**Accelerated Conjugate Gradient Algorithm with Modified Secant Condition for Unconstrained Optimization**

**Neculai ANDREI ^{1,2}**

^{1 }Research Institute for Informatics, Center for Advanced Modeling and Optimization

8-10, Averescu Avenue, Bucharest 1, Romania

^{2 }Academy of Romanian Scientists

54, Splaiul Independentei, Bucharest 5, Romania

**Abstract**: Conjugate gradient algorithms are very powerful methods for solving large-scale unconstrained optimization problems characterized by low memory requirements and strong local and global convergence properties. Over 25 variants of different conjugate gradient methods are known. In this paper we propose a fundamentally different method, in which the well known parameter is computed by an approximation of the Hessian / vector product through modified secant condition. For search direction computation, the method takes both the available gradient and the function values information in two successive iteration points and achieves high-order accuracy in approximating the second-order curvature of the minimizing function. For steplength computation the method uses the advantage that the step lengths in conjugate gradient algorithms may differ from 1 by two order of magnitude and tend to vary in an unpredictable manner. Thus, we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with some conjugate gradient algorithms (including CG_DESCENT by Hager and Zhang [19], CONMIN by Shanno and Phua [29], SCALCG by Andrei [3-5], or LBFGS by Liu and Nocedal [22]) using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that the suggested algorithm outperforms the known conjugate gradient algorithms and LBFGS.

*MSC*: 49M07, 49M10, 90C06, 65K

**Keywords**: Unconstrained optimization, conjugate gradient method, Newton direction, modified secant condition, numerical comparisons.

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**CITE THIS PAPER AS**:

Neculai ANDREI, **Accelerated Conjugate Gradient Algorithm with Modified Secant Condition for Unconstrained Optimization**, *Studies in Informatics and Control*, ISSN 1220-1766, vol. 18 (3), pp. 211-232, 2009.

**Introduction**

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Let us consider the nonlinear unconstrained optimization problem

where is a continuously differentiable function, bounded from below. As we know, for solving this problem starting from an initial guess a nonlinear conjugate gradient method generates a sequence as

where is obtained by line search and the directions are generated as

In (1.3) is known as the conjugate gradient parameter, and . Consider the Euclidean norm and define . The line search in the conjugate gradient algorithms is often based on the standard Wolfe conditions:

(1.4)

where is a descent direction and

The search direction , assumed to be a descent one, plays the main role in these methods. Different conjugate gradient algorithms correspond to different choices for the scalar parameter On the other hand the stepsize guarantees the global convergence in some cases and is crucial in efficiency. The line search in the conjugate gradient algorithms is often based on the standard Wolfe conditions. Plenty of conjugate gradient methods are known and an excellent survey of these methods with a special attention on their global convergence is given by Hager and Zhang [20]. A numerical comparison of conjugate gradient algorithms (1.2) and (1.3) with Wolfe line search (1.4) and (1.5), for different formulae of parameter computation, including the Dolan and Moré performance profile, is given in [6].

In [23] Jorge Nocedal articulated a number of open problems in conjugate gradient algorithms. Two of them seem to be really very important. One refers to the direction computation in order to take into account the problem structure. In particular, when the problem is partially separable the idea is to use the partitioned updating like in quasi-Newton methods [18]. The second one focuses on the step length. Intensive numerical experiments with conjugate gradient algorithms proved that the step length may differ from 1 up to two orders of magnitude, being larger or smaller than 1, depending on how the problem is scaled. Moreover, the sizes of the step length tend to vary in a totally unpredictable way. This is in contrast with the Newton and quasi-Newton methods, as well as with the limited memory quasi-Newton methods, which usually admit the unit step length for most of the iterations and require only very few function evaluations for step length determination.

In this paper we present a conjugate gradient algorithm which address to these open problems. The structure of the paper is as follows. In section 2 we present a conjugate gradient algorithm with modified secant condition. The idea of this algorithm is to use the Newton direction for computation in (1.3). This leads us to a formula for which contains the Hessian of the minimizing function.

In section 3 we present the convergence of the algorithm both for uniformly convex functions and for general nonlinear functions. We prove that under common assumptions and if the direction is a descent one then the method is globally convergent. In section 4 we present an acceleration scheme of the algorithm. The idea of this computational scheme is to take advantage that the step lengths in conjugate gradient algorithms are very different from 1.

Therefore, we suggest we modify in such a manner as to improve the reduction of the function values along the iterations. In section 5 we present the ACGMSEC algorithm and we prove that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons of our algorithm with some other conjugate gradient algorithms including CG_DESCENT by Hager and Zhang [19], CONMIN by Shanno and Phua [29], SCALCG by Andrei [3-5], or limited quasi-Newton LBFGS by Liu and Nocedal [22] are presented in section 6. For this we use a set of 750 unconstrained optimization problems presented in [1], some of them from the CUTE library [10]. We show that the suggested algorithm outperforms the above conjugate gradient algorithms and LBFGS.

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