Pure adaptive search for finite global optimization*

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1 Mathematical Prgramming 69 (1995) Pure adaptive search fr finite glbal ptimizatin* Z.B. Zabinskya.*, G.R. Wd b, M.A. Steel c, W.P. Baritmpa c a Industrial Engineering Prgram, FU-20. University f Washingtn. Seattle, WA 98195, USA b Department fmathematics and Cmputing. Central Queensland University, Rckhamptn. Australia C Department fmathematics and Statistics, University f Canterbury. Christchurch. New Zealand Received 13 December 1993; revised manuscript received 28 June 1994 Abstract Pure Adaptive Search is a stchastic algrithm which has been analyzed fr cntinuus glbal ptimizatin. When a unifnn distributin is used in PAS. it has been shwn t have cmplexity which is linear in dimensin. We define strng and weak variatins f PAS in the setting f finite glbal ptimizatin and prve analgus results. In particular, fr the n-dimensinal lattice {t,...,k}n, the expected number f iteratins t find the glbal ptimum is linear in n. Many discrete cmbinatrial ptimizatin prblems, althugh having intractably large dmains, have quite small ranges. The strng versin f PAS fr all prblems, and the weak versin f PAS fr a limited class f prblems, has cmplexity the rder f the size f the range. Keywrds: Glbal ptimizatin; Discrete ptimizatin; Algrithm cmplexity; Randm search; Markv chains 1. Intrductin Pure adaptive search (PAS) is a randm search methd which has been defined and analyzed fr cntinuus glbal ptimizatin [5,6]. Pure adaptive search generates a sequence f feasible pints accrding t a prbability distributin, with the stipulatin that the pints always have strictly imprving bjective functin values. It has been shwn [6] that pure adaptive search has an encuraging feature: fr cntinuus functins satisfying a Lipschitz cnditin the expected number f PAS iteratins t reach cnvergence is prprtinal t the dimensin. In this paper we examine PAS fr the glbal ptimizatin prblem in which the dmain is a finite set f pints. We begin by intrducing tw variatins f finite PAS, * The authrs wuld like t thank the Department f Mathematics and Statistics at the University f Canterbury fr supprt f this research. Crrespnding authr. zelda@u.washingtn.edu I995-The Mathematical Prgramming Sciety, Inc. All rights reserved SSDI O( 94)

2 444 ZB. Zabinsky et al.lmathematical Prgramming 69 (1995) weak and strng. Fr each variatin, we derive an expressin fr the expected number f iteratins t reach the glbal ptimum, and a simple upper bund. In the special case where the dmain is the n-dimensinal lattice {I,..., k}n, and the vertices are sampled accrding t a unifrm distributin, the expected number f iteratins t exact cnvergence is prprtinal t n, the "dimensin." As in the cntinuus case, strng PAS is inefficient t implement fr general discrete functins. Weak PAS appears t be related clsely t discrete ptimizatin applicatins, fr example, genetic recnstructin via the "Great Deluge" algrithm [2]. Our analysis f finite PAS suggests that such algrithms may have reasnable cmplexity. 2. Analysis f finite PAS 2.1. Terminlgy In this paper we cnsider the fllwing finite glbal ptimizatin prblem: minimize f(x) subject t XES, where f(x) is a real-valued functin n a finite set S. The fllwing algrithms fr finite ptimizatin are cnsidered in this paper. Pure randm search (PRS) [1] samples the dmain at each iteratin accrding t a fixed distributin. Strng pure adaptive search samples frm that part f the dmain that gives a strictly imprving bjective functin value at each iteratin. This is a translatin f PAS frm the cntinuus t the finite prblem. By relaxing the strictness, we can define weak pure adaptive search. This algrithm samples frm that part f the dmain which gives an equal r imprving bjective functin value. Let YI < Y2 <... < YK be the distinct bjective functin values. Ntice that there may be mre than K pints in S. Fr m = 0, 1,..., let the randm variable Y m be the bjective functin value n the mth iteratin f PRS. Nte that Y, l't,... are independent and identically distributed. Given a prbability measure JL n S, we define a prbability measure 'TT = (7TI,..., 7TK) n the range f f as fllws. Let 7Tj be the prbability that any iteratin f pure randm search attains a value f Yj' That is 7Tj = P(Y = Yj) = JL(f-I(Yj») fr j = 1,2,...,K. Thrughut this paper Pj dentes 'L{:::I7Ti the prbability that PRS attains a value f Yj r less. We nw describe the link between PRS and the tw versins f finite PAS. Epch i > is said t be a recrd f the sequence {Y m } fr strng PAS ifli < min {Y,..., li-t} and fr weak PAS if li ~ min{y,...,li-r}. Epch i = is always cnsidered t be a recrd. The crrespnding value li is called a recrd value. Let the randm variable W m be the bjective functin value n the mth iteratin f PAS, and let R(m) be the epch f the mth recrd f PRS. Then as in the cntinuus prblem, PAS cnsists f the recrd values f pure randm search. That is, Wm is stchastically equivalent t YR(m). This is directly analgus t [6, Lemma 3.1] and is prved identically.

3 ZB. Zabinsky et al.lmathematical Prgramming 69 (1995) The stchastic prcess {W m I m = 0,1, } fr either weak r strng PAS can be mdeled as a Markv chain with states YI,, YK, where state Yl represents the glbal ptimum. The initial prbability distributin fr W is given by 'IT. In standard Markv chain terminlgy [4], YI is the absrbing state f this chain and all ther states are transient. Finite PAS cnverges when the chain reaches the absrbing state. The expected number f iteratins t cnvergence can be expressed in terms f the transitin matrix f the Markv chain. The expected number f iteratins t slve the prblem used in this paper des nt include a stpping rule, and thus indicates the average cmputatinal effrt t sample the glbal ptimum but nt necessarily t cnfirm it. We present a simple direct prbabilistic argument here General analysis Therem 1. The expected number f iteratins t slve the finite ptimizatin prblem is (i) 1+ E~2 'TTj/Pj fr strng PAS and (ii) 1 + E~2 'TTj/Pj-l fr weak PAS where Pj = E{=l 'TTi Prf. Let the randm variable X be the number f iteratins required t slve the finite ptimizatin prblem. Then X =1+ X XK, where X j is the number f iteratins spent in state Yj. Thus, E(X) =1 + E(X2) E(XK) = 1 + E(X2 I \-'2)P(\-'2) +." + E(XK I VK )P(VK), where ") is the event that state Yj is visited. Nw, P(")) =P(") n {W > Yj}) + P(") n {W =Yj}) K = L P(") n Bi,j n {W > Yj}) + P(") n {W =Yj}), i=j+l where Bi,j is the event that state Yi is visited immediately befre a state less than r equal t Yj is visited. Since P(") I Bi,j n {W > Yj}) = 'TTj/Pj and P(W =Yj) = 'TTj, we have K PC")) =L P(") I Bi,j n {W > Yj} )P(Bi,j I W > Yj)P(W > Yj) i=j+l + P(") I W =Yj)P(W =Yj) K =(l - Pj) 'TT' P~ " } i=j+1 'TT- L.J P(Bi,j I W > Yj) + 'TTj 'TT' =(l - Pj) -.l... + 'TTj = -.l... Pj Pj

4 446 ZB. Zabinsky ef aumathematical Prgramming 69 (1995) = 1, and E(X) = 1 + E~2 7Tj/Pj as required. Fr Fr strng PAS, E(Xj I ~) weak PAS, nce state Yj is left, it is never re-entered. As with a gemetric distributin, E(Xj I ~i) then equals the inverse f the prbability f leaving state Yj: E(Xj I ~) =(Pj_t/pj)-l =Pj/Pj-l. This yields E( X) = 1+ E~2 7Tj / Pj-l as required. 0 Nte that the hazard rate 7Tj / pj-l appears in the frmula fr the expected time f weak PAS. Fr cmpleteness, the K x K transitin matrix P, in standard frm, having the (i,nth element P [W m = Yj I Wm-l = Yil, fr strng PAS cnsists f the first K rws f the matrix belw. Fr weak PAS, it cnsists f the last K rws: 1 7TI/pI 7TI/p2 7T2/p2 7Tt/PK-I 7T2/PK-I 7T3/PK-I '" 7TK-t/PK-I 0 7Tt/PK 7T2/PK 7T3/PK... 7TK-t/PK 7TK/PK It fllws that P ( Wm =Yi), the prbability f bjective functin value Yi fr weak r strng PAS n the mth iteratin, is the ith entry f 11'pm, with the apprpriate P. The exact expressins given in Therem 1 are valid fr an arbitrary sampling distributin reflected by 7T. We turn t btaining bunds n these expressins. 3. Bunds n perfrmance 3.1. General bunds An upper bund n the expected number f strng PAS iteratins t slve the finite ptimizatin prblem can be stated simply in terms f 7TI, the prbability f sampling the glbal ptimum with pure randm search. Crllary 2. The expected number f strng PAS iteratins t slve the finite ptimizatin prblem is bunded abve by 1 + lge 1/7TI). Prf. Fr 0 < x < 1, x < -lg(1 - x), s fr j =2,..., K, T < -lg (7T) =lg ( _J_ p ). Pj Pj pj-l Therefre frm the therem, the expected number f iteratins is less than 1+lg(p2/PI) +lg(p3/p2) g(PK/PK-}} = g(1/7T}}. 0

5 ZB. Zabinsky et al./mathematical Prgramming 69 (1995) Fr weak PAS the maximum hazard rate is invlved in the bund. Nte the term 'TTj/Pj-I appearing in Therem l(ii) can be written as (l + 'TTj/Pj_})'TTj/Pj, s a similar argument as Crllary 2 gives the fllwing crllary. Crllary 3. The expected number f weak PAS iteratins t slve the finite ptimizatin prblem is bunded abve by 1 + (Mhazard + 1) lg(l/'tt}} where Mhazard = maxj=2...,k'ttj/pj-i Special bunds Many cmbinatrical ptimizatin prblems f interest have extremely large dmains (e.g., pints) but much smaller ranges (e.g., 100 values). In fact, mst purely discrete prblems which are NP-hard fall int this categry. A gd example is the MAXCLIQUE prblem; given a graph G, find the largest number f vertices which induces a clique in the graph. In this case the range (the number f vertices in the graph) is small, but the dmain is the set f all graphs n these vertices and is highly expnential in the range [3]. Clearly, strng PAS has cmplexity f rder K, the size f the range, as it never requires mre than K iteratins, and thus is fast fr these prblems. This cannt be said fr weak PAS (which is clser t practical algrithms). Weak PAS requires a large number f iteratins when 'TTi+tI'TTi is large fr sme i in the prblem. Hwever, an analgus result fr weak PAS hlds n prblems where these ratis are bunded. The bunding factr r becmes the cnstant f prprtinality. Crllary 4. If 'TTi+I/'TTi ~ r, then weak PAS has cmplexity f rder K. The expected number f weak PAS iteratins is bunded abve by 1 + (K - 1)r. The fllwing special case shws the upper limit f Crllary 4 is apprached. Crllary 5. Given that'tti ex: r i fr r > 1, the expected number fiteratins is bunded belw by (i) K(r - 1)/r fr strng PAS and (ii) 1 + (K - l)(r - 1) fr weak PAS. Prf. Fr strng PAS, Therem 1(i) gives the required value f (r-l) /r l:.f=,1 r j / (r j 1). This is at least K(r - 1)/r. Fr weak PAS, Therem l(ii) gives the required value f 1 + l:.f=,2'ttj/pj-i = 1 + rl:.f=,li'ttj/pj. This is at least 1 + (K - l)(r - 1). D 3.3. Unifrm sampling In rder t cmpare the perfrmance f pure adaptive search n a finite ptimizatin prblem with that n a cntinuus prblem, we cnsider the special case in which the distributin n the bjective functin values is unifrm (i.e., 'TTj = 1/K fr all j).

6 448 ZB. Zabinsky et aumathematical Prgramming 69 (1995) Crllary 6. The expected number f iteratins fr finite glbal ptimizatin, given a unifrm distributin n the bjective functin values, is (i) E~l 1/j, bunded abve by 1 + lg K fr strng PAS and (ii) 1 + E~~ 1 1/j, bunded abve by 2 + lg (K - 1) fr weak PAS. T get the analgus linear cmplexity result, cnsider the vertices f an n-dimensinal lattice, {I,..., k}n, with distinct bjective functin values at the vertices. Here K = ~ and Crllary 6 (fr either weak r strng PAS) gives a bund f 2+lg k n = 2+n lg k, an expressin f rder n. 4. Summary Fr an n-dimensinal lattice, the expected number f iteratins fr weak r strng PAS given a unifrm distributin is f rder n, cnsistent with the results fr PAS n a cntinuus ptimizatin prblem. Fr strng PAS, the expected number f iteratins n a finite ptimizatin prblem is bunded by a simple expressin invlving 1Tl, the prbability f randmly sampling the glbal ptimum. A similar bund fr weak PAS is presented which invlves the hazard rate functin. This bund may be clser t that experienced by practical algrithms, and may inspire use f randm search methds fr finite ptimizatin. References [ 1] S.H. Brks, "A discussin f randm methds fr seeking maxima," Operatins Research 6 (1958) [2] D. Dueck and T. Scheuer, "Threshld Accepting: a general purpse ptimizatin algrithm appearing superir t simulated annealing," Jurnal f Cmputatinal Physics 90 (1990) [3] M.R Garey and D.S. Jhnsn, Cmputers and Intractability: A Guide t the Thery fnp-cmpleteness (Freeman, San Francisc, CA, 1979). [4] J.G. Kemeny and J.L. Snell, Finite Markv Chains (Springer, New Yrk, 1976). [5] N.R. Patel, RL. Smith and Z.B. Zabinsky, "Pure adaptive search in Mnte Carl ptimizatin," Mathematical Prgramming 43 (1988) [6] Z.B. Zabinsky and RL. Smith, "Pure adaptive search in glbal ptimizatin," Mathematical Prgramming 53 (1992)

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