Abstract
Convex analysis has wide applications in science and engineering, such as mechanics, optimization and control, theoretical statistics, mathematical economics and game theory, and so on. It offers an analytic framework to treat systems and phenomena that depart from linearity, based on an elegant mathematical characterization of the notion of “duality” (Rockafellar, 1970, 1974, Ekeland and Temam, 1976). Recent work of David Gao (2000) further provided a comprehensive and unified treatment of duality principles in convex and nonconvex systems, greatly enriching the theoretical foundation and scope of applications.
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References
Amari, S. (1985). Differential Geometric Methods in Statistics. Lecture Notes in Statistics 28, Springer-Verlag, New York. Reprinted in 1990.
Amari, S. and Nagaoka, H. (2000). Method of Information Geometry. AMS Monograph, Oxford University Press.
Bauschke, H.H. (2003). Duality for Bregman projections onto translated cones and affine subspaces. J. Approx. Theory 121, 1–12.
Bauschke, H.H. and Combettes, P.L. (2003). Iterating Bregman retractions. SIAM J. Optim. 13, 1159–1173.
Bauschke, H.H., Borwein, J.M., and Combettes, P.L. (2003). Bregman monotone optimization algorithms. SIAM J. Control Optim. 42, 596–636.
Bregman, L.M. (1967). The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Physics 7, 200–217.
Della Pietra, S., Della Pietra, V., and Lafferty, J. (2002). Duality and auxiliary functions for Bregman distances. Technical Report CMU-CS-01-109, School of Computer Science, Carnegie Mellon University.
Eguchi, S. (1983). Second order efficiency of minimum contrast estimators in a curved exponential family. Ann. Statistics 11, 793–803.
Eguchi, S. (1992). Geometry of minimum contrast. Hiroshima Math. J. 22, 631–647.
Ekeland, I. and Temam, R. (1976). Convex Analysis and Variational Problems. SIAM, Amsterdam.
Gao, D.Y. (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic, Dordrecht, xviii+454 pp.
Kurose, T. (1994). On the divergences of 1-conformally flat statistical manifolds. Tohoku Math. J. 46, 427–433.
Matsuzoe, H. (1998). On realization of conformally-projectively flat statistical manifolds and the divergences. Hokkaido Math. J. 27, 409–421.
Matsuzoe, H., Takeuchi, J., and Amari. S (2006). Equiaffine structures on statistical manifolds and Bayesian statistics. Differential Geom. Appl. 24, 567–578.
Nomizu, K. and Sasaki, T. (1994). Affine Differential Geometry – Geometry of Affine Immersions. Cambridge University Press.
Pistone, G. and Sempi, C. (1995). An infinite dimensional geometric structure on the space of all the probability measures equivalent to a given one. Ann. Statistics 33, 1543–1561.
Rockafellar, R.T. (1970). Convex Analysis. Princeton University Press.
Rockafellar, R.T. (1974). Conjugate Duality and Optimization. SIAM, Philadelphia.
Shima, H. (2007). The Geometry of Hessian Structures. World Scientific, Singapore.
Shima, H. and Yagi, K. (1997). Geometry of Hessian manifolds. Differential Geom. Appl. 7, 277–290.
Simon, U., Schwenk-Schellschmidt, A., and Viesel, H. (1991). Introduction to the Affine Differential Geometry of Hypersurfaces. Lecture Notes, Science University of Tokyo.
Takeuchi, J. and Amari, S. (2005). α-Parallel prior and its properties. IEEE Trans. Inf. Theory 51, 1011–1023.
Uohashi, K. (2002). On α-conformal equivalence of statistical manifolds. J. Geom. 75, 179–184.
Zhang, J. (2004). Divergence function, duality, and convex analysis. Neural Comput. 16, 159–195.
Zhang, J. (2006a). Referential duality and representational duality in the scaling of multidimensional and infinite-dimensional stimulus space. In: Dzhafarov, E. and Colonius, H. (Eds.) Measurement and Representation of Sensations: Recent Progress in Psychological Theory. Lawrence Erlbaum, Mahwah, NJ.
Zhang, J. (2006b). Referential duality and representational duality on statistical manifolds. Proceedings of the Second International Symposium on Information Geometry and Its Applications, Tokyo (pp. 58–67).
Zhang, J. (2007). A note on curvature of α-connections on a statistical manifold. Ann. Inst. Statist. Math. 59, 161–170.
Zhang, J. and Hasto, P. (2006). Statistical manifold as an affine space: A functional equation approach. J. Math. Psychol. 50, 60–65.
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Zhang, J., Matsuzoe, H. (2009). Dualistic Riemannian Manifold Structure Induced from Convex Functions. In: Gao, D., Sherali, H. (eds) Advances in Applied Mathematics and Global Optimization. Advances in Mechanics and Mathematics, vol 17. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-75714-8_13
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DOI: https://doi.org/10.1007/978-0-387-75714-8_13
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