In my opinion, the main reason is that steepest ascent and descent method are very simple for these problems. So, they are extremely implementable for engineers.

Nonlinear Programming: Theory and Algorithms 3rd Edition

by Mokhtar S. Bazaraa (Author), Hanif D. Sherali (Author), C. M. Shetty (Author)

These methods are simple to program but will converge very slowly.

This is  an open problem proposed by N. Andrei, I think the steepest descent direction is the only and known information for the iteration.

For smooth functions f(x), there should  be a zero gradient Del f(x)=0 at the desired minimum point x.  Solving for Del f(x)=0 by Newton's method (multivariate) yields a different direction which in most cases leads to an algorithm which converges more rapidly.

Just study a good text about nonlinear programming, the one suggested bu Kudair is a good one. Luenberger also wrote  a nice book about nonlinear  programming.

Dongs answer is completely besides the point!

Steepest ascent and descent methods are important to solve nonlinear programming problems and system of nonlinear equations because its are simple but its converge very slowly.

Dear Boddu,

Steepest descent for minimization (or steepest ascent for maximization)

have some appealing features, for example:

- they use only the 1st order information (gradient of the objective),

- they are simple to implement.

Therefore they might be used to solve some easy optimization problems.

However, they might get desperately slow when applied to anything

but trivial problems. To solve non-trivial problems it is better

to use approaches which employ Newton method because these approaches

capture the 2nd-order information (Hessian of the objective).

To get a better feeling of the problem you might think of a comparison

of the 1st- and the 2nd-order Taylor approximation of the function. The 1st-order uses only the gradient while the 2nd-order uses both

the gradient and the Hessian.

The book of Nocedal and Wright, "Numerical Optimization", Springer 1999

is a good reference.

With best regards,

Jacek Gondzio

Steepest ascent and descent methods are simpler and easy method but its although the convergence  rate is slow..

Since the question is slightly vague providing a quick answer is difficult. And it also differs between systems of nonlinear equations (where you may not have a completely natural objective function) and nonlinear optimization (where an objective function is there by definition of the problem).

If you are not already familiar with the field, you really should read, for example, the books by Bazaraa et al, Luenberger and Ye, Nocedal and Wright, or Bertsekas' book(s) on nonlinear optimization. A fair basic idea about the theory of these problems is MANDATORY, before starting to think about algorithms, and which one to choose for a given problem - which depends not only on properties like convexity and differentiability, but also size, whether derivatives exist and can be computed exactly, and whether constraints are many or few, hard or soft (the latter means that they are allowed to be violated slightly), and so on. 

To summarize - take a course, and read the book. :-) And good luck!

Steepest ascent or steepest descent methods are more important than other similar methods for optimizing the nonlinear problems because they are always convergent; though slow, the descent direction for them is just the minus of the gradient of the objective function and they do not need evaluation of the Hessian.

The steepest ascent or descent methods are two fundamental approaches for non-linear optimization. It is also widely used since it is easy to implement and its convergence has been studied intensively. However, when dealing with practical problem, it might be very difficult to select a good step size. 

Dear Boddu Muralee Bala Krushna,

As an example for solving a constrained optimization problem using a recurrent neural network. By using Steepest ascent and descent models if the solution trajectory is outside the feasible region according to minus gradient of the constrained they will be pushed to feasible region and if the solution trajectory is inside the feasible region by the minus gradient of objective function the solution trajectory will be convergent to the optimal or equilibrium points of the problem. For more details you can see the following papers:

https://www.researchgate.net/publication/312141846_A_projection_type_steepest_descent_neural_network_for_solving_a_class_of_nonsmooth_optimization_problems

https://www.researchgate.net/publication/283448488_A_non-penalty_recurrent_neural_network_for_solving_a_class_of_constrained_optimization_problems

https://www.researchgate.net/publication/236899596_A_New_Steepest_Descent_Differential_Inclusion-Based_Method_for_Solving_General_Nonsmooth_Convex_Optimization_Problems

https://www.researchgate.net/publication/236899619_A_differential_inclusion-based_approach_for_solving_nonsmooth_convex_optimization_problems 

https://www.researchgate.net/publication/236180856_A_recurrent_neural_network_for_solving_a_class_of_generalized_convex_optimization_problems

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Mohammad Munir's answer is incomplete: steepest descent is not always convergent, and when the sequence converges , one needs additional properties about the objective function in order to be sure that the limit point, a critical point, is really a minimum solution. I repeat my first comment: make a serious study of a good book, that is one, that contains the appropriate amount of mathematics, about nonlinear programming!

Dear J. W. Nieuwenhuis,

I have seen your answer. Refer me a good book on above mentioned one.

Thanks to researchers: Ebadi, Gondzio, and all other members for responding to my question on "Why Steepest ascent and descent methods are important to solve nonlinear programming problems and system of nonlinear equations?"

A good book: Nonlinear Programming, Theory and Algorithms, by Bazaraa, Sheraly and Shetty, third edition, 2006. Have a look in the internet!

Thanks J. W. Nieuwenhuis

Thanks to one and all !!