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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

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Research Digest Mathematical Optimization Mathematical approach to pursue the best

Makoto YAMASHITA Department of Mathematical and Computing Science Tokyo Institute of Technology

2017/11/20 1

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 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 !!

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Choosing the best for you

Many fruits The best mix juice

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Mathematical modeling of real-world

ptimization

problems Numerical methods based on mathematical perspective Implementation

f numerical

methods as software

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Three key approaches

Recent research will be introduced from the next slide

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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.

“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

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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.

“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

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 Attach sensors to multiple wild animals in Savanna ⇒ Know the range of living from distance information  Attach sensors to building ⇒Measure aged deterioration  These problems give the same mathematical model.  The proposed method utilizes semidefinite programming to handle more sensor information

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Estimate for sensor positions from distances between sensors

“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

distance distance distance the position of each animal

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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

“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.

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 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

Software development for semidefinite programming

“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)

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 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

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Ongoing research topics

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 Mathematical Optimization  Continuous Optimization  Nonlinear Optimization  Semidefinite Programming  and applications to solve practical problems

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Research keywords

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Mathematical Optimization Enjoy studying

Not for my society, Not for your society, But to improve our society.