Issues on the use of a modified Bunch and Kaufman decomposition for large scale Newton’s equation

In this work, we deal with Truncated Newton methods for solving large scale (possibly nonconvex) unconstrained optimization problems. In particular, we consider the use of a modified Bunch and Kaufman factorization for solving the Newton equation, at each (outer) iteration of the method. The Bunch and Kaufman factorization of a tridiagonal matrix is an effective and stable matrix decomposition, which is well exploited in the widely adopted SYMMBK (Bunch and Kaufman in Math Comput 31:163–179, 1977; Chandra in Conjugate gradient methods for partial differential equations, vol 129, 1978; Conn et al. in Trust-region methods. MPS-SIAM series on optimization, Society for Industrial Mathematics, Philadelphia, 2000; HSL, A collection of Fortran codes for large scale scientific computation, http://www.hsl.rl.ac.uk/; Marcia in Appl Numer Math 58:449–458, 2008) routine. It can be used to provide conjugate directions, both in the case of $$1\times 1$$ 1 × 1 and $$2\times 2$$ 2 × 2 pivoting steps. The main drawback is that the resulting solution of Newton’s equation might not be gradient–related, in the case the objective function is nonconvex. Here we first focus on some theoretical properties, in order to ensure that at each iteration of the Truncated Newton method, the search direction obtained by using an adapted Bunch and Kaufman factorization is gradient–related. This allows to perform a standard Armijo-type linesearch procedure, using a bounded descent direction. Furthermore, the results of an extended numerical experience using large scale CUTEst problems is reported, showing the reliability and the efficiency of the proposed approach, both on convex and nonconvex problems.

OPTIMIZATION OF A SMALL SOLAR CHIMNEY

An optimization study to maximize the exergy efficiency of a small-scale solar chimney was carried out. Optimization variables include collector diameter (Dc), collector height (Hc), tower height (Ht), and tower diameter (Dt). Models from the literature were used to predict environmental and airflow parameters. Exergy efficiency and solar chimney efficiency were determined, on an hourly basis, for a one-year period. The model was simulated using the EES software. Two methods of optimization were used, the method of conjugate directions and the method of variable metric, both providing similar results. Results were compared to the results from an experimental prototype, and it was found that the energetic and exergetic efficiency were significantly improved. The analysis indicated that the height and diameter of the chimney and collector are the most important physical variables in the design of a solar chimney. For both methods, it was found that the maximum exergy efficiency was obtained with a collector height of 0.5 m, a collector diameter of 30 m, a tower diameter of 1 m, and a tower height of 17.8 – 18.8 m. The exergy efficiency was 44 %.

Efficiency Optimization of Four Gas Turbine Power Plant Configurations

Gas turbine power plants fueled by natural gas are common due to their quick start-up operation and low emissions compared with steam power plants that are directly fired. However, the efficiency of basic gas turbine power plant is considered low. Any improvement in the efficiency would result in fuel savings as well as reduction in CO2 emissions. One way to improve the efficiency is to utilize exhaust gas waste heat. Two waste heat utilization options were considered. The first option was to run a steam power plant (i.e. combined cycle power plant) while the other option was to use a regenerator which reduces the size of the combustion chamber. The regenerator utilizes the waste heat to preheat the compressed air before the combustion chamber. In addition, the efficiency can be improved with compressor intercooling and turbine reheating. In this paper, four gas turbine power plant configurations were investigated and optimized to find the maximum possible efficiency for each configuration. The configurations are (1) basic gas turbine, (2) combined cycle, (3) advanced combined cycle and (4) gas turbine with regenerator, intercooler and reheater. The power plants were modeled in EES software and the basic model was validated against vendor’s data (GE E-class gas turbine Model 7E) with good agreement. Maximum discrepancy was only 3%. The optimization was carried out using conjugate directions method and improvements in the baseline design were as high as 25%. The paper presents the modeling work, baseline designs, optimization and analysis of the optimization results using T-s diagrams. The efficiency of the optimized configurations varied from 49% up 65%.