Unified Theory and Algorithm for Solving Challenging Problems in Mathematical Physics and Complex Systems with Applications

Supported by this AOARD grant, the PI and his post-doctor and co-workers have successfully developedimproved a breakthrough canonical duality theory and its associated algorithms for solving a large class of challenging problems in mathematical physics and complex systems. Within one year, he has published 1 book by Springer, 1 journal special issue Springer, and about 29 papers 11 are journal papers. The most significant achievement is the solution to the well-known knapsack problem, which is listed as one of 21 NP complete problems in computer science. By using the canonical duality theory, this 0-1 integer programming problem can be equivalently converted to a non-smooth concave maximization problem with only one dual variable, which can be solved very easily, therefore, this so-called NP-complete problem can be obtained analytically via this canonical dual solution. Application to computational physics leads to a powerful deterministic algorithm for solving the most challenging bi-level mixed integer programming problem in structural topology optimization.