Graduated Weight Approximation (GWA) Algorithm

• Graduated Weight Approximation (GWA) is an iterative method that approximates variable, state-dependent weights to transform nonconvex weighted least-squares problems into convex forms.
• It employs successive refinement via Semidefinite Programming (SDP) to update weight estimates, ensuring global optimality and mitigating local minima issues.
• Empirical evaluations demonstrate GWA’s dramatic reduction in positioning errors, making it a robust tool in signal processing and satellite-based localization.

Graduated Weight Approximation (GWA) Algorithm is an iterative technique for handling variable, state-dependent weights in nonconvex optimization problems, notably in signal processing and positioning frameworks where the cost function’s weighting matrix is dynamically related to the optimization variables. GWA proceeds by approximating unknown weight terms via iterative refinement—starting with a coarse initialization and updating the weights with successive solutions—thereby transforming the original nonlinear or fractional least-squares objective into a form amenable to convex relaxation schemes such as Semidefinite Programming (SDP). This strategy has seen practical deployment in certifiably optimal Doppler positioning using Low Earth Orbit (LEO) satellites, where its interplay with SDP tightness yields global optima without reliance on initialization.

1. Motivation and Foundational Problem Structure

LEO Doppler positioning formulates the localization as a nonlinear weighted least-squares (NWLS) optimization problem. The measurements, derived from Doppler shifts, are functions of the receiver–satellite range variables $\rho_i$, producing an objective involving fractional and nonlinear terms. Specifically, the cost function weights display dependence on unknown state variables, taking the form: $$Q = \text{diag}\left\{\frac{1}{\rho_1}, \dots, \frac{1}{\rho_N}\right\} \cdot \epsilon_D$$ In traditional local search (e.g., Gauss–Newton, Dog–Leg), the nonconvexity and the state-dependent weighting tend to trap solutions in local minima, especially when the initial state estimate deviates greatly from ground truth. GWA is introduced to "approximate" these dynamic weights in a graduated fashion, rendering the NWLS problem suitable for polynomial (POP) reformulation and subsequent SDP relaxation.

2. Algorithmic Formulation and Iterative Weighting

The core mechanism of GWA is to iteratively update the weighting matrix $Q$ based on the most recently estimated state variables. At the initial iteration, $Q^{(0)}$ is set to a constant (commonly the identity or $\epsilon_D$), flattening the weight structure and permitting tractable optimization. After solving the relaxed POP using SDP at iteration $t$, the ranges $\rho_i^{(t)}$ are extracted and $Q^{(t)}$ is refined: $$Q^{(t)} = \text{diag}\left\{\frac{1}{\rho_1^{(t)}}, \dots, \frac{1}{\rho_N^{(t)}}\right\} \cdot \epsilon_D$$ The iterative process continues until convergence is achieved—typically measured by the trace norm difference $\eta^{(t)} = \text{tr}(Q^{(t)} - Q^{(t-1)})$ falling below a pre-specified threshold.

GWA Iteration Schema

Step	Algorithmic Actions	Output/Transition
Initialization	Set $Q^{(0)}$, input max iterations $T$	$Q^{(0)}, \eta = +\infty$, iteration $t=0$
SDP Update	Solve SDP for current $Q^{(t)}$	Obtain $\rho_i^{(t)}$, update $Q^{(t)}$
Convergence	Evaluate $\eta^{(t)}$, check threshold $\bar{\eta}$	If $\eta^{(t)} < \bar{\eta}$, stop

This graduated updating transforms the weighting structure from an initial approximation to values increasingly faithful to the true (state-dependent) weights.

3. Transformation to Polynomial and SDP Relaxation

After the GWA iteration yields a fixed $Q$, the problem is converted into a polynomial optimization problem (POP), then lifted to a quadratically constrained quadratic program (QCQP): $$y = [p_r^\top, c \, \dot{t}_r, \rho_1, \dots, \rho_N, z_1, \dots, z_N]^\top$$ with constraints such as $z_i = c\,\rho_i\,\dot{t}_r$. The QCQP’s rank-one constraint is relaxed via Shor’s SDP relaxation: $$\begin{cases} \min \quad F \bullet Y + l_0 \cdot y + c_0 \ \text{subject to} \quad \left[ \begin{array}{cc} Y & y \ y^\top & 1 \end{array} \right] \succeq 0 \ G_i \bullet Y + l_i \cdot y + c_i = 0,\quad i=1,\dots, 2N \end{cases}$$ The update of $Q$ in each GWA iteration directly induces the construction of $F, l_0$, and the constraint matrices, stabilizing the convex relaxation across iterations.

4. Optimality: Necessary/Sufficient Conditions in Noiseless and Noisy Cases

Upon convergence of GWA and solution of the SDP, strong duality and global optimality are tied to mathematical conditions:

Noiseless Case: Zero duality gap and rank-tightness (i.e., $\text{rank}(Y) = 1$) are guaranteed if:

• Primal feasibility: $g_i(y^*) = 0$
• Dual feasibility: $H(\lambda^*) \succeq 0$
• Complementary slackness: $H(\lambda^*) [1; y^*] = 0$

The problem structure (rank and corank properties of matrices $F$) ensures that under noiseless conditions, the relaxation is tight and the unique global optimum can be extracted from the SDP solution.

Noisy Case: Sufficient bounds for the tightness of relaxation and global optimality are derived using Abadie’s constraint qualification (ACQ) and Weyl’s inequality: $$\frac{1}{\sigma_N} \|\mathcal{G}\|\|\nabla f_\theta(\bar{y})\| + \|F - F_\theta\| < \nu_{N+4}(F_\theta)$$ where variables correspond to singular values, gradients, and eigenspectra. If measurement or model noise remains within derived bounds, the SDP solution remains rank-one and certifiably optimal.

5. Simulation and Experimental Evaluation

Simulation results substantiate the GWA algorithm’s robustness. Standard local search methods perform adequately only when initialization is within 100 km of ground truth, but degrade to multi-hundred or thousand kilometer errors with poor initialization. In contrast, the GWA-SDP approach yields reliably low positioning errors—0.71 km in simulation, and 0.14 km in real Iridium-NEXT satellite tests—without dependence on initialization. Furthermore, when the SDP solution is used to seed local search (SDP-GN, SDP-DL), final 3D positioning errors further decrease (to approximately 130 m).

Optimization Type	Initialization Sensitivity	3D Positioning Error (km)
Gauss-Newton / Dog-Leg	High	Up to 1000+
GWA+SDP (no init)	Low	~0.14
SDP-GN (SDP seed)	Negligible	~0.13

6. Role and Significance of GWA in Convex Relaxation Frameworks

GWA enables tractable convex relaxation for otherwise intractable, variable-weighted NWLS problems by bootstrapping weight estimates and refining them via successive approximations. This iterative refinement stabilizes the SDP relaxation process, fosters rank-tightness, and ultimately permits extraction of certifiably global optima under strong theoretical guarantees. GWA thus bridges domains of polynomial optimization and robust positioning through seamless integration of nonconstant, data-dependent weights, providing a template for addressing similar classes of nonconvex problems.

7. Contextualization and Generalization

Graduated Weight Approximation pivots on iterative weight bootstrap and finds conceptual kinship with other graduated approaches in combinatorial optimization, such as graduated interval partitioning in maximum weight matching (Lingas et al., 2010) and graduated mixture estimation in density modeling (Frisch et al., 2021). Its technical framework—iterative weighting, convex relaxation, and global certification—proves advantageous wherever the original optimization structure inhibits global search via local algorithms or static relaxations. A plausible implication is broader adoption of GWA-driven convexification in GNSS augmentation, resource allocation, and signal inference problems where state-dependent weighting is a core challenge.