A new mathematical analysis sets out a way to track how a set of feasible solutions changes when the parameter in a split-feasibility problem moves. It develops first-order descriptions, called graphical derivatives, of those local changes. In one specially qualified setting, the description is exact. For more general parameter-dependent mappings, the paper supplies only an outer estimate, meaning a set that must contain the true first-order directions. A constructed scalar example shows that the estimate can be wider than reality. The work also gives inner estimates, a local stability bound, and necessary conditions for optimization with implicit constraints.
The object under study is a family of parameterized split-feasibility problems over real Banach spaces, not an empirical participant sample. For each parameter, the solution multifunction, a solution map that can return a set of states, contains the states x that belong to C(p) and whose image under A(p,x) belongs to Q(p). This separates feasibility into two linked checks: one on the state itself and one on its image under a mapping. The central question is which state directions remain available when the parameter changes in a specified direction.
A residual before the derivative
To measure how far a candidate is from meeting both requirements, the paper defines a residual. It adds the distance from A(p,x) to Q(p) to the distance from x to C(p). The residual therefore records violations in the image and state parts of the problem in one quantity. Under lower semicontinuity of C and Q, continuity of A at the reference state as the parameter varies, and the paper's stated regularity condition, nearby parameter values have solutions near the reference state. Within those neighborhoods, the distance from a candidate state to the solution set is no greater than its residual multiplied by the inverse of the positive regularity modulus supplied by that condition.
Exact in one narrow lane
The strongest formula comes with tight conditions. C and Q must have nonempty, closed, convex values; A must be an onto bounded linear map; and an interior qualification must hold for the relevant graphs. When those requirements are met, the graphical derivative of the solution map is exactly the intersection of the derivative of C and the inverse-image, under A, of the derivative of Q. Put less formally, for every parameter direction, a state direction is admitted precisely when it passes both linearized feasibility tests.
Once A is allowed to depend on the parameter and is only assumed to be Frechet differentiable at the reference pair, the exact equality becomes an outer inclusion. Every true graphical-derivative direction must belong to an intersection built from the derivative of C, the derivative of Q, and the derivative of A. That intersection is a useful container for the answer, but it can also include directions that the solution map does not actually take. The result is therefore a bound on possible first-order behavior, not a general exact formula.
The paper demonstrates the gap with a scalar constructed example rather than empirical data. At the reference parameter-state pair (0,0), every positive parameter direction has an empty true graphical derivative, while the corresponding outer set is the nonpositive half-line. In that case, the approximation admits candidates even though no actual first-order state direction is present. The example shows why inclusion and equality must be kept separate.
Lower bounds and local stability
The analysis also derives two inner inclusions under lower semicontinuity, Frechet differentiability, and the stated regularity condition. An inner inclusion works in the other direction: the directions in the constructed set are guaranteed to lie inside the true graphical derivative. One construction combines the contingent derivative of C with the interior-displacement derivative of Q. The second swaps those roles, pairing the interior-displacement derivative of C with the contingent derivative of Q. The paper cautions that these estimates may say little when an interior-displacement derivative is empty.
For local stability, the paper gives a derivative-based upper bound for the Aubin property, a Lipschitz-like condition on how a set-valued solution map responds to nearby parameter changes. The exact Aubin bound of the solution multifunction is no larger than a quantity built from the derivative approximations of C, A and Q. If that quantity is finite, the paper treats finiteness as sufficient for the Aubin property, subject to local closedness and the stated regularity assumptions.
What the tools say about optimization
The application is to optimization problems whose constraints are defined by the split-feasibility solution map. When the objective is locally Lipschitz and the error-bound assumptions apply, the constrained problem has a local exact-penalty formulation. The replacement objective adds a multiple of the split-feasibility residual to the original objective, and the penalty parameter must reach the threshold set by the inverse regularity modulus multiplied by the objective's local Lipschitz constant. This lets the local constrained problem be treated through an unconstrained penalized objective within the proposition's assumptions.
The paper then states a necessary graphical optimality condition. At a local constrained solution, every upper subgradient of the objective must have a nonnegative pairing with every direction in the graph of the solution map's contingent derivative. In ordinary language, each allowed first-order parameter-state movement must pass a nonnegative directional test for every such subgradient. The condition is necessary, not sufficient, so it does not by itself certify that a candidate is a local solution.
With additional support-function qualifications, a second condition separates the parameter component of an upper subgradient from its state component through a subdifferential expression involving support functions of the two derivative sets. Support functions summarize how those sets extend in a given direction. This adds structure to the necessary test, but it remains conditional on the extra qualifications.
A theorem set, not a field test
The supplied document is an arXiv version-one preprint and states that it was updated on August 31, 2026. Its unit of analysis is a family of parameterized split-feasibility problems over real Banach spaces, with constructed mathematical examples rather than an empirical participant sample. The exact formula is limited to the special assumptions; in the general differentiable case, the paper gives an outer inclusion that can be strict. The inner estimates depend on their regularity conditions and can be uninformative when interior-displacement derivatives are empty. The optimization results provide a penalty construction and necessary conditions, not a sufficiency theory.
Paper data and sources
Original title: First-order approximations of multifunctions defined implicitly by "split" feasibility problems with applications to optimization
Authors: Amos Uderzo
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text