Research Digest Mathematical Optimization Mathematical approach to pursue the best Makoto YAMASHITA Department of Mathematical and Computing Science Tokyo Institute of Technology 1 2017/11/20
Choosing the best for you Many fruits The best mix juice Lots of “optimization problems” in our daily lives How to store as much as possible on the shelves? What is the best route to explore all the sightseeing spots? Solve them with a mathematical approach !! 2017/11/20 2
Three key approaches Mathematical Implementation modeling of of numerical real-world methods as optimization software problems Numerical methods based on mathematical perspective Recent research will be introduced from the next slide 2017/11/20 3
“Using semi-definite programming to optimize unequal deployment of genotypes to a clonal seed orchard,” J Ahlinder, T. J. Mullin, M. Yamashita Tree Genetics & Genomes , DOI:10.1007/s11295-013-0659-z, September, 2013 Optimal contribution in tree breeding Many genotypes Highest profit considering genetic diversity A good seed-orchard should include as many profitable trees as possible keep genetic diversity to improve long-term performance This problem is solved through an optimization problem called semidefinite programming. 2017/11/20 4
“A Successive LP Approach with C-VaR Type Constraints for IMRT Optimization,” S. Kishimoto, M. Yamashita To Appear in Operations Research for Health Care , DOI:10.1016/j.orch.2017.09.007, September, 2017 Beam intensity in radiotherapy For cancer treatment, beam intensity should be controlled so that higher radiation dose for cancer cells lower radiation dose for normal tissues around cancer The proposed method solves linear programming iteratively and enables to avoid the serious problem that a specific part receives extraordinary doses. 2017/11/20 5
“Algorithm 920: SFSDP: a Sparse Version of Full SemiDefinite Programming Relaxation for Sensor Network Localization Problems ,” S. Kim, M. Kojima, H. Waki and M. Yamashita ACM Transactions on Mathematical Software , Volume 38 Issue 4, Article No. 27, August, 2012 Estimate for sensor positions from distances between sensors distance Attach sensors to multiple wild animals in Savanna ⇒ Know the range of living distance distance from distance information Attach sensors to building the position of each animal ⇒ Measure aged deterioration These problems give the same mathematical model. The proposed method utilizes semidefinite programming to handle more sensor information 2017/11/20 6
“The second-order reduced density matrix method and the two-dimensional Hubbard model,” J. S. M. Anderson, M. Nakata, R. Igarashi, K. Fujisawa, M. Yamashita Computational and Theoretical Chemistry , 1003 , 22—27, 2013. Electronic structure calculation in quantum chemistry The most stable structure of molecules/atoms is very important to know the energy of molecules/atoms To understand what kind of chemical reactions is likely to cause Mathematical model through semidefinite programming 2017/11/20 7
“Latest developments in the SDPA Family for solving large-scale SDPs,” M. Yamashita, K. Fujisawa, M. Fukuda, K. Kobayashi, K. Nakta, M. Nakata in "Handbook on Semidefinite, Cone and Polynomial Optimization: Theory, Algorithms, Software and Applications“ edited by Miguel F. Anjos and Jean B. Lasserre, Springer, NY, USA, Chapter 24, pp. 687--714 (2011) Software development for semidefinite programming Semidefinite programming (SDP) is applied to Control theory Combinatorial Optimization Quantum Computation Tree breeding and so on We are developing an SDP solver called SDPA https://sdpa.sourceforge.net/ Free software package on Windows/Linux Many users from various contries 2017/11/20 8
Ongoing research topics Application side Efficient method for Pooling Problem Mathematical model for Bikesharing Theoretical side Improvement on primal-dual interior-point method Theoretical analysis on semidefinite programming 2017/11/20 9
Research keywords Mathematical Optimization Continuous Optimization Nonlinear Optimization Semidefinite Programming and applications to solve practical problems 2017/11/20 10
Research Concept Mathematical Optimization Mathematical approach to pursue the best Not for my society, Not for your society, But to improve our society. Enjoy studying 2017/11/20 11
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