By S.-C. Fang, J. R. Rajasekera, H.-S. J. Tsao (auth.)
Entropy optimization is an invaluable mix of classical engineering idea (entropy) with mathematical optimization. The ensuing entropyoptimization models have proved their usefulness with profitable purposes in parts similar to photo reconstruction, development attractiveness, statistical inference, queuing concept, spectral research, statistical mechanics, transportation making plans, city and neighborhood making plans, input-output research, portfolio funding, details research, and linear and nonlinear programming.
whereas entropy optimization has been utilized in assorted fields, various appropriate answer tools were loosely built with no enough mathematical therapy. a scientific presentation with right mathematical therapy of this fabric is required by means of practitioners and researchers alike in all software parts. the aim of this booklet is to satisfy this want. Entropy Optimization andMathematical Programming deals views that meet the wishes of numerous user communities in order that the clients can practice entropyoptimization techniques with whole convenience and simplicity. With this attention, the authors specialise in the entropy optimization difficulties in finite dimensional Euclidean house such that just some easy familiarity with optimization is needed of the reader.
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Extra resources for Entropy Optimization and Mathematical Programming
Ym] with corresponding probabilities pi} (, page 544). In other words, i, j = 1, ... , n, and m = 1, ... 64) with M L pi} = 1, V i, j = 1, ... , n. j = Ym] = qi}, for i, j = 1, ... , n, m = 1, ... t. j=l m=l = 1, ... 67) xjYmPi} = U;, i xjYmPi} = Vj, j = 1, ... M n LL ;=1 m=l 2 M L pi} = 1, m=l pi} 2: 0, i, j = 1, ... , n, i, j = 1, ... , n, and m = 1, ... , M. 59) respectively. 7, but they are very similar in the sense that both have only linearly constrained equalities. 8 can be generalized when more complex probability distributions for coefficients a;j occur.
58) j = 1, ... j = 0, i, j = 1, ... , n. 7 is equivalent to a regular cross-entropy minimization problem. 60) LJ-lj(LXjaij - Vj), j=1 i=l where Ai and J-lj, i, j = 1, ... 59), respectively. 61 ) where ri = exp[>"Xj-~) and Sj = exp[l'jXj- ~), for i, j = 1, ... , n. jsjXj), i = 1, ... jXj ), j = 1, ... , n. 63). 63) by an iterative procedure. The name RAS originated from the matrix notation given to the n x n diagonal matrices defined as: R = diag[ri], and S = diag[sj]. It can be shown that (, page 310) A = RAoS.
5, 1971, pp. 122-140. , "Traffic Distribution and Entropy," Nature, Vol. 220, 1968, pp. 974-976. , "Practices in Input-Output Table Compilation," Regional Science and Urban Economics, Vol. 24, 1994, pp. 27-54. , "Multicriterion Maximum Entropy Image Reconstruction from Projections," IEEE Transactions on Medical Imaging, Vol. 11, 1992, pp. 70-75. , "A Statistical Theory of Spatial Distribution Models," Transportation Research, Vol. 1, 1967, pp. 253-269. , Entropy in Urban and Regional Planning, Pion, London, 1970.
Entropy Optimization and Mathematical Programming by S.-C. Fang, J. R. Rajasekera, H.-S. J. Tsao (auth.)