## An improved multi-step gradient-type method for large scale optimization.(English)Zbl 1222.90044

Summary: We propose an improved multi-step diagonal updating method for large scale unconstrained optimization. Our approach is based on constructing a new gradient-type method by means of interpolating curves. We measure the distances required to parameterize the interpolating polynomials via a norm defined by a positive-definite matrix. By developing on implicit updating approach we can obtain an improved version of Hessian approximation in diagonal matrix form, while avoiding the computational expenses of actually calculating the improved version of the approximation matrix. The effectiveness of our proposed method is evaluated by means of computational comparison with the BB method and its variants. We show that our method is globally convergent and only requires $$O(n)$$ memory allocations.

### MSC:

 90C26 Nonconvex programming, global optimization 65K10 Numerical optimization and variational techniques 90C52 Methods of reduced gradient type

minpack
Full Text:

### References:

 [1] Barzilai, J.; Borwein, J.M., Two point step size gradient methods, IMA J. numer. anal., 8, 141-148, (1988) · Zbl 0638.65055 [2] Sun, W.; Yuan, Y., () [3] Hassan, M.A.; Leong, W.J.; Farid, M., A new gradient method via quasi-Cauchy relation which guarantees descent, J. comput. appl. math., 230, 300-305, (2009) · Zbl 1179.65067 [4] Leong, W.J.; Hassan, M.A.; Farid, M., A monotone gradient method via weak secant equation for unconstrained optimization, Taiwanese J. math., 14, 2, 413-423, (2010) · Zbl 1203.90148 [5] Dennis, J.E.; Wolkowicz, H., Sizing and least change secant method, SIAM J. numer. anal., 30, 1291-1313, (1993) · Zbl 0802.65081 [6] Farid, M.; Leong, W.J.; Hassan, M.A., A new two-step gradient method for large-scale unconstrained optimization, Comput. math. appl., 59, 3301-3307, (2010) · Zbl 1198.90395 [7] Ford, J.A.; Moghrabi, L.A., Alternating multi-step quasi-Newton methods for unconstrained optimization, J. comput. appl. math., 82, 105-116, (1997) · Zbl 0886.65064 [8] Ford, J.A.; Tharmlikit, S., New implicite updates in multi-step quasi-Newton methods for unconstrained optimization, J. comput. appl. math., 152, 133-146, (2003) · Zbl 1025.65035 [9] Andrei, N., An unconstrained optimization test functions collection, Adv. model. optim., 10, 147-161, (2008) · Zbl 1161.90486 [10] Moré, J.J.; Garbow, B.S.; Hillstorm, K.E., Testing unconstrained optimization software, ACM trans. math. softw., 7, 17-41, (1981) · Zbl 0454.65049 [11] Dolan, E.D.; More, J.J., Benchmarking optimization software with perpormance profiles, Math. program., 91, 201-213, (2002) · Zbl 1049.90004 [12] Dai, Y.H.; Yuan, J.Y.; Yuan, Y., Modified two-point stepsize gradient methods for unconstrained optimization, Comput. optim. appl., 22, 103-109, (2002) · Zbl 1008.90056 [13] Dai, Y.H.; Yuan, Y., Alternative minimization gradient method, IMA J. numer. anal., 23, 373-393, (2003)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.