Major scientific results

During the last two decades, Nonconvex Programming (differentiable /nondifferentiable) knowed spectacular developments all the world over. The explanation for this explosion is simple: modern convex analysis and convex programming widely studied since the beginning of the 1960's are forced to a logical and natural extension to nonconvexity/nondifferentiability. Because most practical problems are of nonconvex nature, and in many areas of industrial, economic and financial applications, the requirements of-the-moment lead to replacements of old convex models by nonconvex ones, more complex but more reliable (because they represent more accurately the nature of studied problems) and particularly more economic and competitive.

 

The most innovative and applicative part of my research activities in Optimization is concerned with DC (Difference of Convex functions) and DCA (DC Algorithms), which constitute the backbone of Nonconvex Programming and Global Optimization. We would like to make an extension of convex programming, not too large to still allow using the arsenal of powerful tools in convex analysis and convex optimization but sufficiently wide to cover most real world nonconvex optimization problems. It turns out that the key idea for designing DCA is to generate an appropriate sequence of approximate convex programs to DC programs. My works concern both local and global approaches in nonconvex programming with a more important part devoted to local approaches.

  

DC programming and DCA are introduced in 1985 by Pham Dinh Tao in their preliminary form and extensively developed by Le Thi Hoai An & Pham Dinh Tao since 1994, to become now classics and increasingly popularized by researchers and practitioners all the world over, for modeling and solving nonconvex programs in different fields: Transport-Logistics, Telecommunication, Genomique, Finance, Data Mining and Machine Learning, Cryptology, Mechanics, Image Processing, Robotic & Computer Vision, Petrochemistry, Optimal Control, Inverse Problems and Ill-Posed Problems, Multiobjective Programming, Variational Inequalty Problems (VIP), etc. . Their popularity is due to their robustness and efficiency compared to existing methods, their adaptation to the structures of treated problems and their ability to solve large-scale real world nonconvex programs.

 

In particular, one important part of my works focuses on DC programming and DCA for machine learning and data mining. And besides these works, in the last decade, a variety of machine learning methods based on DCA was developed by other researchers who demonstrated the effectiveness and the scalability of DCA. The emergence of DC programming and DCA in the community of researchers and practitioners is mainly due to their usefulness and efficiency in the field of machine learning and data mining.

 

In addition to our (my colleagues and myself) work, you can find on http://lita.sciences.univ-metz.fr/~lethi/index.php/en/research/dc-programming-and-dca.html an incomplete list of the work of researchers worldwide who use our Approach "DC Programming and DCA" in various topics, especially data mining, image processing and communication networks.

 

In applied research, I have developed new powerful methods in  DC programming and global optimization approaches for several large scale problems in various fields mentioned above. For instance (see "DC programming and DCA" for more complete list)

 

  • Machine learning and data mining: various areas of unsupervised learning, supervised learning, semi-supervised learning, learning with sparsity / uncertainty
  • Transport Logistics 
  • Production management: Supply Chain Design, Scheduling, production and maintenance, etc
  • Communication systems: Routing, Cross-layer Optimization in Multi-hop Networks TDMA, Power Allocation in Wireless Networks, Optimal spectrum balancing in multi-users DSL Networks, etc
  • Genomics and Bioinformatics: molecular conformation by distance geometry, minimizing Lennard-Jones / Morse potential energy functions,  multiple sequence alignment,  clustering of genes, gene selection, etc
  • Data security: Cryptography, Cryptanalysis, Anomaly detection
  • Image processing and computer vision: image restoration, image segmentation, intelligent camera, discrete tomography, compressed sensing, sparse image recovery, etc
  • Finance: several issues of optimal portfolio management:  risk management, portfolio optimization with transaction cost, worst cases analysis, etc