Consider an unconstrained optimization problem (UP)
where is a continuously differentiable function. In general, the iterative algorithms for solving (UP) usually take the form:
where and are current iterative point, a positive step length and a search direction, respectively. For simplicity, we denote by and by .
The main task in the iterative formula (2) is to choose search direction and determine step length along the direction. There are many classic methods to choose search direction , such as the steepest descent methods, Newton-type methods, Variable metric methods (see  ), and conjugate gradient methods
where is a parameter (see    ). For step length , it is usually determined by line search procedure, such as exact line search, Wolfe line search, Armijo line search, and so on. However, these line search procedures may involve extensive computation of objective functions and its gradients, which often becomes a significant burden for large-scale problems. Evidently, it is a good idea that line search procedure is avoided in algorithm design in order to reduce the evaluations of objective functions and gradients.
Based on the above consideration, some authors have started to study the algorithms without line search. Recently, some conjugate gradient algorithms without line search were investigated. In  , Sun and Zhang studied some well-known conjugate gradient methods without line search, for instance, Fletcher-Reeves method, Hestenes-Stiefel method, Dai-Yuan method, Polak- Ribière method and Conjugate Descent method. In  , Chen and Sun researched a two-parameter family of conjugate gradient methods without line search. In   , Wang and Zhu put forward to conjugate gradient path methods without line search. Shi, Shen and Zhou proposed descent methods without line search in  and  , respectively. Further, Zhou presented the steepest descent algorithm without line search in  .
Inspired by the above literatures, in this paper we will extend the descent algorithm without line search of  to more general case, and discuss its global convergence. The rest of this paper is organized as follows. In Section 2, we describe the extended descent algorithm without line search. In Section 3, we analyze its global convergence. Further, we generalize the search direction to more general form, and obtain global convergence of corresponding algorithm. Finally, numerical results are reported in Section 4.
2. Extended Descent Algorithm
To proceed, we first assume that 
(H1) The function f has lower bound on , where is available.
(H2) The gradient g is Lipschitz continuous in an open convex set that contains , i.e., there exists such that
Now we give the extended algorithm.
Algorithm 2.1. Given a starting point , a positive constant , three
parameters and such that , . Let .
Step 1. If , then stop; otherwise go to Step 2.
Step 2. Compute
Step 3. Set search direction
Step 4. Compute step length by the following rule. When , is determined by Wolfe line search, i.e., it satisfies that
where satisfies that and is a positive sequence which has a sufficient large upper bound.
Step 5. Set next iterative point
Step 6. Set , and go to Step 1.
Remark 2.1. Note that the formula of and in Algorithm 2.1 are the generalized forms of those in  .
3. Global Convergence
Lemma 3.1. If Algorithm 2.1 generates an infinite sequence , then all search directions are descent, and , it holds that
Proof. If , it is obvious that . If , by (5) and (6), we have
This completes the proof. ,
Lemma 3.2 (Mean value theorem, see  ). Suppose that the objective function is continuously differentiable on an open convex set , then
where , . If is twice continuously differentiable on , then
Lemma 3.3. ,
Proof. Where , it holds that by (5). Then , we have
Using induction principle and noting that , it yields that
Therefore (16) holds. The proof is completed. ,
Theorem 3.1. If (H1), (H2) hold, and Algorithm 2.1 generates an infinite sequence , then
Proof. When , from (13), (4), Lemma 3.1, Lemma 3.3 and , it yields that
which implies that is a decreasing sequence. And it is clear that the sequence generated by Algorithm 2.1 is contained in by (H1), and there exists a constant such that . Therefore
which combining with (19) yields
Since has an upper bound, (17) holds.
On the other hand, by (9) and Lemma 3.1, we have
By the same analysis as the above proof, (18) holds. The proof is completed. ,
Lemma 3.4 (see  ). If the conditions in Theorem 3.1 hold and , then both the sequence and have a bound.
Theorem 3.2. If the conditions in Theorem 3.1 hold, then
Proof. Suppose , then there exists a positive such that
In the following, we carry out our proofs in two cases.
Case 1. We complete the proof by utilizing reduction to absurdity. Suppose that . By (17), we have
From Lemma 3.4, we know that there exists such that . Combining (23), we have
It is known that
which contradicts with (24). Therefore (22) holds.
Case 2. When , the proof is the same as that in  and here is omitted.
It follows from the proofs of Case 1 and Case 2 that (22) holds. This completes the proof. ,
Remark 3.1. Search direction of Algorithm 2.1 can be extended to more general form as follows:
where the function satisfies the following conditions(see  ):
a) It is continuous and strictly monotone increasing when ;
b) and ;
c) is continuous, strictly monotone increasing when , and
Evidently, there are many functions satisfying the conditions (a)-(c). For
example, , , , etc (see  ). We denote Algorithm
2.1 in which is determined by (26) as Algorithm 3.1. By using proof technique of above Theorem 3.2, it is easy to get its convergence theorem.
4. Numerical Results
In this section, we report some preliminary numerical experiments. The test problems and their initial values are drawn from  .
In numerical experiment, we take the parameter ,and stop the iteration if the inequality is satisfied. The detailed numerical results
Table 1. Numerical results.
are reported in Table 1, in which NI, NF and NG denote the total number of iterations, the total number of function evaluations and the total number of gradient evaluations, respectively. From Table 1, we can see the extended algorithm has good numerical results.
In this paper, we extended the descent algorithm without line search of  to more general case, and got its global convergence. Compared with  , the extended algorithm has more effective numerical perfermance, so it is effective. In the future, we will further research the descent algorithms without line search, and try to get some new descent algorithms without line search, which not only convergence globally, but also have good numerical results.
We gratefully acknowledge the scholarship fund of education department of Guangxi Zhuang autonomous region, Guangxi basic ability improvement project fund for the middle-aged and young teachers of colleges and universities (2017KY0068, KY2016YB069), Guangxi higher education undergraduate course teaching reform project fund (2017JGB147), NNSF of China (11761014), Guangxi natural science foundation (2017GXNSFAA198243).
 Chen, X.D. and Sun, J. (2002) Global Convergence of a Two-Parameter Family of Conjugate Gradient Methods without Line Search. Journal of Computational and Applied Mathematics, 146, 37-45.
 Wang, J.Y. and Zhu, D.T. (2016) Conjugate Gradient Path Method without Line Search Technique for Derivative-Free Unconstrained Optimization. Numerical Algorithms, 73, 957-983.
 Wang, J.Y. and Zhu, D.T. (2017) Derivative-Free Restrictively Preconditioned Conjugate Gradient Path Method without Line Search Technique for Solving Linear Equality Constrained Optimization. Computers and Mathematics with Applications, 73, 277-293. https://doi.org/10.1016/j.camwa.2016.11.025