Bouligand Stationarity in Constrained Optimization

• Bouligand stationarity is a geometric optimality condition using the Bouligand tangent cone to assess feasibility on nonsmooth and stratified sets.
• It unifies tangent and normal cone structures to analyze low-rank matrices, variational inequalities, and nonsmooth control problems in constrained optimization.
• Numerical methods like projected gradient descent and rank-adaptive schemes converge to Bouligand-stationary points by reducing the norm of the projected negative gradient.

Bouligand stationarity, also known as contingent stationarity, is a geometric first-order necessary optimality condition for constrained optimization problems—especially those posed on nonsmooth or stratified sets such as algebraic varieties, variational inequalities, and closed unions of manifolds. It represents the strongest stationarity requirement generically attainable by first-order methods on such sets, unifying the treatment of precise tangent and normal cone structures and supporting the analysis of both smooth and nonsmooth problems, with or without explicit structure.

1. Bouligand Tangent Cone and Definition of Stationarity

Let $C\subset \mathbb{R}^n$ be a nonempty closed set and $f:\mathbb{R}^n\to\mathbb{R}$ continuously differentiable on $C$. At a point $x\in C$, the Bouligand (or contingent) tangent cone is defined as

$$T_C(x) = \Big\{ d\in\mathbb{R}^n \;\Big|\; \exists\, x_k\to x,\, t_k\downarrow 0: \ (x_k - x)/t_k \to d,\, x_k\in C \Big\}.$$

The Bouligand stationarity (B-stationarity) condition for minimizing $f$ over $C$ is then

$$\langle \nabla f(x), d\rangle \geq 0, \quad \forall d\in T_C(x),$$

or, equivalently, $-\nabla f(x)\in \hat N_C(x)$, where $\hat N_C(x)$ is the regular (Fréchet) normal cone—the negative polar of $T_C(x)$. For matrix varieties, the tangent cones admit explicit, computable forms. For example, on the determinantal variety

$$R_{≤r}^{m\times n} = \{X\in\mathbb{R}^{m\times n}: \operatorname{rank}(X)\leq r\},$$

the tangent cone at $X$ of rank $\rho$ is given by

$$T_{R_{≤r}}(X) = T_{R_{\rho}}(X) + [\,N_{R_{\rho}}(X) \cap R_{≤r-\rho}^{m\times n}\,],$$

with $$T_{R_{\rho}}(X) = \{AX + XB : A\in\mathbb{R}^{m\times m}, B\in\mathbb{R}^{n\times n}\}$$ and $N_{R_{\rho}}(X)$ the normal space to the smooth stratum of rank-$\rho$ matrices (Olikier et al., 2024, Olikier et al., 2022, Olikier et al., 2024).

2. Relationships Among Stationarity Concepts

Bouligand stationarity is part of a hierarchy of necessary optimality notions for nonconvex sets. For a closed set $C$:

• Proximal stationarity: $-\nabla f(x)\in N_C^p(x)$, the proximal normal cone.
• Bouligand (regular) stationarity: $-\nabla f(x)\in \hat N_C(x)$, the Fréchet (regular) normal cone.
• Mordukhovich (limiting) stationarity: $-\nabla f(x)\in N_C(x)$, the Mordukhovich (limiting) normal cone.

These satisfy the inclusion $N_C^p(x) \subseteq \hat N_C(x) \subseteq N_C(x)$, so proximal $\implies$ Bouligand $\implies$ Mordukhovich stationarity. Bouligand stationarity is the strongest first-order necessary condition generally attainable by descent algorithms on singular sets, unless additional regularity (such as prox-regularity or local smoothness) allows for proximal stationarity (Olikier et al., 2024).

3. Stationarity Measures and Numerical Certification

A widely used quantitative measure of Bouligand stationarity is the norm of the projected negative gradient: $$s(x;f,C) := \|\operatorname{Proj}_{T_C(x)}(-\nabla f(x))\|.$$ Bouligand stationarity at $x$ is characterized by $s(x;f,C) = 0$. Since $T_C(x)$ is a closed cone, the metric projection is always well defined. For matrix rank varieties, projection formulas reduce via SVD to explicit low-dimensional computations for each rank stratum, with block structure detailed in (Olikier et al., 2024, Olikier et al., 2022). The use of $s(x;f,C)$ as a stopping and progress measure is now standard in both theory and large-scale implementation.

4. Algorithms and Convergence to Bouligand Stationary Points

First-order methods achieving Bouligand stationarity operate by generating sequences within the feasible set $C$ whose accumulation points satisfy $-\nabla f(x^*)\in \hat N_C(x^*)$. Notable methods include:

• Projected Gradient Descent (PGD) for closed sets $C$: every accumulation point is Bouligand stationary, and, with locally Lipschitz gradients, even proximally stationary. PGD is optimal for constraint sets without further structure (Olikier et al., 2024).
• Extended Riemannian-like descent (CRFDR) for low-rank matrix varieties $R_{≤r}^{m\times n}$: accumulation points are Bouligand stationary, all SVDs involve matrices of dimension at most $r$, and a standard stationarity measure $s(X;f,R_{≤r})$ converges to zero at $O(1/\sqrt{i})$ along bounded subsequences. This method combines tangent and normal step strategies and backtracking for non-smooth stratified sets (Olikier et al., 2024).
• Projected-projected gradient and rank-reduction schemes: these methods add rank adaptation by occasionally projecting to lower-rank strata, avoiding "apocalypse" (trapping at nonstationary points caused by tangent cone discontinuity across strata) and guaranteeing Bouligand stationarity (Olikier et al., 2022).

For discrete variational inequalities and control problems, trust-region and epigraphical approximations can deliver accumulation at Bouligand-stationary points under conditions on the directional differentiability of the solution operator (Reyes, 12 Apr 2025, Cui et al., 2023).

5. Bouligand Subdifferentials and Stationarity in Nonsmooth and Optimal Control Problems

Bouligand stationarity extends naturally to variational inequalities, nonsmooth problems, and optimal control under generalized differentiability. Given a solution mapping $S$ (e.g., the state operator in a variational inequality or a PDE with nondifferentiable nonlinearities), the Bouligand subdifferential $\partial_B S(u)$ comprises all strong limits of directional or Fréchet derivatives at nearby smooth points. For a reduced objective $f(u) = J(S(u),u)$, $u$ is Bouligand-stationary if the directional derivative $f'(u;h)\ge 0$ for all $h$ in the Bouligand tangent cone to the admissible set at $u$. This is encoded in primal conditions or, under further assumptions, strong stationarity systems introducing multipliers (adjoint states, Lagrange multipliers, and complementary slackness). These characterizations are fundamental for discrete and continuous VI-constrained controls, PDE-constrained optimization, and nonsmooth semi-linear problem classes (Nhu et al., 2023, Reyes, 12 Apr 2025, Brokate et al., 2022).

6. Rank-Adaptation, Strong Stationarity, and Algorithmic Enhancements

Modern Bouligand-stationary methods address rank-adaptivity by wrapping inner iterations at a fixed rank within an outer loop that adaptively increases the rank parameter until stationarity is reached at a suitable reduced rank. This ensures efficient convergence and prevents overfitting to an overestimated rank. For cases where solutions lie on strata with further regularity (e.g., smooth points of an algebraic variety or regions with strict complementarity in VIs), strong stationarity systems can be derived that subsume Bouligand stationarity and incorporate adjoint multipliers with explicit complementarity and sign conditions. This allows full equivalence between primal (directional derivative) conditions and system-multiplier formulations, facilitating efficient algorithmic implementation and improved theoretical guarantees (Olikier et al., 2024, Nhu et al., 2023, Reyes, 12 Apr 2025).

7. Applications and Illustrative Examples

Bouligand stationarity arises across a spectrum of nonconvex and nonsmooth optimization models:

• Low-rank matrix optimization: matrix completion, collaborative filtering, signal recovery, and dimensionality reduction (Olikier et al., 2024, Olikier et al., 2022).
• Variational inequalities and TV constraints: image processing, Bingham flow control, and discrete/continuous VIs (Reyes, 12 Apr 2025).
• Nonsmooth PDE-constrained control: semilinear elliptic equations with nondifferentiable nonlinearities, rate-independent evolution problems, and boundary/distributed controls (Nhu et al., 2023, Brokate et al., 2022).
• Discontinuous and piecewise problems: structured sparse optimization, piecewise continuous program classes, where pseudo-Bouligand stationarity provides the strongest attainable criticality results (Cui et al., 2023).

In all these contexts, Bouligand stationarity forms the central tool for analyzing the limiting performance of first-order schemes and for certifying necessary conditions at iterates or computed solutions, especially in the absence of convexity or full smoothness.