Active Set and Interior Methods for Nonlinear Optimization · 1998-opt-9

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

Verdict: unresolved.

The exact problem asks for a practical selection or update rule for the scalar parameter in the nonsmooth $\ell_1$ exact-penalty formulation of general smooth, large-scale nonlinear programs, with robust and effective behavior while avoiding the small-parameter and large-parameter failure modes. The literature does not completely resolve this broad question. Existing convergence theorems and numerical studies cover particular algorithmic frameworks, constraint classes, and regularity assumptions; they do not prove a universally reliable and efficient rule for every problem in the stated class, nor do they establish a uniform notion of large-scale robustness and effectiveness.

Key literature progress:

• Richard H. Byrd, Jorge Nocedal, and Richard A. Waltz, “Steering Exact Penalty Methods for Nonlinear Programming,” Optimization Methods and Software 23(2), 197–213 (2008), DOI: 10.1080/10556780701394169. This paper introduces dynamic steering rules that choose the penalty parameter by solving an auxiliary subproblem to balance linear feasibility and objective progress. It analyzes trust-region SQP and SLQP contexts and reports that the parameter choice is no longer purely heuristic. The result is restricted to those frameworks; the paper itself notes limitations for unconstrained penalty minimization and line-search merit-function use.

• Richard H. Byrd, Gabriel López-Calva, and Jorge Nocedal, “A Line Search Exact Penalty Method Using Steering Rules,” Mathematical Programming 133(1–2), 39–73 (2012), DOI: 10.1007/s10107-010-0408-0. This work extends steering to a line-search SQP method, dynamically adjusts the penalty parameter, proves favorable global convergence properties, and gives numerical experiments. It resolves the particular line-search extension identified as open in the 2008 paper, but not the original general large-scale question.

• Zhongwen Chen and Yu-Hong Dai, “A Line Search Exact Penalty Method with Bi-Object Strategy for Nonlinear Constrained Optimization,” Journal of Computational and Applied Mathematics 300, 245–258 (2016), DOI: 10.1016/j.cam.2015.12.033. The paper proposes a nonmonotone penalty update based only on current-iterate information, separately targeting feasibility and objective reduction, and proves global convergence under weaker assumptions with numerical tests. This is another particular algorithmic resolution, not a complete theory for all smooth large-scale nonlinear programs.

• James V. Burke, Frank E. Curtis, Hao Wang, and Jiashan Wang, “Inexact Sequential Quadratic Optimization with Penalty Parameter Updates within the QP Solver,” SIAM Journal on Optimization 30(3), 1822–1849 (2020), DOI: 10.1137/18M1176488. The authors develop a dynamic exact-penalty update inside an inexact, matrix-free SQP subproblem solver for large-scale nonlinear optimization, prove convergence under stated assumptions, and report numerical benefits. This directly addresses scalability, but only for the proposed SQO framework and its hypotheses.

• James V. Burke, Frank E. Curtis, Hao Wang, and Jiashan Wang, “Matrix-Free Solvers for Exact Penalty Subproblems,” arXiv:1402.1917 (2014), https://arxiv.org/abs/1402.1917. It gives globally convergent matrix-free methods for certain convex-composite exact-penalty subproblems and an $O(1/\varepsilon^2)$ iteration bound, with numerical experiments. The problem class and subproblem setting are narrower than the general nonlinear program in the statement.

• M. V. Dolgopolik, “Exact Penalty Functions with Multidimensional Penalty Parameter and Adaptive Penalty Updates,” arXiv:2102.13632 (2021), https://arxiv.org/abs/2102.13632. This develops exactness conditions and adaptive-update convergence theory for vector-valued penalty parameters, including infinite-dimensional settings. It broadens the theory but does not supply a complete practical scalar-parameter resolution for the stated large-scale nonlinear-programming problem.

Thus, the established progress consists of dynamic and adaptive rules with convergence guarantees and encouraging computations in several important subclasses. The remaining gap is a general, practically validated and theoretically comprehensive answer covering the full smooth large-scale problem class and the requested robustness/effectiveness tradeoff.