本篇論文主要在研究推廣型平均函數的凸性。特別地，我們對於凸性指標函數特別感興趣，因為其可以被利用來選取一對內、外函數來合成凸的推廣型平均函數。我們發現這個凸性指標也出現在數學規劃的其他領域，比如說牛頓法及自協和函數。此外，藉著研究自正則函數，我們可以構造出許多凸性指標大於一的函數。最後我們提及推廣型平均函數及(h,θ)凸函數之間的關係。

The purpose of this thesis is to study the convexity of generalized mean function. In particular, we are most interested in the index function of convexity which can be used to choose a pair of inner and outer functions so that the composite function is convex. We found that the index function appears in many other areas of mathematical programming such as the Newton’s method and the self-concordance functions. In addition, by studying the self-regular functions, we construct many functions whose index function of convexity is greater than one. The relationship between the generalized mean function and (h,θ) convex functions is also mentioned.

[1] M. Avriel, “Nonlinear Programming” Analysis and Methods, Englewood Cliffs, New Jersey, pp. 101~181, 1976.
[2] M. S. Bazaraa, C. M. Shetty, “Nonlinear Programming” Theory and Algorithms, Georgia, USA, pp. 78~91, 1979.
[3] A.Ben-Tal and M. Teboulle, “A smoothing technique for nondifferentiable optimization problems.” Lecture Notes in Mathematics 1405, Springer-Verlag, Varetz, 1988.
[4] S. Boyd and L. Vandenberhe, “Introduction to Convex Optimization with Engineering Applications,” Stanford university, Stanford, CA, 1997.
[5] N. Cameron, “Introduction to Linear and Convex Programming,” Cambridge University, Cambridge, pp. 83~95, 1985.
[6] M. Chiang. “Geometric Programming for Communication Systems,” 2004. To appear in Trends and Foundations in Communication and Information Theory.
[7] S.C. Fang, “An unconstrained convex programming view of linear programming,” Mathematical Methods op Operations Research, 36, pp.149-161, 1992.
[8] S. C. Fang, J. R. Rajasekera, and H. S. J. Tsao, “Entropy Optimization and Mathematical Programming.” Boston: Kluwer Academic, 1997.
[9] S. C. Fang And H. S. J. Tsao, “On the entropic perturbation and exponential penalty methods for linear programming,” Journal of Optimization Theory and Applications, 89, pp. 461-466, 1996.
[10] W. Fenchel, “Convex set, Cones and Functions,” Lecture in Preceton University, New Tersey, 1953.
[11] G.. Hardy, J. L. Littlewood and G.. Pólya, “Inequalities,” Cambridge University Press, 1934.
[12] Y. Nesterov and A. Nemirovsky, “Interior-point Polynomial Methods in Convex Programming,” SIAM Studies 13, Philadelphia, PA, 1994.
[13] J. Peng, C. Roos, and T. Terlaky, “Self-Regular Proximities and New Search Directions for Linear and Semidefinite Optimization,” Technical report, Department of Computing and Software, McMaster University, Ontario, Canada, March 2000.
[14] J. Peng, C. Roos, and T. Terlaky, “Self-Regularity: A new paradigm for primal-dual interior point algorithms, Princeton University Press, 2002.
[15] R. T. Rockafellar, “Convex Analysis,” Princeton , New Jersey, pp. 23~31, 1972.
[16] C.E. Shannon, “A Mathematical Theory of Communication,” Bell system Technical Journal, 27, pp. 379-423, 1948.
[17] Y. B. Zhao, F. C. Fang and D. Li, “Constructing Generalized Mean Functions using Convex Functions with Regularity Conditions,” accepted for publication in SIAM Journal on Optimization, 17(1), pp.35-57, 2006.