## Abstract

Consider the BFGS quasi-Newton method applied to a general non-convex function that has continuous second derivatives. This paper aims to construct a four-dimensional example such that the BFGS method need not converge. The example is perfect in the following sense: (a) All the stepsizes are exactly equal to one; the unit stepsize can also be accepted by various line searches including the Wolfe line search and the Arjimo line search; (b) The objective function is strongly convex along each search direction although it is not in itself. The unit stepsize is the unique minimizer of each line search function. Hence the example also applies to the global line search and the line search that always picks the first local minimizer; (c) The objective function is polynomial and hence is infinitely continuously differentiable. If relaxing the convexity requirement of the line search function; namely, (b) we are able to construct a relatively simple polynomial example.

This is a preview of subscription content, log in to check access.

## References

- 1
Byrd R.H., Nocedal J., Yuan Y.X.: Global convergence of a class of quasi-Newton methods on convex problems. SIAM J. Numer. Anal.

**24**, 1171–1190 (1987) - 2
Broyden, C.G.: The convergence of a class of double rank minimization algorithms: 2. The new algorithm. J. Inst. Math. Appl.

**6**, 222–231 (1970) - 3
Dai Y.H.: Convergence properties of the BFGS algorithm. SIAM J. Optim.

**13**(3), 693–701 (2002) - 4
Davidon W.C.: Variable metric methods for minimization. SIAM J. Optim.

**1**, 1–17 (1991) - 5
Dennis J.E., Moré J.J.: Quasi-Newton method, motivation and theory. SIAM Rev.

**19**, 46–89 (1977) - 6
Fletcher R.: A new approach to variable metric algorithms. Comput. J.

**13**, 317–322 (1970) - 7
Fletcher R.: An Overview of Unconstrained Optimization. In: Spedicato, E. (ed.) Algorithms for Continuous Optimization: The State of Art, pp. 109–143. Kluwer, Dordrecht (1994)

- 8
Fletcher R., Powell M.J.D.: A rapidly convergent descent method for minimization. Comput. J.

**6**, 163–168 (1963) - 9
Goldfarb D.: A family of variable metric methods derived by variational means. Math. Comp.

**24**, 23–26 (1970) - 10
Lasserre J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim.

**11**(3), 796–817 (2001) - 11
Li D.H., Fukushima M.: On the global convergence of the BFGS method for nonconvex unconstrained optimization problems. SIAM J. Optim.

**11**, 1054–1064 (2001) - 12
Mascarenhas W.F.: The BFGS algorithm with exact line searches fails for nonconvex functions. Math. Program.

**99**(1), 49–61 (2004) - 13
Nocedal J.: Theory of algorithms for unconstrained optimization. Acta Numer.

**1**, 199–242 (1992) - 14
Powell M.J.D.: On the convergence of the variable metric algorithm. J. Inst. Math. Appl.

**7**, 21–36 (1971) - 15
Powell M.J.D.: Some global convergence properties of a variable metric algorithm for minimization without exact line searches. In: Cottle, R.W., Lemke, C.E. (eds.) Nonlinear Programming, SIAM-AMS Proceedings Vol. IX, pp. 53–72. SIAM, Philadelphia (1976)

- 16
Powell M.J.D.: Nonconvex minimization calculations and the conjugate gradient method. In: Griffiths, D.F. (Ed.) Numerical Analysis, Lecture Notes in Math. 1066., pp. 122–141. Springer, Berlin (1984)

- 17
Powell M.J.D.: On the convergence of the DFP algorithm for unconstrained optimization when there are only two variables. Math. Program. Ser. B

**87**, 281–301 (2000) - 18
Shanno D.F.: Conditioning of quasi-Newton methods for function minimization. Math. Comp.

**24**, 647–650 (1970) - 19
Shor N.Z.: Quadratic optimization problems. Sov. J. Comput. Syst. Sci.

**25**, 1–11 (1987) - 20
Sun W.Y., Yuan Y.X.: Optimization Theory and Methods: Nonlinear Programming. Springer, New York (2006)

- 21
Yuan, Y.X.: Numerical Methods for Nonlinear Programming. Shanghai Scientific and Technical Publishers, Shanghai (1993); (in Chinese)

## Author information

## Rights and permissions

## About this article

### Cite this article

Dai, Y. A perfect example for the BFGS method.
*Math. Program.* **138, **501–530 (2013). https://doi.org/10.1007/s10107-012-0522-2

Received:

Accepted:

Published:

Issue Date:

### Keywords

- Unconstrained optimization
- Quasi-Newton method
- Non-convex function
- Global convergence

### Mathematics Subject Classification

- 49M37
- 90C30