A mathematical study identifies conditions under which the value of an optimization problem and its set of best solutions remain stable when both the objective and the feasible choices are sequentially perturbed. Its central guarantee is one-sided: under the paper’s convergence assumptions, the outer graphical limit of the approximate solution mappings is contained in the solution mapping of the limiting problem.
The analysis is not based on participants or an empirical dataset. Its units are abstract parametric optimization problems, each defined by a parameter, an objective function and a feasible mapping, the rule that describes which choices are available. The paper follows these ingredients through sequential perturbations and studies their limiting behavior.
To make that comparison precise, the framework combines lower and upper continuous convergence, epi- and hypo-convergence for functions, and lower and upper continuous plus graphical convergence for multifunctions, or set-valued mappings. In plain English, these are formal tests for whether changing objectives and changing sets of allowed choices continue to line up as the parameter moves.
The guarantee is deliberately one-sided
The central result puts a boundary around what an approximate optimizer can do. If a sequence of approximate solution mappings has a graphical limit, the theorem says that limit remains within the solution mapping of the limiting model, provided the objective and feasible data meet the stated convergence conditions. It does not, by itself, say that every solution of the limiting model must be reached by approximate solutions.
Boundedness is a key part of the next stability result. With eventual local boundedness of the approximate feasible mappings—roughly, their allowable choices do not escape without bound near the parameter—the approximate value functions converge continuously and the approximate solution mappings are eventually locally bounded.
Stronger assumptions produce a stronger existence result. With strong eventual local boundedness, closed-valued feasible mappings and upper-semicontinuous approximate objectives, approximate solution mappings are strongly eventually locally bounded. Their outer graphical limit is nonempty and bounded-valued, and the limiting solution mapping is nonempty-valued.
Boundedness does much of the work
The paper also studies a uniform version of the argument. Under its graphical-convergence stability framework, eventual uniform boundedness of the approximate feasible mappings is inherited by the approximate solution mappings, and the limiting solution map is bounded and compact-valued.
Another result concerns recovery along parameter sequences. With eventual local boundedness, approximate values converge and solution mappings inherit eventual local boundedness. For every limiting parameter, a convergent sequence of parameters can realize value convergence while the outer limit of the associated solution mappings lies inside the limiting solution set.
The framework reaches games and dynamic programs
In the generalized Nash equilibrium application, approximate equilibrium sets have an outer limit contained in the limiting equilibrium set. If every player’s feasible-map sequence is eventually locally bounded and eventually nonempty, each player’s approximate value function converges.
In finite-horizon dynamic programming, the paper tracks both approximate policy mappings, which record the choices prescribed at each stage, and approximate value functions. Under local-boundedness conditions, approximate policy mappings have outer-limit containment, approximate values converge, and the limiting policy map is nonempty-valued, locally bounded and outer-semicontinuous.
Under stronger graphical outer and inner conditions on the feasible mapping, dynamic-programming values and policy mappings retain variational stability. Additional regularity makes policy outer limits nonempty and compact and ensures that they intersect the limiting policy set.
The assumptions are part of the result
The assumptions are not decorative. A counterexample shows that eventual local boundedness cannot be omitted from the relevant stability theorem. Another shows that continuous convergence of objective functions cannot be replaced by pointwise or uniform convergence.
Nor does the framework generally give two-sided continuity. Even under the theorem’s continuous-data assumptions, the solution mapping need not be inner semicontinuous or continuous. That means the results control solution behavior from the approximating problems toward the limiting problem, but do not generally guarantee that every limiting solution is recovered.
Because the dynamic-programming application is finite-horizon, its policy and value conclusions are scoped to that setting and to the theorem-specific local-boundedness and graphical assumptions.
The manuscript is labeled an arXiv version-one preprint. It reports ANID–Chile funding through Fondecyt projects 1220687 for López and 1200525 for Fierro.
Paper data and sources
Original title: Sequential Stability of the Value Function and the Solution Mapping in Berge's Maximum Theorem via Variational Convergence
Authors: John Cotrina, Raúl Fierro, Rubén López
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text