Exact line searches along each **steepest** **descent** direction converge very slowly. Barzilai and Borwein suggested two stepsizes that ensures superlinear convergence and performs quite well. Barzilai-Borwein **method** is not monotone, thus it is not easy to be generalized for general nonlinear functions. A new stepsize enables fast convergence and possesses monotone property is proposed by Yuan. The new stepsize is modified to obtain modified new **steepest** **descent** **method**, which is for convex quadratic problems only is proposed by Yuan. The new **steepest** **descent** **method** uses the new stepsize after every m exact line search iterations. An algorithm for m=2 is proposed in this paper.

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the setting of the inﬁnite-dimensional Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient **method**, the viscosity approximation **method** [], Mann’s it- eration **method**, and the hybrid **steepest**-**descent** **method**. Our aim is to prove that the it- erative algorithm converges strongly to a common element of these sets, which also solves some hierarchical variational inequality. We observe that related results have been derived in [, , , ].

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In this paper, we introduce a new iterative scheme in a Hilbert space H which is a mixed iterative scheme of (.) and (.). We prove that the sequence converges strongly to a common element of the set of solutions of the system of equilibrium problems and the set of common ﬁxed points of an inﬁnite family of strictly pseudo-contractive mappings by using a viscosity hybrid **steepest**-**descent** **method**. The results obtained in this paper improved and extended the above mentioned results and many others. Finally, we give a simple numerical example to support and illustrate our main theorem in the last part.

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In the setting of Hilbert spaces, we introduce a relaxed hybrid **steepest** **descent** **method** for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions of a variational inequality for an inverse strongly monotone mapping and the set of solutions of generalized mixed equilibrium problems. We prove the strong convergence of the **method** to the unique solution of a suitable variational inequality. The results obtained in this article improve and extend the corresponding results.

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In this paper, we combine the gradient projection algorithm and the hybrid **steepest** **descent** **method** and prove the strong convergence to a common element of the equilibrium problem; the null space of an inverse strongly monotone operator; the set of ﬁxed points of a continuous pseudocontractive mapping and the minimizer of a convex function. This common element is proved to be the unique solution of a variational inequality problem.

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developed a hybrid **steepest**-**descent** **method** for solving VI(F, K) [7,8], but choosing an efficient and implementable nonexpansive mapping is still a difficult problem. Subse- quently, Xu and Kim [9] and Zeng et al. [10] proved the convergence of hybrid stee- pest-**descent** **method**. Noor introduced iterations after analysis of several three-step iterative methods [18]. Ding et al. provided a three-step relaxed hybrid **steepest**-**descent** **method** for variational inequalities [11] and Yao et al. [19] provided a simple proof of the **method** under different conditions. Our group has described a modified and relaxed hybrid **steepest** **descent** (MRHSD) **method** that makes greater use of historical information and minimizes information loss [20].

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5 J. T. Oden, Qualitative Methods on Nonlinear Mechanics, Prentice-Hall, Englewood Cliﬀs, NJ, USA, 1986. 6 I. Yamada, “The hybrid **steepest** **descent** **method** for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8 of Studies in Computational Mathematics, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001.

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[11] I. Yamada, “The hybrid **steepest**-**descent** **method** for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, and S. Reich, Eds., vol. 8, pp. 473–504, North-Holland, Amsterdam, The Netherlands, 2001. [12] H. K. Xu and T. H. Kim, “Convergence of hybrid **steepest**-**descent** methods for variational in-

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Since all the four chirps have same k,τ and the time-domain amplitude considered to be 1 same as in **Steepest** **descent** **method**. Now with this we can justify that the individual spectra will have amplitudes proportional to the square root of their individual BT product , so therefore to model the spectrum as rectangle it is necessary to normalize all he spectrum w.r.t square root of BT product which will be convenient for plotting.

3. Yao, JC, Chadli, O: Pseudomonotone complementarity problems and variational inequalities. In: Crouzeix, JP, Haddjissas, N, Schaible, S (eds.) Handbook of Generalized Convexity and Monotonicity, pp. 501-558 (2005) 4. Kirk, WA: Fixed point theorem for mappings which do not increase distance. Am. Math. Mon. 72, 1004-1006 (1965) 5. Combettes, PL: A block-iterative surrogate constraint splitting **method** for quadratic signal recovery. IEEE Trans. Signal

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The classical variational inequality problem with a Lipschitzian and strongly monotone operator on a nonempty closed convex subset in a real Hilbert space was studied. A new finite-step relaxed hybrid **steepest**-**descent** **method** for this class of variational inequalities was introduced. Strong convergence of this **method** was established under suitable assumptions imposed on the algorithm parameters.

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Classical gradient methods and evolutionary algorithms represent two very different classes of optimization techniques. In optimization, a problem is typically specified by a set of parameters and an objective function, which is also called a fitness function in the context of evolutionary algorithms. The goal of the optimization process is to find a set of variables such that the objective function is optimum. In the special case of continuous parameter optimization in which all parameters are real valued, Newton developed the gradient **method**, which is also known as the **method** of **steepest** **descent**. In unimodal functions, the optimum can be found by moving along the local gradients, which leads to the following formulation of the **steepest**-**descent** **method**. It is obvious that **steepest**-**descent** algorithms can be applied only to continuously differentiable objective functions. If either the objective function is not continuously differentiable or if the function is not (completely) given due to limited knowledge, which often occurs in real-world applications, the designer has to resort to other methods, such as evolutionary algorithms. Evolutionary algorithms are a class of stochastic optimization and adaptation techniques that are inspired by natural evolution. Each evolutionary algorithm is designed along a different methodology. Despite their differences, all evolutionary algorithms are heuristic population-based search procedures that incorporate random variation and selection.

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2. Yamada, I: The hybrid **steepest**-**descent** **method** for the variational inequality problem over the intersection of fixed- point sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. pp. 473 – 504. Kluwer Academic Publishers, Dordrecht, Netherlands (2001)

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Abstract—In this paper, we shall introduce an itera- tive algorithm by multi-step implicit hybrid **steepest**- **descent** **method** for finding a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of solutions of a finite family of variational inequalities for inverse strongly monotone mappings and the set of fixed points of a countable family of nonexpansive mappings in a real Hilbert space. We also prove strong and weak conver- gence theorems for the proposed iterative algorithm under appropriate conditions. Our results improve and extend the earlier and recent results in the liter- ature.

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Yamada, I: The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings.. In: Butnariu, D, Censor, Y, Rei[r]

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In many practical scenarios, it is observed that we are required to filter a signal whose exact frequency response is not known. A solution to such problem is an adaptive filter. It can automatically acclimatize for changing system requirements and can be modelled to perform specific filtering and decision-making tasks. This paper primarily focusses on the implementation of the two most widely used algorithms for noise cancelling which form the crux of adaptive filtering. The empirical explanation of **steepest** **descent** **method** is elucidated along with its simulation in MATLAB by taking a noise added signal and applying the ingenuity of this algorithm to get the desired noise-free response. Furthermore, this paper also sheds light on a more sophisticated algorithm which is based on the underlying criteria of minimum mean square error called as the Least Mean Square (LMS). Additionally, there are various applications of adaptive filtering including system identification which is briefly explained to emphasize the instances where it can be used.

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When Fix(T ) is the set of all minimizers of a convex, continuously Fréchet diﬀerentiable functional f over H, that algorithm () is the **steepest** **descent** **method** [] to minimize f over H. Suppose that the gradient of f , denoted by ∇ f , is Lipschitz continuous with a constant L > and deﬁne T f : H → H by

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For all x S . The convergence of the **steepest** **descent** **method** with Armijo-type search is proved under very general conditions in (Andrei, N., 2009). On the other hand, in (Dai, Y. and Y. Yuan, 2000) it is proved that, for any conjugate gradient **method** with strong Wolfe line search, the following general result holds. Lemma 4.2.1:

In this paper, we introduce a new explicit iteration **method** based on the **steepest** **descent** **method** and Krasnoselskii-Mann type **method** for ﬁnding a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common ﬁxed points for a ﬁnite family of nonexpansive mappings in a real Hilbert space.

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In this paper, we propose a new iteration **method** based on the hybrid **steepest** **descent** **method** and Ishikawa-type **method** for seeking a solution of a variational inequality involving a Lipschitz continuous and strongly monotone mapping on the set of common ﬁxed points for a ﬁnite family of Lipschitz continuous and

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