An Examination of Tunable, Random Search Landscapes - CiteSeerX
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Abstract This paper carefully considers random landscapes related to Kauffman’s NK model. In particular, it considers a superset of this model (the NKP model) recently suggested in the GA-analytic literature. Landscapes are exhaustively examined for both the distribution of local optima relative to the global optima, and for characteristics that would effect juxtapositional (building block based) search. The later is accomplished through a Walsh-based analysis. The results indicate that K and P have distinct effects on peak distribution, K controlling peak placement, and P affecting relative peak height. The results indicate that P has little effect on a landscape’s expected juxtapositional complexity. Moreover, the results suggest that landscapes developed by the NKP procedure are unlikely to have substantial juxtapositional complexity. A Walsh-based procedure that embodies the flavor of the NKP procedure, but can allow for greater expected juxtapositional complexity, is suggested. ,QWURGXFWLRQ
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x The landscapes have two tunable parameters, (N and K) that can be used to control characteristics (e.g., landscape size and epistasis, respectively) of landscapes. x For given values of N and K, one can generate a large number of landscapes, at random. 7KHVHDGYDQWDJHVDOORZRQHWRHDVLO\JHQHUDWHDODUJHQXPEHURI(&WHVWSUREOHPV DQG H[DPLQH (& SHUIRUPDQFH YHUVXV WKH SDUDPHWHUV 1 DQG . +RZHYHU WKH PHDQLQJIXOQHVV RI VXFK H[DPLQDWLRQV GHSHQGV RQ WKH UHODWLRQVKLSE HWZHHQ WKHVH SDUDPHWHUVDQGWKHGLIILFXOW\RIWKHODQGVFDSHVIRUVHDUFKLQJHQHUDODQG(&VHDUFK LQ SDUWLFXODU .DXIIPDQ©V ZRUN SURYLGHVV RPH LQGLFDWLRQR I WKH UHODWLRQVKLS EHWZHHQ WKH SDUDPHWHUV DQG FHUWDLQ DVSHFWV RI ODQGVFDSHF RPSOH[LW\ 7KLV FRPSOH[LW\PD\ RUPD\QRW LQGLFDWHVHDUFKGLIILFXOW\ 7KHF RPSOHWH UHODWLRQVKLSE HWZHHQ WKH 1. SDUDPHWHUV DQG WKH GLIILFXOW\ RI WKH VHDUFK ODQGVFDSHV WKH\ LQGXFH LV QRW HQWLUHO\ FOHDU IRU VHYHUDO UHDVRQV )LUVW .DXIIPDQ©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Abstract This paper carefully considers random landscapes related to Kauffman’s NK model. In particular, it considers a superset of this model (the NKP model) recently suggested in the GA-analytic literature. Landscapes are exhaustively examined for both the distribution of local optima relative to the global optima, and for characteristics that would effect juxtapositional (building block based) search. The later is accomplished through a Walsh-based analysis. The results indicate that K and P have distinct effects on peak distribution, K controlling peak placement, and P affecting relative peak height. The results indicate that P has little effect on a landscape’s expected juxtapositional complexity. Moreover, the results suggest that landscapes developed by the NKP procedure are unlikely to have substantial juxtapositional complexity. A Walsh-based procedure that embodies the flavor of the NKP procedure, but can allow for greater expected juxtapositional complexity, is suggested. ,QWURGXFWLRQ
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x The landscapes have two tunable parameters, (N and K) that can be used to control characteristics (e.g., landscape size and epistasis, respectively) of landscapes. x For given values of N and K, one can generate a large number of landscapes, at random. 7KHVHDGYDQWDJHVDOORZRQHWRHDVLO\JHQHUDWHDODUJHQXPEHURI(&WHVWSUREOHPV DQG H[DPLQH (& SHUIRUPDQFH YHUVXV WKH SDUDPHWHUV 1 DQG . +RZHYHU WKH PHDQLQJIXOQHVV RI VXFK H[DPLQDWLRQV GHSHQGV RQ WKH UHODWLRQVKLSE HWZHHQ WKHVH SDUDPHWHUVDQGWKHGLIILFXOW\RIWKHODQGVFDSHVIRUVHDUFKLQJHQHUDODQG(&VHDUFK LQ SDUWLFXODU .DXIIPDQ©V ZRUN SURYLGHVV RPH LQGLFDWLRQR I WKH UHODWLRQVKLS EHWZHHQ WKH SDUDPHWHUV DQG FHUWDLQ DVSHFWV RI ODQGVFDSHF RPSOH[LW\ 7KLV FRPSOH[LW\PD\ RUPD\QRW LQGLFDWHVHDUFKGLIILFXOW\ 7KHF RPSOHWH UHODWLRQVKLSE HWZHHQ WKH 1. SDUDPHWHUV DQG WKH GLIILFXOW\ RI WKH VHDUFK ODQGVFDSHV WKH\ LQGXFH LV QRW HQWLUHO\ FOHDU IRU VHYHUDO UHDVRQV )LUVW .DXIIPDQ©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7KLVSDSHUSURYLGHVDQH[WHQVLYHDQDO\VLVRI1.3ODQGVFDSHV7KH1.ODQGVFDSHV DUHD VXEVHWRIWKH1.3 ODQGVFDSHV DQG WKHUHIRUH WKLV LPSOLHV D PRUHH [WHQVLYH DQDO\VLVRI1.ODQGVFDSHVDVZHOO7KHSDSHUDWWHPSWVWRDQVZHUVHYHUDOTXHVWLRQV x Does the P parameter change the complexity characteristics of the NK landscapes examined by Kauffman? If so, how? x Does an exhaustive examination of several NKP landscapes yield complexity characteristics that agree with those reported by Kauffman? x How does the difficulty of NKP landscapes for juxtapositional operators vary with N, K, and P (e.g., how are N, K, and P related to deception and other GAanalytic measures of difficulty)? 7KHSUDFWLFDOLPSOLFDWLRQVRIWKHVHH[DPLQDWLRQVDUHWZRIROG
x They provide greater understanding of the expected difficulty of EC test problems generated via the NKP (and therefore, the NK) landscape procedure. x They suggest possible alternative methods for generating random landscapes with tunable difficulty for use as EC test problems. 0RUHRYHU WKHVHH [DPLQDWLRQV PD\ SURYLGH IXUWKHU LQVLJKW LQWR WKH QDWXUH RI WKH VHDUFKSUREOHPVLQGXFHGE\UHDOZRUOGSUREOHPV
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