Nonlinearly Constrained Augmented Lagrangian

• Nonlinearly Constrained Augmented Lagrangian (NCL) is a two-level optimization framework that combines quadratic penalties for nonlinear equality constraints with multiplier updates.
• It employs a proximal block-coordinate descent inner loop to enable distributed, parallel updates in large-scale or agent-based optimization problems.
• NCL guarantees convergence to KKT points under semi-algebraic conditions, making it robust for solving challenging nonconvex nonlinear programs.

A Nonlinearly Constrained Augmented Lagrangian (NCL) method is a two-level optimization framework for solving nonconvex nonlinear programs with nonlinear equality constraints and possibly coupled objective/constraint structure across distributed agents. NCL is distinguished by its use of a quadratic penalty for the nonlinear constraints (while treating inequalities as hard constraints), a multiplier (dual) update at the outer level, and an inner block-coordinate descent (BCD) loop with proximal regularization. Convergence is established under semi-algebraicity (implying the Kurdyka–Łojasiewicz property), ensuring approach to KKT points even in the presence of nonconvexity and general constraint coupling. This approach is particularly suited to distributed or large-scale settings such as distributed NMPC or agent-based optimization with significant constraint or objective coupling (Hours et al., 2014).

1. Problem Structure and Formulation

The NCL method targets nonconvex nonlinear programs defined by

$$\min_{z \in Z} \; J(z) = \sum_{i=1}^N J_i(z_i) + Q(z_1, \ldots, z_N) \quad \text{s.t.} \quad H(z) = 0,$$

where the variable $z = (z_1, \ldots, z_N)$ is partitioned by agent, the coupling term $Q$ ties together the local variables, $H(z)$ is a vector of nonlinear equalities typically comprising individual agent constraints $F_i(z_i)$ and a global coupling constraint $G(z_1,\ldots, z_N)$, and the feasible set $Z$ is the product of box or polyhedral (inequality) constraints (polytopic sets) for each agent.

• Only the nonlinear equalities in $H(z)$ are penalized.
• Inequality constraints are enforced as hard constraints defining $Z$.
• The problem is structured for distributed solution in agent-based settings.

2. Augmented Lagrangian and Two-Level Algorithm

Outer Level: Multiplier Update

The augmented Lagrangian is

$$L_\rho(z,\mu) = J(z) + \mu^\top H(z) + \frac{\rho}{2} \| H(z) \|^2,$$

with $\rho > 0$ and multiplier estimate $\mu$. At each outer iteration $k$, the following occur:

• Solve (approximately) $\min_{z \in Z} L_{\rho_k}(z, \mu_k)$ to inexact first-order accuracy.
• Update dual variables: $\mu_{k+1} = \mu_k + \rho_k H(z_{k+1})$.
• Increase penalty: $\rho_{k+1} = \beta \rho_k$, decrease tolerances for inner solves.

Inner Level: Proximal Block-Coordinate Descent (BCD)

Given the coupled structure, $L_\rho(z,\mu)$ is not block-separable. The algorithm applies an inexact, proximal regularized BCD update: $$z_i^{(l+1)} = \arg\min_{z_i \in Z_i} \bigg\{ \nabla_{z_i} L_\rho(\cdots)^\top (z_i - z_i^{(l)}) + \frac{1}{2}(z_i - z_i^{(l)})^\top (B_i^{(l)} + \alpha_i^{(l)} I)(z_i - z_i^{(l)}) \bigg\}$$ where $B_i^{(l)}$ approximates the block Hessian and $\alpha_i^{(l)}$ is a diagonal regularization. This quadratic surrogate gives computational tractability for each block update under $Z_i$ (often allowing for convex QP subproblems per agent).

The inner loop continues until a first-order accuracy tolerance is achieved: $$d\big(0, \nabla_z L_\rho(z, \mu) + N_Z(z) \big) \leq \varepsilon,$$ where $N_Z(z)$ is the normal cone to $Z$ at $z$.

3. Convergence under Semi-Algebraicity

A key theoretical component is the semi-algebraicity of the problem data: all $J_i$, $Q$, $F_i$, and $G$ are assumed twice continuously differentiable and semi-algebraic.

• Semi-algebraicity ensures satisfaction of the KL property.
• The framework leverages KL to guarantee global convergence of the sequence $(z^k)$, even for nonconvex programs.
• The convergence proof relies on "sufficient decrease" and "relative error" properties established for the inner BCD (as in proximal alternating minimization literature), and on classical multiplier method theory for the outer updates.

It is shown that, under mild regularity and proximity to an isolated KKT point, the iterates satisfy: $$z_k \rightarrow z^*, \quad \tilde{\mu}_k \rightarrow \mu^*, \quad \tilde{\mu}_k := \mu_k + \rho_k H(z_k)$$ for $(z^*, \mu^*)$ a KKT point.

4. Distributed and Decomposed Computation

The inner BCD loop aligns with distributed computation:

• Each block $z_i$ can be updated by a dedicated agent, provided its local QPs are solved.
• The global coupling terms in $Q$ and $G$ are accounted for via coordination in gradient and Hessian computations.
• Sparsity and weak coupling allow for partial parallelization; for example, in certain chain-structured problems, the BCD can be arranged to have $N/2$ parallel updates per cycle.

5. Numerical Insights and Scalability

A representative numerical example involves minimizing a coupled sum of quadratic forms (possibly indefinite) subject to quadratic equalities $\| x_i \|_2^2 = a^2$ and box constraints $-b \le x_{i,j} \le b$: $$\min \sum_{i=1}^N x_i^\top H_i x_i + \sum_{i=1}^{N-1} x_i^\top H_{i,i+1} x_{i+1} \quad \text{s.t.} \quad \|x_i\|_2^2 = a^2, \; \forall i;\; -b \leq x_{i,j} \leq b.$$ The system allows inner BCD updates to be grouped for parallel solution according to problem sparsity. Key observations from experiments include:

• Achieving feasibility to $10^{-3}$ in equality constraints generally requires $\mathcal{O}(100)$ iterations.
• Tighter feasibility tolerances require more BCD cycles; performance is strongly dependent on the inner-outer iteration interplay.

The two-layer architecture (outer multiplier update, inner regularized BCD) is effective for distributed, nonconvex problems and can be implemented partially in parallel with low communication costs.

6. Relation to Other Augmented Lagrangian and Decomposition Methods

NCL, as described here, is closely related to the LANCELOT method, but with explicit coordination-decomposition features for agent-based problems and a proximal BCD based inner solver. In contrast to fully separable augmented Lagrangians, the NCL approach can handle arbitrary nonlinear coupling in both objectives and constraints across blocks. The proximal BCD distinguishes it from classical multiblock multiplier methods by delivering sufficient decrease and leveraging semi-algebraicity for convergence.

Distinctive aspects:

• Blockwise quadratic regularization ensures robust inner convergence even when the underlying Lagrangian is nonconvex and coupled.
• Only equality constraints are penalized, preventing loss of feasibility from hard inequality constraints.

7. Typical Algorithmic Workflow

Stage	Key Operation	Parallelization
Outer iteration	Update multipliers, penalty, accuracy target	Serial with possible asynchrony
Inner iteration	Proximal BCD updates for $z_i$	Block-wise parallel or sequential
Local QP solve	Solve for $z_i$ under $Z_i$	Can exploit agent-level solvers

The outer level is bottlenecked by global constraint/cost coupling but is computationally light, while the inner loop dominates the computational complexity and benefits from the BCD decomposition.

8. Summary and Impact

The NCL method offers a robust, theoretically sound, and distributed-compatible framework for nonconvex programs with nonlinearly constrained (equality) structure. Its convergence to KKT points is guaranteed under the semi-algebraic setting, leveraging modern proximal BCD methodology and multiplier updates. The method is particularly impactful for agent-based and distributed optimization scenarios, such as those arising in distributed NMPC or large-scale, coupled engineering systems (Hours et al., 2014).