Source: https://www.nvidia.in/object/iitb-in.html
Timestamp: 2019-04-20 05:08:46+00:00

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Global optimization is the problem of finding the global minimum of a function over a given search space. Global optimization problems occur widely in almost all areas of science and engineering. Many methods for solving global optimization problems are available , , and they find the minimum value of a function on a given domain. Such techniques include Generalized Benders Decomposition, Branch-and-Bound, Outer Approximation, and Generalized Outer Approximation. However these techniques do not guarantee that they can always find the global minimum of the function.
The Bernstein polynomial method is a recent technique for finding the global minimum of polynomial functions. The Bernstein method is based on the idea that if a polynomial is written in the Bernstein form , then the range is bounded by the minimum and maximum Bernstein coefficients. Moreover, the bounds are sharp if and only if these coefficients are also the vertex coefficients. Using these beautiful properties and the tool of domain subdivision, the Bernstein approach to global polynomial optimization is being developed at IIT Bombay. The remarkable feature of the Bernstein optimization approach is that it guarantees finding the global minimum. Further, no initial guess is needed for starting the optimization; an initial search box bounding the domain of interest will do. Recently, a free to use SCILAB based toolbox based on this approach was developed and released by the IIT Bombay group for worldwide use. The software is available from //atoms.scilab.org/toolboxes/Global_Optim_toolbox. More than 1000 downloads of this software have taken place till now.
On the other hand, NVIDIA has been developing the CUDA (Compute Unified Device Architecture)  parallel computing architecture. This architecture enables dramatic increases in computing performance by harnessing the power of the GPU (Graphics Processing Unit). CUDA gives developers access to the virtual instruction set and memory of the parallel computational elements in CUDA GPUs. CUDA provides both a low level API and a higher level API. The next generation GPU (code named Fermi) is a standard on NVIDIA's GPU; it has been designed from the ground up to natively support more programming languages such as C++.
CUDA has been enthusiastically received in several areas of scientific research. However, a survey of the literature and the internet reveals that global optimization approaches specially designed for CUDA GPUs are not yet available.
We have initiated development of new global optimization methods based on the Bernstein polynomial approach for CUDA GPUs. We believe that the new Bernstein polynomial approaches will be able to more efficiently solve global optimization problems, especially where number of variables are large.
The first task in the Bernstein approach is to compute the coefficients of the Bernstein form of polynomials. There are various methods to compute Bernstein coefficients, see . Computation of the Bernstein coefficients in turn involves computation of binomial coefficients. For the latter task, a popular method is based on the routine nchoosek of MATLAB . An alternate, perhaps more efficient method, is Comb6 . We programmed these methods for CUDA C, as both serial and parallel CUDA C versions. We then applied them for constructing large matrices of binomial coefficients. The matrices were of up to 1024 x 1024 size. The computations were done on Intel's 64 bit i7 3.07 GHz processor, MS-Windows 7.0, VC++ 2010, CUDA 3.2 toolkit, with NVIDIA's Fermi Tesla C2050 single GPU.
With the parallel CUDA C version of Comb6, we obtained speedup by factors of 175, 359, and 23030 over the serial CUDA Comb6 version, nchoosek method in C, and nchoosek method in MATLAB, respectively.
A forthcoming paper in the Second International Conference on Meta Computing, Goa, India, Dec. 2011 (see www.icomec.org) describes this work.
In the near future, we plan to develop a complete method based on CUDA GPUs in the Bernstein optimization framework. This is a challenging task for multivariate polynomials, and more so when the number of variables is large. Of course, we also will have to develop an efficient CUDA based software for applying this method to application problems.
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Tharewal Sachin Shrihari , "Automated Synthesis of QFT Controllers and Prefilters using Interval Global Optimization Techniques", IIT Bombay, 2005.
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Bhagyesh V. Patil, “Global optimization of mixed-integer nonlinear polynomial programming problems using the Bernstein form”, IIT Bombay, Joined in Jan 2008.
Priyadarshan S. Dhabe, “A new approach to global optimization based on Bernstein polynomial and GPU computing”, IIT Bombay, Joined in July 2010.
Jeyasenthil R., “Robust Control and GPU Computing”, IIT Bombay, Joined in Jan 2011.

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