A stabilized sequential quadratic programming method for degenerate nonlinear optimization problems on Riemannian manifolds

Abstract: We propose a stabilized sequential quadratic programming (SQP) method for degenerate constrained optimization problems on Riemannian manifolds. The problem considered in this study is a Riemannian nonlinear programming problem (RNLP) with equality and inequality constraints, where classical constraint qualifications may fail. While existing Riemannian SQP methods guarantee global convergence only under constraint qualifications, their convergence behavior is not ensured for degenerate problems. To address this limitation, we extend the stabilized SQP framework from Euclidean spaces to Riemannian manifolds. Without assuming any constraint qualification, we prove that the generated sequence has an accumulation point that is a Karush--Kuhn--Tucker (KKT) point, an approximate KKT (AKKT) point, or a stationary point of an associated feasibility problem. Finally, we conduct numerical experiments to confirm the effectiveness of the proposed method for degenerate problems.

• The paper introduces RSSQP, the first stabilized sequential quadratic programming method for Riemannian manifolds, using proximal perturbations, augmented-Lagrangian merit steps, and safeguarded multiplier updates to maintain solvable subproblems under degeneracy.
• The convergence analysis requires smooth data, coercive stabilized Hessian operators, bounded iterates, and manifold completeness, and guarantees accumulation at a KKT point, an AKKT point, or a stationary point of the feasibility problem without standard constraint qualifications.
• Numerical tests show RSSQP solved all five degenerate spherical MPECs within 64–84 iterations and residuals below 9.8×10⁻⁷, while standard RSQP reached its 500-iteration limit, although local convergence rates remain unproved.

Problem setting and motivation

The paper considers the Riemannian nonlinear programming problem (RNLP)

$$\min_{x \in M} f(x) \quad \text{s.t.} \quad g_i(x)=0,\; h_j(x)\ge 0,$$

where $M$ is a $d$-dimensional connected, complete Riemannian manifold and all data are twice continuously differentiable. The central difficulty addressed is degeneracy: classical constraint qualifications (CQs) such as LICQ, MFCQ, and the Robinson CQ (RCQ) may fail at feasible points, in which case local optima need not satisfy the Karush–Kuhn–Tucker (KKT) conditions. Existing Riemannian SQP methods—most notably that of Obara, Okuno, and Takeda [(Cai et al., 2022)-style work cited as Obara:2022]—guarantee global convergence only under CQs and therefore offer no convergence guarantee on degenerate instances. The authors extend Wright's stabilized SQP (SSQP) framework from Euclidean spaces to manifolds, yielding what they state is the first stabilized SQP method for constrained optimization on Riemannian manifolds.

Algorithmic design

The method, termed RSSQP, iteratively solves a stabilized quadratic subproblem over $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$:

$$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$

subject to linearized constraints perturbed by proximal terms $\sigma_k(\zeta - \widehat y_k)$ and $\sigma_k(\eta - \widehat z_k)$. After eliminating equality constraints, the subproblem reduces to a strongly convex program in $(\xi,\eta)$ with Hessian-like operator $M_k = H_k + \frac{1}{\sigma_k}Dg(x_k)^*Dg(x_k)$; its coercivity (Assumption (A2)) guarantees a unique solution satisfying its own KKT conditions regardless of degeneracy of the original problem—a structural advantage over ordinary RSQP subproblems, which may be infeasible.

The merit function is the augmented Lagrangian

$$F(x;y,z,\sigma) = f(x) + \frac{1}{2\sigma}\|\sigma y - g(x)\|^2 + \frac{1}{2\sigma}\|[\sigma z - h(x)]_+\|^2,$$

and steps are accepted by an Armijo backtracking line search on $M$0. The distinctive component is the multiplier/parameter update rule, which maintains two bounded multiplier sequences—the main sequence $M$1 and auxiliary sequence $M$2 confined to compact convex sets $M$3 and $M$4 via projections—and branches among three cases:

• Case 1 (Steps 4.1/4.2): if the KKT residual measures $M$5 or $M$6 halve, accept the SQP multipliers directly.
• Case 2 (Step 4.3): if $M$7, update multipliers by augmented-Lagrangian rules with projection onto $M$8, halve $M$9 and $d$0.
• Case 3 (Step 4.4): otherwise leave multipliers unchanged.

The analysis partitions iteration indices into sets $d$1, $d$2, $d$3 according to which case fires, and shows $d$4 cannot be eventually always active when $d$5 is bounded.

Global convergence without constraint qualifications

Under Assumption (A1) ($d$6 data) and (A2) (uniform coercivity of $d$7), plus boundedness of $d$8, the main theorem establishes that every accumulation point $d$9 of the generated sequence satisfies one of:

1. KKT point of the RNLP (when $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$0);
2. AKKT point, with a subsequence of $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$1 forming an AKKT sequence (when $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$2 and feasibility holds);
3. Stationary point of the feasibility problem $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$3, i.e., $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$4, possibly at an infeasible point.

Moreover, if $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$5 satisfies the extended Robinson CQ (ERCQ), case 3 is excluded and the AKKT point upgrades to a genuine KKT point with bounded multipliers. This trichotomy mirrors the strongest known CQ-free guarantees for Euclidean augmented-Lagrangian and SSQP schemes, transferred to the manifold setting. Two technical points deserve emphasis. First, the proof that Step 4.4 cannot persist indefinitely requires a compactness argument on the tangent bundle using local charts, since boundedness of $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$6 alone does not trivially imply sequential compactness of $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$7 in $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$8. Second, the completeness assumption on $$\mathcal{V}_k = T_{x_k}M \times \mathbb{R}^m \times \mathbb{R}^\ell$$9 is essential: it ensures closed metric balls are compact, underpinning both well-definedness and extraction of convergent subsequences.

A limitation stated explicitly by the authors is that the analysis covers global convergence only; no local rate (superlinear or quadratic) is established, in contrast to the Euclidean SSQP literature where such rates are known under second-order sufficient conditions. Additionally, the compactness requirement on the multiplier boxes $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$0 and $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$1 introduces algorithmic parameters whose influence on practical performance is not analyzed theoretically.

Numerical evidence

Experiments were implemented in MATLAB R2025a with Manopt, solving each QP subproblem with quadprog, terminating at residual $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$2 or 500 iterations, with $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$3 set to the Riemannian Hessian (eigenvalues floored at $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$4).

Degenerate MPECs on the sphere. On mathematical programs with equilibrium constraints over $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$5—degenerate in that RCQ fails at every feasible point—the contrast is stark:

Method	Iterations	Final residual
RSSQP	64–84 across all five sizes	$$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$6
RSQP	500 (max) in all cases	$$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$7 to $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$8

RSSQP attained tolerance on all five instances up to $$\min_{(\xi,\zeta,\eta)} \langle grad f(x_k),\xi\rangle + \tfrac12\langle H_k[\xi],\xi\rangle + \tfrac{\sigma_k}{2}\|\zeta\|^2 + \tfrac{\sigma_k}{2}\|\eta\|^2$$9, while RSQP failed everywhere. This is the paper's strongest empirical claim supporting the stabilization mechanism under total CQ failure.

Nonnegative low-rank matrix completion. On fixed-rank manifolds $\sigma_k(\zeta - \widehat y_k)$0 (which, the authors note, are not complete, so theory does not strictly apply), both methods solved all five instances. RSSQP was competitive overall and superior on the two largest instances—for $\sigma_k(\zeta - \widehat y_k)$1: 86 vs. 141 iterations and 80.7 s vs. 129.2 s; for $\sigma_k(\zeta - \widehat y_k)$2: 74 vs. 135 iterations and 314 s vs. 572 s. The authors attribute this partly to the guaranteed feasibility of the stabilized subproblem even from infeasible iterates.

Limitations and open questions

Three caveats bear directly on the results. First, the convergence guarantee is asymptotic and requires boundedness of $\sigma_k(\zeta - \widehat y_k)$3, which is assumed rather than proven; unbounded iterates fall outside the theory. Second, the guarantee permits termination at stationary points of the feasibility problem—infeasible accumulation points—which is unavoidable without CQs but means the method offers no certificate of primal feasibility in the worst case. Third, the numerical validation is limited to two problem classes with a single instance per size, and the matrix-completion experiments violate the manifold completeness assumption, so the favorable results there are empirical rather than theoretically grounded. The open question the authors themselves identify is establishing local superlinear or quadratic convergence of RSSQP under appropriate second-order sufficient conditions and regularity assumptions.

Conclusion

This paper supplies the first stabilized SQP framework for constrained optimization on Riemannian manifolds, combining proximal-perturbed quadratic subproblems, an augmented-Lagrangian merit function, and safeguarded dual updates. Its principal theoretical contribution is a CQ-free global convergence result delivering a KKT point, an AKKT point, or a feasibility-problem stationary point—with ERCQ upgrading the outcome to KKT—and its principal empirical contribution is demonstrating reliable solution of fully degenerate spherical MPECs where standard RSQP fails entirely. The absence of any local convergence rate remains the main gap relative to the Euclidean stabilized SQP literature.