# 臺灣博碩士論文加值系統

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 在眾多領域方面，最佳化的優劣往往佔了決定性的因素。好的最佳化方法，常常是一件事情成敗的關鍵。在應用數學方面，舉凡”組合最佳化” (combinatorial optimization)、 ”線性規劃”(linear programming)、”整數規劃” (integer programming)、”極值問題”(extreme-value problems)..等等，皆是最佳化的範疇。在實際應用層面上，如”積體電路設計”(VLSI design)、”流程規劃”、”訊號處理”(signal processing)，最佳化的應用更是不可或缺。本文敘述了如何在霍普非爾隨機網路的架構下，以神經活化及關聯性的觀點，來推導平均場退火法(naïve mean field annealing)。推導的方式共分成二部分，第一部份以固定關聯性的觀點來計算神經活化，另一部份以固定活化的觀點來計算神經關聯性。推導出的方法，在平均場退火法(naïve mean field annealing)上具有二個不同的穩定態，並以自由能函數上的鞍點(saddle points)為特徵，同步地在所構成的分佈函數對個別及對耦的神經變數編碼。我們將在之後展示對耦平均場退火法的效能。
 Optimization is a key factor in many areas. It plays a decisive role to the success. Combinatorial optimization, linear programming, integer programming, and extreme-value problems are such examples in applied mathematics. Moreover, in real applications, such as VLSI design, demand chain optimization, signal processing, and railway schedule, optimization is unavoidable. This thesis derives mean field approximation of neural activations and correlations for a stochastic Hopield neural network. The task is decomposed into two subtasks, each composed of the constrained naive mean field approximation, one for mean activations subject to fixed correlations and the other estimating neural correlations subject to fixed mean activations. As a result, the equilibrium state is represented by two sets of mean field equations, which characterize saddle points of a novel free energy function, simultaneously encoding underlying distributions of individual and pairwise neural variables. We explore the capability of the coupled mean field equations for neural optimizations.
 1.Naive mean field theory2.Mean field approximation of neural activations and correlations3.Numerical Simulation and Conclusions
 Hopfield, J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences of the USA, 79(2554).D. S. Johnson, C. R. Aragon, L. A. McGeoch, and C. Schevon. Optimization by simulated annealing: An experimental evaluation; part ii, graph coloring and number partitioning. Operations Research, 39:378--406, 1991.P. Merz and B. Freisleben, Memetic Algorithms and The Fitness Landscape of the Graph Bi-Partitioning Problem, Lecture Notes in Computer Science, No. 1,498, Springer-Verlag, Heidelberg, Germany, 1998, pp. 765--774.S. Boettcher and A.G. Percus, "Nature's Way of Optimizing," Artificial Intelligence, Vol. 119, Nos. 1--2, May 2000, pp. 275--286.R. Battiti and A. Bertossi, "Greedy, Prohibition, and Reactive Heuristics for Graph Partitioning," IEEE Transactions on Computers, in press, 1999. Preprint version available at: http://rtm.science.unitn.it/~battiti/archive/gp.ps.gz.Fundamentals of Statistical Mechanics, Manuscript and Notes of Felix Bloch, Prepared by John Dirk Walecka, Stanford University Press, Stanford CA (1989); 302 pages.M. Mezard, G. Parisi and M. A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).J. Lin. Divergence measures based on the Shannon entropy. IEEE Trans. lnf. Theory, 37(1):145 151, Jan. 1991.S. Kullback and R.A. Leibler (1951). "On information and sufficiency," Ann. Math. Statist. 22,79-86.Peterson C, Soderberg B (1989) A new method for mapping optimization problems onto neural networks. Int J Neural Syst 1:3.C.Peterson and J.R.Anderson, Neural Networks and NP-complete Optimization Problems; A Performance Study on the Graph Bisection Problem, Complex Systems, 2 (1988) 59--89.Thang N. Bui and Byung-Ro Moon. "Genetic Algorithm and Graph Partitioning," IEEE Transactions on Computers, Vol. 45, No. 7, pp. 841-855, 1996.George L. Nemhauser and Laurence A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, New York, 1988.R. Brayton, G. Hachtel, C. McMullen, and A. Sangiovanni-Vincentelli..Logic Minimization Algorithms for VLSI Synthesis. Kluwer Academic Publishers, 1984.
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