小林 玄宙

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小林, 玄宙

コバヤシ, ヒロミチ

Chuo University Academic Information Repository / 中央大学学術リポジトリ

(2012 )Finding All Solutions of Piecewise-Linear Resistive CircuitsUsing Integer ProgrammingHiromichi KOBAYASHI1.[1], [2][3][4], [5]CPLEX2.nf(x)4= Pg(x) +Qx− r = 0 (1)1x = (x1, x2, · · · , xn)T ∈ Rnng(x) = [g1(x1), g2(x2), · · · , gn(xn)]TP Q n×n rn,f(x) = [f1(x), f2(x), · · · , fn(x)]T3.0-1δ3. 1 ‹[6] “Model ”δ1 i (i = 1, . . . , N)δx1Py +Qx− r = 0xi = ai0 +K∑j=1δijyi = bi0 +K∑j=1bij − bij−1aij − aij−1 δij∆i1µi1 ≤ δi1 ≤ ∆i1∆ijµij ≤ δij ≤ ∆ijµij−10 ≤ δiK ≤ ∆iKµiK−1µij ∈ {0, 1} (j = 1, 2, . . . ,K − 1)(2)3. 2λij(i =1, 2, . . . , N ; j = 0, 1, 2, . . . ,K) 1i (i = 1, . . . , N)x1Py +Qx− r = 0xi = ai0 + (ai1 − ai0)λi1+K∑j=2(aij − 2aij−1 + aij−2)λijyi = bi0 + (bi1 − bi0)λi1+K∑j=2(bij − 2bij−1 + bij−2)λijµi1 ≤ λi1 − λi2 ≤ 1µij ≤ λij − λij+1 ≤ µij−10 ≤ λiK ≤ µiK−1µij ∈ {0, 1} (j = 1, . . . ,K − 1)(3)λij xij−1xi jλλik − λik+1 = 1(k < j)λik − λik+1 = 0(k > j)(4)xij−2xij−1+xij−2 04.g(x) 0-1g(x),x0-1yi, xi yi, xi4. 1x1fh(x) =N∑i=1b′i0 +N∑i=1(b′i1 − b′i0)λi1+N∑i=1K∑j=2(b′ij − 2b′ij−1 + b′ij−2)λij − r = 0µi1 ≤ λi1 − λi2 ≤ 1µij ≤ λij − λij+1 ≤ µij−10 ≤ λiK ≤ µiK−1µij ∈ {0, 1} (j = 1, . . . ,K − 1)(5)h(h = 1, . . . ,M) b′ijPg(x) +Qx xij(3) yi, xi4. 2 ‹δx1fh(x) =N∑i=1b′i0 +N∑i=1K∑j=1b′ij − b′ij−1aij − aij−1 δij − r = 0δi1 ≤ ∆i1δij ≥ ∆ijµijδij ≤ ∆ijµij−1δiK ≥ 0µij ∈ {0, 1} (j = 1, . . . ,K − 1)(6)(2) yi, xi5. IBM ILOG CPLEXIBM ILOG CPLEX( CPLEX ) IBM1112IBMCPLEX10NPCPLEX (solution pool)6.6. 1 1n [3]2.5x3i − 10.5x2i + 11.8xi − i+n∑j=1xj = 0 (7)i = 1, 2, . . . , n.K = 10δS 11 1 ( )δn S10 7 0.1 0.1 0.3 120 7 0.8 0.1 1 230 7 1 0.1 3 240 7 0.1 0.1 4 350 9 9 0.1 9 1260 9 8 0.1 24 1970 9 8 0.1 26 1680 7 9 0.1 17 1290 9 42 0.2 90 39100 9 43 0.2 78 48..................150 11 265 5 329 243200 9 740 5 777 320250 9 894 3 762 1021300 11 1601 4 1403 1366350 9 3106 4 2311 1877400 3 3488 6 3108 2901450 3 6201 9 6375 5147500 3 8358 8 7513 56937. 2[3], [7] 5.1–5.82([−20, 0.5], . . . , [−20, 0.5])TK = 10[3], [7]2 2 ( )δS1 9 0.15 0.14 0.22 0.232 3 0.05 0.04 0.1 0.353 11 0.08 0.17 0.79 1.334 1 1.22 0.3 1.28 1.6810K4K5K 0.5K0.5K10.1K10.1K4K4K30K+12V+12V4K30K30K2V10V30K2 112V4K8K0.1K8K4K0.1K30K1K 4K 10K1K0.1K10K3 25K2.7K2.7K 1K 3K15K 15K1K 11K1K15K15K1K3K 11K505012V15K4 38.11K18V1010012K470150K4.7K68K150K 1.8K25V10 1001.8K1.2K22K 5.6K56K22K330K2K10120K 101K 120K1.5K 27033K5 4δ[1] K. Yamamura and K. Suda, An efficient algorithm for find-ing all solutions of separable systems of nonlinear equations,BIT-Numerical Mathematics, vol.47, no.3, pp.681-691, Sept.2007.[2] K. Yamamura, K. Suda, and N. Tamura, “LP narrowing: Anew strategy for finding all solutions of nonlinear equations, ”Applied Mathematics and Computation, vol.215, no.1, pp.405-413, Sept. 2009.[3] “,” , NLP2009-68, CAS2009-32, pp.47-52, Sept.2009.[4] . 502005.[5] , / /vol.50[6] M. W. Padberg, “Approximating separable nonlinear func-tions via mixed zero-one programs,” Operations Research Let-ters, vol. 27, no. 1, pp.1–5, Aug. 2000.[7] “GLPK,” , NLP2009-144, pp.25-30, Jan. 2010.[8] H. D. Sherali, “On mixed-integer zero-one representationsfor separable lower semicontinuous piecewise-linear functions,”Operations Research Letters, vol. 28, no. 4,pp.155–160, May. 2001.[9] E. M. L. Beale and J. J. H. Forrest, Global Optimization UsingSpecial Ordered Sets, Mathematical Programming, pp.52–69,1976.[10] C.H. Huang, “An Effective Linear Approximation Method forSeparable Programming Problems, ” Applied Mathematicsand Computation, pp.1496–1506, Jul. 2009.

Finding All Solutions of Piecewise-Linear Resistive Circuits Using Integer Programming

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