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Constrained Optimization From a Control Perspective via Feedback Linearization
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Establishes global convergence rates, an SQP equivalence, inequality-constraint extension, and a momentum-accelerated variant for feedback-linearization-based constrained optimization.
Problem
Theoretical understanding of feedback linearization (FL) for constrained optimization is limited: global convergence rates, connection to existing optimizers, inequality constraints, and acceleration remain open.
Approach
Views constrained optimization as a control problem where Lagrange multipliers are control inputs; uses feedback linearization to drive constraint violation to zero exponentially, shows forward-Euler discretization equals SQP, extends to inequalities via a QP-based multiplier, and adds a momentum term (FL-momentum).
Key results
- Global KKT-gap convergence rate O(1/sqrt(T)) for FL on equality-constrained problems, with exponential decay of constraint violation.
- Forward-Euler discretization of FL with K = (1/eta) I is shown equivalent to SQP (equality and inequality cases).
- FL-momentum gets an O(1/sqrt(T)) rate in the nonconvex equality setting and converges faster than FL-proximal empirically, though no rate improvement is proven.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use feedback linearization for equality constraints | Constrained optimization · Equality-constrained optimization | KKT-gap: O\left(\frac{1}{\sqrt{T}}\right) | Helped |
| Use the inequality-constrained FL controller | Constrained optimization · Inequality-constrained optimization | KKT-gap: O\left(\frac{1}{\sqrt{T}}\right) | Helped |
| Add momentum | FL-proximal · Heterogeneous logistic regression | convergence speed | Helped |
| Use second-order Hessian information | FL-proximal · Heterogeneous logistic regression | convergence speed | Helped |
| Add momentum | FL-proximal · AC OPF, IEEE-39 bus | convergence speed | Helped |
| Add momentum | FL-proximal · AC OPF, IEEE-118 bus | convergence speed | No clear effect |
| Use second-order Hessian information | FL-proximal · AC OPF, IEEE-39 and IEEE-118 bus systems | convergence speed | Helped |
| Add an integral controller term (ablation) | Approximated FL · Quadratic programming with approximate inverse | convergence to optimal solution | Helped |
| Use momentum | FL-proximal · Constrained optimization comparison with PDGD and ALM | convergence speed | Helped |
Surprising findings
- Approximated FL (with diagonal approximation of the inverse) fails to converge on a random QP, while the PI-augmented variant converges.
Limitations
- Convergence is to KKT points, not distinguishing local/global optima or saddle points.
- Requires LICQ-type assumptions (J_h J_h^T invertible with bounded inverse) and uniform Lipschitz bounds.
- Momentum variant has same O(1/sqrt(T)) rate as non-accelerated version; analysis is continuous-time only for FL-momentum.
Experimental setup
Datasets: synthetic heterogeneous client logistic regression data IEEE-39 bus system IEEE-118 bus system random quadratic programs
Benchmarks: heterogeneous logistic regression (C=5 clients) AC OPF IEEE-39 and IEEE-118
Metrics: KKT gap
Baselines: FL-proximal FL-Newton PDGD ALM Projected gradient descent
Builds on: feedback linearization sequential quadratic programming primal-dual gradient dynamics Nesterov acceleration
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