Levitin-Polyak Well-Posedness in Optimization

• Levitin-Polyak well-posedness is a stability concept in optimization that requires asymptotically optimal approximate solutions to converge (or have convergent subsequences) to exact solutions.
• It is characterized through the behavior of approximating sequences in convex bilevel optimization, variational inequalities, and equilibrium problems to ensure residual control and near-feasibility.
• By linking feasibility, optimality, and compactness of the solution set, this concept underpins robust convergence and stability even under perturbations and inexact iterative processes.

Searching arXiv for papers on Levitin-Polyak well-posedness and closely related optimization/VI formulations. Levitin-Polyak well-posedness is a notion of stability for optimization, variational inequality, and equilibrium-type problems that requires asymptotically optimal approximate solutions to converge, or at least to possess convergent subsequences whose limits are exact solutions. In the materials considered here, the notion appears in several technically distinct but structurally related forms: as a subsequential compactness property for convex bilevel optimization (Giang-Tran et al., 22 Sep 2025), as convergence of generalized approximating sequences for elliptic variational inequalities (Gariboldi et al., 2023), as a perturbation-stable metric property for split equilibrium problems (Dey et al., 2022) and split multivalued variational inequalities (Dey et al., 2023), and as an extended or infeasibility-measure-based concept for constrained optimization in infinite-dimensional spaces (Dolgopolik, 22 Aug 2025). Across these settings, the common theme is that near-feasibility together with near-optimality, or the analogue appropriate to the problem class, is sufficient to recover exact solvability in a strong asymptotic sense.

1. Conceptual core and basic definitions

In convex bilevel optimization, the problem has the form

$$\begin{aligned} \text{(Inner problem):} \quad & g^* := \min_{x \in X} g(x) \ \text{(Outer problem):} \quad & f^* := \min_{x \in X_g} f(x), \quad \text{where } X_g := \arg\min_{z \in X} g(z), \end{aligned}$$

where $X$ is a closed convex set, and $f,g:X\to\mathbb{R}$ are convex (Giang-Tran et al., 22 Sep 2025). If $X^* := \arg\min_{x\in X_g} f(x)$, then three notions are distinguished.

LP Well-Posedness: For any sequence $\{x_t\}\subset X$ with $\operatorname{Dist}(x_t,X_g)\to 0$ and $f(x_t)\to f^*$, there exists a subsequence $\{x_{t_k}\}$ and $\bar x\in X^*$ such that $x_{t_k}\to \bar x$ (Giang-Tran et al., 22 Sep 2025).

Generalized LP Well-Posedness: For any sequence $X$0 with $X$1 and $X$2, there exists a subsequence converging to some $X$3 (Giang-Tran et al., 22 Sep 2025).

Strong Generalized LP Well-Posedness: For any sequence $X$4 with $X$5 and $X$6, there exists a subsequence converging to some $X$7 (Giang-Tran et al., 22 Sep 2025).

In constrained optimization in infinite-dimensional spaces, the extended formulation is stated for

$X$8

with feasible region $X$9 (Dolgopolik, 22 Aug 2025). The problem is Levitin-Polyak well-posed in the extended sense if $f,g:X\to\mathbb{R}$0 is nonempty and every sequence $f,g:X\to\mathbb{R}$1 satisfying $f,g:X\to\mathbb{R}$2 and $f,g:X\to\mathbb{R}$3 has a subsequence converging to some $f,g:X\to\mathbb{R}$4 (Dolgopolik, 22 Aug 2025). The weak version requires only $f,g:X\to\mathbb{R}$5 (Dolgopolik, 22 Aug 2025).

A further generalization replaces the distance-to-feasibility by an infeasibility measure $f,g:X\to\mathbb{R}$6, with $f,g:X\to\mathbb{R}$7 feasible. Then Levitin-Polyak well-posedness with respect to $f,g:X\to\mathbb{R}$8 requires $f,g:X\to\mathbb{R}$9 and that any sequence with $X^* := \arg\min_{x\in X_g} f(x)$0 and $X^* := \arg\min_{x\in X_g} f(x)$1 has a subsequence converging to some $X^* := \arg\min_{x\in X_g} f(x)$2 (Dolgopolik, 22 Aug 2025). This generalizes both the extended Levitin-Polyak and Tykhonov versions when $X^* := \arg\min_{x\in X_g} f(x)$3 for closed $X^* := \arg\min_{x\in X_g} f(x)$4 (Dolgopolik, 22 Aug 2025).

These definitions show that Levitin-Polyak well-posedness is not a single formal condition but a family of asymptotic compactness requirements adapted to the native approximation model of each problem class. This suggests that its unifying content is best understood through the types of approximating sequences it controls.

2. Approximating sequences and convergence mechanisms

For elliptic variational inequalities in a real Hilbert space $X^* := \arg\min_{x\in X_g} f(x)$5, the problem is to find $X^* := \arg\min_{x\in X_g} f(x)$6 such that

$X^* := \arg\min_{x\in X_g} f(x)$7

where $X^* := \arg\min_{x\in X_g} f(x)$8 is nonempty, closed, and convex, $X^* := \arg\min_{x\in X_g} f(x)$9 is strongly monotone and Lipschitz, and $\{x_t\}\subset X$0 is convex and locally Lipschitz (Gariboldi et al., 2023). A sequence $\{x_t\}\subset X$1 is a Levitin-Polyak generalized approximating sequence if there exist $\{x_t\}\subset X$2 with $\{x_t\}\subset X$3, and $\{x_t\}\subset X$4, such that

$\{x_t\}\subset X$5

and

$\{x_t\}\subset X$6

(Gariboldi et al., 2023). The variational inequality is well-posed in the sense of Levitin-Polyak if every such approximating sequence converges strongly in $\{x_t\}\subset X$7 to the unique solution $\{x_t\}\subset X$8 (Gariboldi et al., 2023).

The same paper formulates a more general necessary and sufficient convergence criterion. If $\{x_t\}\subset X$9 is the unique solution, then for any sequence $\operatorname{Dist}(x_t,X_g)\to 0$0,

$\operatorname{Dist}(x_t,X_g)\to 0$1

if and only if there exist $\operatorname{Dist}(x_t,X_g)\to 0$2 such that both

$\operatorname{Dist}(x_t,X_g)\to 0$3

and

$\operatorname{Dist}(x_t,X_g)\to 0$4

hold (Gariboldi et al., 2023). The classical LP-approximating sequence is a special case of this criterion, because $\operatorname{Dist}(x_t,X_g)\to 0$5 with $\operatorname{Dist}(x_t,X_g)\to 0$6 implies $\operatorname{Dist}(x_t,X_g)\to 0$7 (Gariboldi et al., 2023). Corollary 17 therefore yields Levitin-Polyak well-posedness under the standard assumptions (Gariboldi et al., 2023).

In split multivalued variational inequalities, generalized approximating sequences are similarly built from controlled infeasibility and vanishing residual inequalities. If $\operatorname{Dist}(x_t,X_g)\to 0$8 is the solution set, the problem is LP well-posed when $\operatorname{Dist}(x_t,X_g)\to 0$9 is a singleton and every generalized approximating sequence converges to that unique solution; it is LP well-posed in the generalized sense when $f(x_t)\to f^*$0 is nonempty and every generalized approximating sequence has a subsequence converging to some element of $f(x_t)\to f^*$1 (Dey et al., 2023). In split equilibrium problems under perturbations, generalized approximating sequences likewise allow small infeasibility and perturbed equilibrium inequalities, and LP well-posedness by perturbations requires convergence of every such sequence to the unique solution when perturbations vanish (Dey et al., 2022).

A recurring structural feature is that Levitin-Polyak well-posedness is formulated for inexact trajectories rather than exact iterates. In that respect it is a natural notion for iterative computation, perturbation analysis, and asymptotic stability.

3. Convex bilevel optimization: duality, compactness, and LP well-posedness

The convex bilevel setting in "Convergence, Duality and Well-Posedness in Convex Bilevel Optimization" makes the role of Levitin-Polyak well-posedness particularly explicit (Giang-Tran et al., 22 Sep 2025). The paper identifies two challenges: strong duality is not guaranteed due to the lack of Slater constraint qualification, and convergence of algorithms is not guaranteed even when usual suboptimality gap bounds are present, because super-optimal solutions can occur (Giang-Tran et al., 22 Sep 2025).

The central duality statement is Theorem 3.1: $f(x_t)\to f^*$2 Thus strong duality for the Lagrangian dual of the value function formulation is equivalent to the claim that asymptotically vanishing primal suboptimality in the sense $f(x_t)\to f^*$3 and $f(x_t)\to f^*$4 forces exact convergence of outer function values, excluding persistent super-optimality (Giang-Tran et al., 22 Sep 2025).

Two sufficient conditions are then separated. Condition 4.1 is the proximity or metric subregularity property

$f(x_t)\to f^*$5

This is an error bound on $f(x_t)\to f^*$6, connected to metric subregularity or error bounds, and it guarantees strong duality and hence $f(x_t)\to f^*$7 (Giang-Tran et al., 22 Sep 2025). However, it does not guarantee convergence in distance to the optimal solution set $f(x_t)\to f^*$8; the paper gives an example where sequences converge in function value but do not approach $f(x_t)\to f^*$9 in distance (Giang-Tran et al., 22 Sep 2025).

Condition 4.2 is compactness: $\{x_{t_k}\}$0 This includes cases where $\{x_{t_k}\}$1 is bounded, or when a coercivity condition holds on $\{x_{t_k}\}$2 or on $\{x_{t_k}\}$3 (Giang-Tran et al., 22 Sep 2025). Under this condition, strong duality holds and convergence in both function value and distance to $\{x_{t_k}\}$4 is ensured (Giang-Tran et al., 22 Sep 2025).

Proposition 4.4 then establishes a precise equivalence: compactness of $\{x_{t_k}\}$5, generalized LP well-posedness, and strong generalized LP well-posedness are equivalent; standard LP well-posedness also implies compactness (Giang-Tran et al., 22 Sep 2025). This is one of the clearest available statements of the relation between LP well-posedness and the geometry of the optimal set. In this setting, LP well-posedness is not merely a qualitative stability label; it is equivalent to compactness of the solution set and underwrites convergence beyond function values.

Corollary 4.5 sharpens this point quantitatively. Under compactness, there exists a continuous function $\{x_{t_k}\}$6 with $\{x_{t_k}\}$7 such that

$\{x_{t_k}\}$8

Hence any sequence with $\{x_{t_k}\}$9 and $\bar x\in X^*$0 must also satisfy $\bar x\in X^*$1 (Giang-Tran et al., 22 Sep 2025). This is a modulus-type error bound derived from generalized LP well-posedness via Theorem 2.2 of Huang and Yang, 2006, as cited in the paper (Giang-Tran et al., 22 Sep 2025).

4. Metric characterizations and perturbation frameworks

In split equilibrium problems in real Banach spaces, LP well-posedness is characterized through approximate solution sets (Dey et al., 2022). If $\bar x\in X^*$2 denotes the solution set and $\bar x\in X^*$3 the $\bar x\in X^*$4-approximate solution set, then Theorem 1 states that the split equilibrium problem is LP well-posed by perturbations if and only if $\bar x\in X^*$5 is nonempty and

$\bar x\in X^*$6

(Dey et al., 2022). Under upper semicontinuity assumptions on the perturbed bifunctions, this is equivalently stated as

$\bar x\in X^*$7

(Dey et al., 2022).

For generalized LP well-posedness by perturbations, the relevant metric criterion is

$\bar x\in X^*$8

together with nonempty compact $\bar x\in X^*$9, where $x_{t_k}\to \bar x$0 denotes the Hausdorff metric (Dey et al., 2022). In finite-dimensional settings, generalized LP well-posedness is equivalent to

$x_{t_k}\to \bar x$1

where $x_{t_k}\to \bar x$2 is the Kuratowski measure of noncompactness (Dey et al., 2022). Theorem 5 further states that in finite-dimensional Banach spaces, under suitable monotonicity, continuity, and convexity assumptions, LP well-posedness by perturbations is equivalent to existence and uniqueness of the solution (Dey et al., 2022).

The split multivalued variational inequality setting has closely parallel metric characterizations. Let $x_{t_k}\to \bar x$3 denote the $x_{t_k}\to \bar x$4-approximate solution set. Then the problem is LP well-posed if and only if

$x_{t_k}\to \bar x$5

(Dey et al., 2023). Generalized LP well-posedness is equivalent to

$x_{t_k}\to \bar x$6

(Dey et al., 2023). In finite-dimensional Hilbert spaces, if $x_{t_k}\to \bar x$7 is nonempty and bounded for some $x_{t_k}\to \bar x$8, then LP well-posedness is equivalent to uniqueness of the solution, while generalized LP well-posedness is equivalent to nonemptiness of the solution set (Dey et al., 2023).

These metric characterizations expose a common logic: LP well-posedness is equivalent to collapse of approximate solution sets. In the singleton case the collapse is measured by vanishing diameter; in the nonunique case it is measured by Hausdorff convergence to a compact solution set or by vanishing measure of noncompactness. This suggests a general interpretation of LP well-posedness as an asymptotic concentration principle for approximate solutions.

5. Relations to Tykhonov well-posedness and extended variants

The constrained optimization framework in infinite-dimensional spaces explicitly compares Tykhonov and Levitin-Polyak notions (Dolgopolik, 22 Aug 2025). Tykhonov well-posedness is the analogous property restricted to feasible sequences $x_{t_k}\to \bar x$9, whereas Levitin-Polyak well-posedness allows approximate feasibility through $X$00 (Dolgopolik, 22 Aug 2025). The paper states that Levitin-Polyak well-posedness generalizes Tykhonov well-posedness in this sense, and that if the feasible set $X$01 is bounded and $X$02 is uniformly continuous on bounded sets, the two notions are equivalent (Dolgopolik, 22 Aug 2025).

The same source introduces the infeasibility-measure-based extension. The implication chain is summarized as

$X$03

but not conversely; specifically, Tykhonov well-posedness does not imply Levitin-Polyak well-posedness (Dolgopolik, 22 Aug 2025).

A further structural property appears in Proposition 2 and Proposition 6 of that paper: Levitin-Polyak well-posedness in the extended sense is equivalent to the weak version plus compactness of $X$04, and the same is true for the $X$05-based extension (Dolgopolik, 22 Aug 2025). This is closely aligned with the convex bilevel result that generalized and strong generalized LP well-posedness are equivalent to compactness of the optimal set $X$06 (Giang-Tran et al., 22 Sep 2025). Across these settings, compactness is not incidental; it is the mechanism that upgrades weak asymptotic closeness to subsequential convergence.

The elliptic variational inequality paper adds a distinct perspective. There, the authors introduce T-well-posedness through T-approximating sequences satisfying

$X$07

and show that this coincides exactly with the necessary and sufficient condition of Theorem 8 (Gariboldi et al., 2023). In that treatment, classical LP and Tykhonov notions are only sufficient, whereas the new criterion is necessary and sufficient (Gariboldi et al., 2023). A plausible implication is that, in some variational inequality settings, LP well-posedness is best viewed as one member of a larger taxonomy of approximation-sensitive convergence notions rather than as the maximal possible characterization.

6. Applications and broader significance

The source materials attach LP well-posedness to several applied and algorithmic contexts. In elliptic variational inequalities, two applications are highlighted: a heat transfer problem and an elastic frictionless contact problem (Gariboldi et al., 2023). In the contact setting, the general criterion is used to show that solutions to perturbed approximating problems converge strongly to the solution of the original frictionless problem (Gariboldi et al., 2023). In the heat transfer setting, approximating the boundary condition yields a family of problems whose solutions converge to the original as prescribed by the general criterion, and LP well-posedness is guaranteed (Gariboldi et al., 2023).

In split equilibrium problems, the metric characterizations imply that any iterative process whose outputs form generalized approximating sequences converges, or has convergent subsequences, to true solutions under LP well-posedness (Dey et al., 2022). The paper explicitly interprets this as convergence and stability under perturbations from discretization, errors, or modeling (Dey et al., 2022). Similarly, in split multivalued variational inequalities, LP well-posedness is specialized to split feasibility, split minimization, split variational inequalities, and split mixed variational inequalities (Dey et al., 2023). The theory is designed to cover exact and approximate computation in Hilbert spaces with multivalued operators and split constraints (Dey et al., 2023).

In constrained optimization, LP well-posedness with respect to an infeasibility measure $X$08 is used to formulate conditions for exact penalty functions and global saddle points of augmented Lagrangians (Dolgopolik, 22 Aug 2025). If the problem is weakly Levitin-Polyak well-posed with respect to $X$09, then the penalty function

$X$10

is globally exact if and only if three conditions hold: zero duality gap,

$X$11

existence of global minimizers of $X$12 over $X$13 for all $X$14, and uniform local exactness at global minimizers (Dolgopolik, 22 Aug 2025). Under additional assumptions, analogous localization principles characterize the global exactness of penalty functions and the existence of global saddle points of augmented Lagrangians (Dolgopolik, 22 Aug 2025). Here LP well-posedness functions as a stability hypothesis that allows local information to propagate to global conclusions in infinite-dimensional settings.

7. Common themes, limitations, and points of interpretation

Several common themes recur across the cited problem classes.

Theme	Manifestation	Source
Approximate feasibility	Measured by $X$15, $X$16, or analogous split constraints	(Dolgopolik, 22 Aug 2025, Gariboldi et al., 2023, Dey et al., 2022, Dey et al., 2023)
Approximate optimality or residual control	Function values approach optima or residual inequalities vanish	(Giang-Tran et al., 22 Sep 2025, Gariboldi et al., 2023, Dey et al., 2022, Dey et al., 2023)
Compactness as decisive property	Compactness of optimal/solution set is equivalent to generalized LP variants in several settings	(Giang-Tran et al., 22 Sep 2025, Dolgopolik, 22 Aug 2025)
Metric characterization	Vanishing diameter, Hausdorff convergence, or vanishing noncompactness of approximate solution sets	(Dey et al., 2022, Dey et al., 2023)

A central limitation emphasized in convex bilevel optimization is that strong duality and convergence of function values do not by themselves guarantee convergence to the optimal solution set in distance (Giang-Tran et al., 22 Sep 2025). Condition 4.1 ensures strong duality and $X$17, but not $X$18 (Giang-Tran et al., 22 Sep 2025). Only the stronger compactness-based condition yields both. This addresses a common misconception that asymptotically vanishing primal gaps are sufficient for full convergence of iterates in bilevel problems; the paper explicitly exhibits failure due to super-optimal solutions and lack of approach to $X$19 (Giang-Tran et al., 22 Sep 2025).

Another limitation is dimensional. For split equilibrium problems, the equivalence between LP well-posedness and uniqueness is fully established in finite-dimensional Banach spaces; for infinite-dimensional spaces, the paper states that this remains an open question (Dey et al., 2022). This indicates that some of the neat metric equivalences rely materially on finite-dimensional compactness mechanisms.

The overall picture is that Levitin-Polyak well-posedness is a convergence-stability concept for inexact solution processes. Its strongest formulations combine three ingredients: approximate feasibility, approximate optimality or residual decay, and compactness of the exact solution set. In convex bilevel optimization, these ingredients connect directly to strong duality and distance convergence (Giang-Tran et al., 22 Sep 2025). In variational inequalities and split equilibrium models, they yield necessary and sufficient metric convergence criteria and perturbation-stable solution behavior (Gariboldi et al., 2023, Dey et al., 2022, Dey et al., 2023). In constrained optimization in infinite-dimensional spaces, extended and $X$20-based versions support exact penalization and augmented Lagrangian theory (Dolgopolik, 22 Aug 2025). Taken together, these results position Levitin-Polyak well-posedness as a unifying stability principle across contemporary nonlinear analysis and optimization.