- assgnments by varable fxng, together wth the value assgnments by a default B&B, form the complete solutons to the orgnal problem. The best soluton n to problems then are solved by a default B&B. Value the, together wth the value assgnments by varable fxng approxmates an optmal soluton n to the orgnal problem. PROBLEM FORMULATION Consder that an nvestor s holdng an ntal portfolo that consstss of a set of n assets. To respond to the changes n the market, the nvestor must revew ts current portfolo, wth the vew to carry out a number of transactons. It s assumed that the new portfolo wll be held for a fxed tme perod. The nvestor s goall s to mnmze both the transacton costs occurred and the rsk of the assets n the portfolo at the end of the nvestment perod, whle satsfyng a set of constrants. These constrants typcally nclude meetng the target return, the mnmum poston sze, and the mnmum tradng sze. Let w be the percentage of captal nvested n assett, =1,,n. We shall use a weght vector w ( w, w,..., w ) T to denote an ntal portfolo. The 1 2 percentage amount transacted n each asset a s specfed by weght vector x = (xx 1, x 2, x n ) T, x < 0 means ng and x >0 means ng. A weght vector w denotes the portfolo after the revson. After the transacton, the adjusted portfolo s w = w 0 + x, andd s held for a fxed perod of tme. We denote the return n of asset at the end of the nvestment perod as r and the expected return of the portfolo as R. We denote the covarance between assetss and j n return as σ j. We further defne ( x) as the sum of ndvdual transacton costs assocated wth each x. Based on the basc MV model, the portfolo selecton problem wth transacton costs can thus be modeled ass follows: n j n j w w j (x) 1 j 1 mn s.t. n r w 1 n R (1) (2) objectve problem where (1)) s the sum of two objects wth the same weghts. w The transactonn cost s the sum of the transacton costss assocated wthh the assets traded: In ths paper, we w consder a model that ncludes a fxed fee plus a lnear cost, thus leads to a non-convex functon, as shown n Fg.. 1. Ths functon s also appled n (Lobo, Fazel et al. 2007). The fxed feee charged for ng and ng asset s denoted as and, and the t varable costs assocated to ng w j and ng asset are denoted by and. The transacton costt functon s gven n (4), and shown n Fg. 1: 0, x 0; ( x) x, x 0; (4) x, x 0; Fg.1 The transacton cost functon (Lobo, Fazel et al. 2007) Problem Model wth Transacton Cost and Tradng Constrants Parameter n The total number of assets The ndex of assets, =1,,n 0 n ( x) ( x) 1 Intal poston p of thee portfolo Covarance betweenn assets and j w = w 0 + x F (3) where objectve (1) s to mnmze the rsk of the portfolo and the transacton costs ncurred. (2) ensures the expected return. F n (3) represents a set of feasble portfolos subject to all the related constrants. These constrants nclude the mnmum poston sze, the mnmum tradng sze, etc., whch wll be detaled next. In ths paper, we model the problem as a sngle r R Return perod of asset at the end of the nvestment Expected return of the portfolo Fxed cost c for ng or ng asset Varable cost rate for ng or ng asset

3 w Mnmum hold poston mn x Mnmum tradng amount mn k Varable w Number of assets n the portfolo after transacton Revsed poston of the portfolo after transacton Feature Decson varable x Amount of ng asset Decson varable x Amount of ng asset Decson varable z Hold asset or not n the revsed portfolo Auxlary varable z Buy asset or not Auxlary varable z Sell asset or not Auxlary varable There are two groups of varables n the formulaton of the problem, as denoted by the feature column. w, x, x are decson varables. z, z and z are auxlary varables whch are used to formulate the constrants. The column core varable denotes whch varables are core varables. The selecton of the core varables s problem dependent. Several researchers have ponted out that the cardnalty constrant presents the greatest computatonal challenge to the problem (Benstock 1996, Jobst, Hornman et al. 2001, Stoyan and Kwon 2010, Stoyan and Kwon 2011). Actually, the PSP wth cardnalty constrant has been recognzed to be NP-complete (Benstock 1996, Mansn and Speranza 1999). To elmnate the cardnalty constrant, we dentfy varables z whch defne the cardnalty n constrant z k as a set of core varables. 1 Based on the model PSP, we wll ntroduce two addtonal reduced models (PSP basc, PSP sub) as follows whch wll be appled to evaluate the neghbourhood n the local search and to calculate the lower bound: mn ww + ( x) (1) (PSP) st.. n 1 n jn j j 1 j (2) w w x x, 1,... n (3) n rw R w mn n ( x) 1 n (5) w z, 1,... n (6) w z w, 1,... n (7) n 1 z k mn mn (8) x z x, 1,... n (9) x z x, 1,... n (10) x z, 1,... n (11) x z, 1,... n (12) z z, 1,... n (13) z z 1, 1,... n (14) 0 w 1, 1,... n (15) 0 x 1, 1,... n (16) 0 x 1, 1,... n (17) z, z, z {0,1}, 1,... n (18) mn ww (1) (PSP basc) st.. n 1 n jn 1 j1 j j (2) w z, 1,... n (6) rw R w z w, 1,... n (7) mn n 1 (8) 0w 1, 1,... n (15) z wth assgnments n {0,1}, 1,... n (18) z k n jn mn ww + ( x) (1) (PSP sub) j j 1 j1 1 st.. (2) (17) n zwthassgnments n{0,1}, 1,... n (18) z, z {0,1}, 1,... n LS-B&B TO PSP ALGORITHM (19)

4 In ths secton, we propose a new hybrd search, named LS-B&B to PSP accordng to the property of the problem. To the PSP wth bnary varable z we are dealng wth, we know that exactly k out n bnary varables wll be assgned to 1 n the feasble and optmal solutons. Wth ths knowledge, we can apply varable fxng on a set of varables at one tme, resultng nto smplfed. A local search s performed on these set of varables to generate a sequence of s, and the best soluton wll be dentfed among them. Framework of LS-B&B to PSP We present the framework of LS- B&B to PSP, as shown n Fg.2. LS-B&B conssts of four man components. The frst component s the ntalzaton phase (lne 1). In ths phase, varable fxng s appled to the core varables to generate a smplfed. Lower bound and upper bound of the problem are also ntalzed n ths phase. The second component s a default B&B search (lne 7). It s called to solve the s to optmalty. Ths soluton to the together wth the varable assgnments by varable fxng, forms the soluton to the orgnal problem. The thrd component s a local search (lne 9) whch s performed on set Z of varable z to update sets S and. Wth the updated S, the s updated correspondngly. Therefore, we state that ths local search generates a sequence of s. The fourth component s an overall search procedure (the whle loop). In ths search procedure, a local search, varable fxng and a default B&B work together to dentfy the best soluton among the subproblems by prunng nferor s and solvng the promsng s to optmalty. We present explanatons of these components next. Components of LS-B&B to PSP Varable fxng (Hard) varable fxng has been used n MIP context to dvde a problem nto s. It assgns values to a subset of varables of the orgnal problem. That s, certan varables are fxed to the gven values. Based on the defnton of varable fxng n (Bxby, Fenelon et al. 2000, Lazc, Hanaf et al. 2009), we apply ths varable fxng to smplfy the orgnal problem nto s n the followng way. We frst denote a subsets S on the bnary varable set B: S B. Then we fx varables n subsets S to 1, to obtan s P as follows: T Psub :mnc x y st.. Ax b; xj 1, j S B xj [0,1], jc In ths way, we smplfy the orgnal problem to a subproblem. One selecton of the subsets S can generate one possble smplfed of the orgnal problem. Therefore, we apply varable fxng together wth a local search to generate a sequence of subproblems where we wll search for the best soluton. LS- B&B LB: lower bound; UB: upper bound; (h, x, w, z): a soluton (x, w, z) of the problem wth a correspondng objectve value h; solveb&b: a default B&B solver; Z: set of z ; S: subset of Z; P org : the orgnal problem defned by model (PSP); P : defned by varable fxng; 1: Intalzaton phase 2: whle (the number of teratons not met) 3: If (LB ( P ) UB) 4: prune the P ; 5: go to lne 9; 6: Else 7: (h, x, w, z) = solveb&b( P ) ; 8: f h <UB set UB =h ; 9: perform a Local search on set Z; 10: generate s by varable fxng: P = P org (z = 1), z S; 11: set (x*, w*, z*) as the best soluton among all (x, w, z) and h* be the correspondng objectve value; Fg. 2 The LS-B&B algorthm to PSP Intalzaton phase The man task of the ntalzaton phase s the generaton of a s P by varable fxng on varables z on sets S. From the defnton of P, we can state that P s P sub org wth the ntalzaton of y varables n S to 1. In the ntalzaton phase, the lower bound s obtaned by solvng the contnuous relaxaton of the sub-

5 problem P based on model (PSP sub), and the upper bound s set as. Default B&B search As we stated n the framework of LS-B&B, each of the s tself s stll a MIQP problem due to the presence of bnary varables z and z. However, due to the assgnments of varable z by varable fxng, the sze of the s much smaller comparng to the orgnal one. Therefore, s can be handled by the default B&B. In ths paper, the default B&B algorthm n the MIQP solver n CPLEX s appled to solve the promsng s (when LB ( P ) < UB ) to optmalty. What s more, the nherent smlar structures of the s enable a very successful reuse of soluton nformaton, so the reparng heurstcs embedded n solveb&b are evoked to mprove the search. Overall search procedure The overall search explores the sequence of subproblems. Ths s shown n the whle loop n Fg.2. In ths search, the lower bound of the P s computed by a general QP solver, whch relaxes the to a contnuous problem,.e. model PSP sub (lne 3 n Fg.2). Here, the computaton of the lower bound s dfferent from the evaluaton of a soluton n the local search, whch s based on model PSP basc. The objectve value of the feasble soluton to the concerned P serves as the upper bound of the orgnal problem. If the lower bound of a s above the current upper bound found so far, we can dscard ths durng the search (lne 4 n Fg.2). Otherwse, these promsng subproblems are solved exactly by a default B&B (lne 7 n Fg.2). The solutons to the s together wth the assgnments of core varables consst of the feasble solutons to the complete orgnal problem. These s are solved n sequence, and the best soluton among them, together wth the varable assgnments done by varable fxng, approxmates the optmal soluton to the orgnal problem. The whole procedure termnates by a pre-defned number of teratons n the local search. Therefore, the search s an ncomplete search. It cannot guarantee optmalty of the soluton due to the nature of the local search on core varables z. The local search together wth varable fxng creates a sequence of s whch have very smlar structures. They only dffer n the coeffcent or the rght-hand sde of constrants whch are related to z. When solvng ths sequence of s, the soluton nformaton such as the bass lst and bass factors from ts smplex tableau (.e., we apply the extended tableau smplex algorthm n the default MIQP solver) for the current problem are stored, and ths can be retreved and appled to the successve subproblems. Ths means the soluton nformaton (.e., bass lst and bass factors) of the problem P can thus be reused to obtan soluton to P ', so that P does ' not need to be solved agan from scratch. Ths soluton nformaton reusng thus can evoke the reparng heurstcs embedded n the default B&B solver. Ths soluton nformaton reusng has shown to be extremely effcent. EXPERIMENTAL RESULTS To evealuate our algorthm on more general benchmark nstances, we also concern n ths paper the portfolo optmsaton nstances publcly avalable n the OR lbrary (ORlbrary), wth addtonal constrants derved from the above real-world problem. Sx problem nstances are used to test the algorthm proposed n ths paper, whch can be found at (He and Qu, 2014). We set the mnmum proporton of wealth to be nvested n an asset, w mn, to 0.01%, and the mnmum transacton amount, x mn, to 0.01%. We also set the parameters n the transacton cost functon α to and ß to for all the assets. Other values of k n the cardnalty constrant have been tested, rangng from 10 to 150 for dfferent szes of portfolos. Evaluatons on the LS-B&B algorthm In LS-B&B, after fxng values for varables z by varable fxng and the local search, the resultng MIQP s are created. If the lower bound of a subproblem s not greater than the current upper bound (we say t s a promsng, otherwse t wll be pruned), t wll be solved by the default B&B n CPLEX12.0. Therefore, when these s are processed, n concluson four possble stuatons could emerge: (1) a could be solved by B&B to optmalty; (2) the reparng heurstc mechansm mbedded n CPLEX could be evoked and appled to a to obtan a feasble soluton heurstcally; (3) a could be pruned; ths wll happen f the optmal soluton under contnuous relaxaton on model PSP sub s larger than the current upper bound; and (4) the soluton of a could be nfeasble. Table 1 llustrates the behavor of the above four stuatons durng the processng of s. The total CPU tme of the algorthm s dependent upon the CPU tme needed for each stuaton.

6 Table 1. Informaton of processng. Instance total CPU tme solved Number CPU Numb tme/p repared pruned CPU Number tme/p CPU Numb tme/p nfeasble CPU tme/p If the reparng heurstc s evoked and succeed; The total CPU tme requred. Table 2. Comparsons of default B&B and LS- B&B. + denotes that the reparng heurstcs are succeed. All the CPU tme s measured n seconds. Socété Générale HangS DAX FTSE S&P Nkke Table 1 clearly ndcates that the CPU tme for dentfyng nfeasblty s neglgble. The CPU tme for prunng the nferor s qute effcent. Therefore, the more s pruned, the more effcent the search s. It can be nterpreted from Table 1 that solvng s wth reparng heurstcs s qute effcent. These reparng heurstcs are the results of soluton nformaton reuse n the B&B solver. Solvng s exactly s the most tme consumng stuaton comparng wth the other three stuatons. Comparsons wth the default B&B n CPLEX It s worth notng that LS-B&B s a heurstc approach to the problem. It cannot prove optmalty of the soluton due to the nature of the local search on core varables z, although the s can be measured by the optmalty gap. In order to evaluate the qualty of the solutons we obtaned from LS-B&B, we compare t aganst the optmal soluton to the problem. It s however very dffcult, f not mpossble, to obtan and prove the optmal soluton to the problems concerned. We therefore calculate the approxmate optmal soluton to the problem concerned by runnng the default B&B algorthm n CPLEX12.0 for an extensve amount of tme. In the comparson presented n Table 2, we am to demonstrate the effectveness of the reparng heurstc evoked n our proposed LS- B&B. Therefore, we present the characterstcs of the s beng repared by heurstc aganst the characterstcs of the default B&B. We compare LS-B&B wth the default B&B n Table 2 n terms of the followng crtera: The number of nodes beng processed n B&B to obtan the best nteger feasble soluton; The gap between optmalty and the qualty of the best feasble soluton; In Table 2, n LS-B&B, the number of nodes processed s the average of nodes beng processed wth reparng heurstcs. From Table 2 we can see that by smplfyng the problem through varable fxng, the reparng heurstcs succeed n LS- B&B approach. The reparng heurstcs cannot be evoked by the default B&B whle solvng the orgnal problem. Wthout the smplfcaton, the default B&B needs to explore a much larger number of nodes n the search to obtan feasble solutons, whle LS-B&B wth smplfcaton requres much less tme, shown n Table 2. For example, for the largest nstance Nkke, more than 35,500 nodes have been explored n the default B&B to obtan a feasble soluton wth a gap of 0.44%. The optmalty gap of soluton obtaned by LS- B&B s calculated by gap = (f LS f R ) / f R, where f LS s the objectve value obtaned by LS- B&B, and f R s the objectve value of contnuous relaxaton. Table 2 shows that, to acheve solutons of smlar qualty (as measured by the optmalty gap), the CPU tme needed by the default B&B s much greater than that requred by LS-B&B (e.g. 180 CPU seconds as opposed to seconds for the nstance Nkke). The comparson of LS-B&B wth the default B&B can be more clearly llustrated n Fg. 3, whch plots of the objectve values of LS-B&B and the approxmate optmal values obtaned by the default B&B wth extensve runtme. It can be seen that LS-B&B converges very well for nstances Socété Générale, Hang Seng and Nkke, where the gap between the objectve values of LS- B&B and approxmate optmal s very small. For nstance DAX, the best soluton of LS-B&B s even better than the approxmate optmal value. For nstances FTSE and S&P, the gap s slghtly larger. However, t should be noted that LS-B&B spends

7 sgnfcantly less tme (3-79 seconds) than the default B&B (180 and 600 seconds). Fg. 3 The gap between LS-B&B and the approxmate optmal by the default B&B CONCLUSIONS In ths paper, we have ntroduced the hybrd LS-B&B method to solve the portfolo selecton problem wth practcal tradng constrants and transacton costs. We have analysed a specfc PSP problem whch s modelled as MIQP. The hybrd method closely ntegrates local search wth B&B. It mplements an ncomplete search whch ams to seek near optmal solutons n a lmted computatonal tme. It smplfes the problem nto much smaller s, whch are much easer to solve than the orgnal complete problem, hence can be searched ntensvely by B&B. It has been demonstrated by our experments that the reparng heurstcs are evoked by soluton nformaton reusng n solvng s, thus the successve s can be solved more effcently. The heurstc ntalzaton of the core varables n our problem provdes a tght upper bound to prune more s. REFERENCES Bxby, R., M. Fenelon, Z. Gu, E. Rothberg and R. Wunderlng (2000). MIP:Theory and practce--closng the gap. System Modellng and Optmzaton: Methods,Theory and Applcatons 174: Benstock, D. (1996). Computatonal study of a famly of mxed-nteger quadratc programmng problems. Mathematcal Programmng 74(2): Jobst, N. J., M. D. Hornman, C. A. Lucas and G. Mtra (2001). Computatonal aspects of alternatve portfolo selecton models n the presence of dscrete asset choce constrants Quanttatve Fnance 1(5): Hansen, P., N. Mladenovc and D. Urosevc (2001). Varable neghborhood search: Prncples and applcatons. European Journal of Operatonal Research 130(3): He, F. and R. Qu, A two-stage stochastc mxed-nteger program modellng and hybrd soluton approach to portfolo selecton problems. Informaton Scences, 289: , Konno, H. and A. Wjayanayake (2001). Portfolo optmzaton problem under concave transacton costs and mnmal transacton unt constrants. Mathematcal Programmng 89(2): Konno, H. and A. Wjayanayake (2002). Portfolo optmzaton under D.C. transacton costs and mnmal transacton unt constrants. Journal of Global Optmzaton 22(1): Lazc, J., S. Hanaf, N. Mladenov and D. Urosevc (2009). Varable neghbourhood decomposton search for 0-1 mxed nteger programs. Computers &Operatons Research 37(6): Mansn, R. and M. G. Speranza (1999). Heurstc algorthms for the portfolo selecton problem wth mnmum transacton lots. European Journal of Operatonal Research 114(2): Stoyan, S. and R. Kwon (2010). A two-stage stochastc mxed-nteger programmng approach to the ndex trackng problem. Optmzaton and Engneerng 11(2): Stoyan, S. J. and R. H. Kwon (2011). A Stochastc-Goal Mxed-Integer Programmng approach for ntegrated stock and bond portfolo optmzaton. Comput. Ind. Eng. 61(4): AUTHOR BIOGRAPHIES Fang He was born n Chna and obtaned her PhD degree from The Unversty of Nottngham, U.K. And she worked as a Research Fellow at the same unverty for 3 years on modellng and optmsaton for combnatoral optmsaton problem n real-world applcatons. Ths work s condunt durng that tme. Now she s a lecturer n the Department of Computer Sccence, Unversty of Westmnster. Rong Qu s an Assocate Professor of Computer Scence, at the School of Computer Scence, The Unversty of Nottngham, U.K. Her research nterests are on the modellng and optmsaton algorthms (meta-heurstcs, mathematcal approaches and ther hybrdsatons) to real-world optmsaton and schedulng problems.