An overview of nonlinear optimization · 2022-opt-19

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

Status: unresolved.

The exact question—global convergence of the canonical Davidon–Fletcher–Powell (DFP) method with a Wolfe line search for every uniformly convex unconstrained objective—has not been completely resolved. The strongest sources located continue to describe it as open, including a 2026 paper that explicitly says the standard DFP method under the weak Wolfe–Powell line search remains open. Results for modified/projected DFP schemes do not answer the canonical-method question.

Key references and progress:

• Jorge Nocedal, “Theory of algorithms for unconstrained optimization,” Acta Numerica 1 (1991), 199–242. DOI: https://doi.org/10.1017/S0962492900002270. This survey identified the DFP-with-Wolfe convergence question for uniformly convex functions as an open problem, while contrasting it with the then-established BFGS result.
• M. J. D. Powell, “On the convergence of the variable metric algorithm,” IMA Journal of Applied Mathematics 7 (1971), no. 1, 21–36. DOI: https://doi.org/10.1093/imamat/7.1.21. Powell proved convergence for the variable-metric method under stronger curvature assumptions (strictly positive-definite second derivative everywhere); this does not settle the stated Wolfe-line-search problem for all uniformly convex functions.
• Richard H. Byrd, Jorge Nocedal, and Ya-Xiang Yuan, “Global Convergence of a Class of Quasi-Newton Methods on Convex Problems,” SIAM Journal on Numerical Analysis 24 (1987), no. 5, 1171–1190. DOI: https://doi.org/10.1137/0724077. They proved global and superlinear convergence for the restricted Broyden class under standard sufficient decrease, explicitly excluding DFP; hence their theorem leaves the exact case unresolved.
• Ya-Xiang Yuan, “Convergence of DFP algorithm,” Science in China (Series A) 38 (1995), 1281–1294. Direct author bibliography: https://lsec.cc.ac.cn/~yyx/worklist.html. The paper gives several sufficient conditions for global convergence on uniformly convex objectives, but not an unconditional Wolfe-line-search theorem.
• Guanghui Liu and Dachuan Xu, “A new sufficient condition for the convergence of the Davidon–Fletcher–Powell algorithm with Wolfe line search,” Journal of Systems Science and Complexity 9 (1996), no. 3, 259–269. Direct bibliographic record: https://www.iaorifors.com/paper/27373. It supplies an additional sufficient condition, not a resolution of the unrestricted problem.
• Dachuan Xu, “Global Convergence Analysis of DFP Method,” Mathematica Numerica Sinica 19 (1997), no. 3, 287–292. DOI: https://doi.org/10.12286/jssx.1997.3.287. It proves global convergence under some extra conditions for uniformly convex objectives; it does not cover every uniformly convex function with a Wolfe line search.
• Ya-Xiang Yuan, “Problems on convergence of unconstrained optimization algorithms,” ICM-98-028, April 1998. Direct URL: https://lsec.cc.ac.cn/pub/yyx/papers/p983.pdf. The survey records conditional DFP convergence results and states that the unrestricted Wolfe-line-search question remains unknown even for uniformly convex objectives.
• Yu-Hong Dai, “An overview of nonlinear optimization,” Proceedings of the International Congress of Mathematicians 2022, vol. 7, 5290–5313 (2022). DOI: https://doi.org/10.4171/ICM2022/72. Dai again lists this as an open question and notes Yuan’s significant but incomplete progress.
• Gonglin Yuan, Peifeng Zhou, and Hongtruong Pham, “A projection DFP quasi-Newton algorithm and its applications in Muskingum model and machine learning,” Numerical Algorithms (2026). DOI: https://doi.org/10.1007/s11075-025-02284-6. The authors explicitly state that global convergence of the normal DFP algorithm under weak Wolfe–Powell search for uniformly convex functions remains open; their theorem concerns a newly modified projected DFP algorithm instead.

Thus the literature establishes only conditional/special-case convergence and convergence of modified methods. No citable result found through July 31, 2026 proves or disproves the exact unrestricted statement.