Binary Anchor Optimization Algorithm

• The algorithm is a convex framework that minimizes active anchor count while enforcing CRB constraints for high-accuracy sensor localization.
• It uses sparse optimization techniques with ℓ1 relaxation and iterative reweighted schemes to efficiently handle one-way TOA ranging in both transmission modes.
• Numerical results indicate that the iterative reweighted approach notably reduces anchor usage (up to 45% fewer) while satisfying localization performance via LMIs.

The Binary Anchor Optimization Algorithm is a convex optimization-based framework for anchor placement in sensor network localization via one-way time-of-arrival (TOA) ranging. The algorithm exploits the inherent sparsity in anchor deployment, targeting minimal-anchor or minimal-energy solutions while ensuring that localization performance—quantified through the Cramér-Rao bound (CRB)—meets prescribed accuracy specifications throughout the sensor region. The approach is rigorously developed for both scenarios where anchors transmit (OW-A) or receive (OW-S) ranging signals, and involves a sequence of sparse optimization, convex relaxation, and solution refinement steps to yield provably feasible and sparse anchor placements (Chepuri et al., 2013).

1. System Formulation and Optimization Variables

Consider a deployment area with $M$ candidate anchor locations $a_1,\ldots,a_M\in \mathbb{R}^2$ in "anchor area" $\mathcal{A}$ and an unknown sensor position $s\in\mathcal{S}\subset\mathbb{R}^2$ discretized on a grid. The distance $d(a_m, s)$ is computed as $\|a_m - s\|_2$. Two operational modes are distinguished:

• OW-A (Anchors Send): Each anchor $m$ transmits a ranging pulse of energy $e_m$, resulting in TOA noise variance $\sigma_{m,s}^2 = (\rho_a/\gamma^2)/e_m$, where $\rho_a$ depends on pulse and noise spectral density, and $\gamma^2 = \alpha d(a_m,s)^{-\beta}$ models path-loss (with known $\alpha$, $\beta$).
• OW-S (Sensor Sends): The sensor transmits once with fixed energy $e_s$; each anchor $m$ observes TOA with variance $\sigma_{s,m}^2 = (\rho_s/\gamma^2)/e_s$.
• Selection Vectors: In OW-A, vector $e=[e_1,\ldots,e_M]^T$ (with $e_m\ge 0$) simultaneously selects and sets transmit energy per anchor. In OW-S, vector $w=[w_1,\ldots,w_M]^T\in\{0,1\}^M$ selects the set of utilized anchors (pure selection).

This formulation subsumes a sparse-selection paradigm, as optimal performance is typically achieved with only a subset of anchors being active.

2. Fisher Information Matrix and Cramér-Rao Bound Constraints

Localization performance is governed by the Fisher Information Matrix (FIM):

• OW-A: $J_a(e,s) = \sum_{m=1}^M e_m F_{a,m}(s)$
• OW-S: $J_s(w,s) = \sum_{m=1}^M w_m F_{s,m}(s)$

with per-anchor Fisher information components

$$F_{a,m}(s) = \alpha\rho_a^{-1} d(a_m,s)^{-(\beta+2)} (s-a_m)(s-a_m)^T,$$

$$F_{s,m}(s) = e_s \alpha \rho_s^{-1} d(a_m,s)^{-(\beta+2)} (s-a_m)(s-a_m)^T.$$

The unconstrained CRB is $$\mathrm{CRB}(e, w; s) = \mathrm{trace}(J(e, w; s)^{-1})$$, but in practice a constraint is imposed requiring the smallest FIM eigenvalue to satisfy $\lambda_{\min}(J(e,w;s)) \geq \lambda$, for all $s\in\mathcal{S}$, ensuring maximum variance in any direction does not exceed $2/\lambda$. Equivalently, the set of convex matrix inequalities (LMIs):

• For OW-A: $\sum_{m} e_m F_{a,m}(s) \succeq \lambda I_2$ for all $s\in \mathcal{S}$
• For OW-S: $\sum_{m} w_m F_{s,m}(s) \succeq \lambda I_2$ for all $s\in \mathcal{S}$

3. Combinatorial $\ell_0$-based Formulation

The anchor optimization objective is to minimize the support of the selection vector ($\ell_0$ norm or cardinality), subject to CRB (LMI) constraints:

• OW-A:

$$\min \|e\|_0 \quad \text{s.t.} \quad \sum_{m=1}^M e_m F_{a,m}(s) \succeq \lambda I_2 ~ \forall s\in\mathcal{S};\, 0 \leq e_m \leq e_b$$

• OW-S:

$$\min \|w\|_0 = 1^T w \quad \text{s.t.} \quad \sum_{m=1}^M w_m F_{s,m}(s) \succeq \lambda I_2 ~ \forall s\in\mathcal{S};\, w_m \in \{0,1\}$$

Both formulations are NP-hard and combinatorial in $M$.

4. Convex Relaxation via $\ell_1$ and SDP

A standard convex surrogate is employed:

• OW-A: The $\ell_0$-norm on $e$ is replaced by the $\ell_1$ norm, yielding a semidefinite program (SDP):

$$\min_{e\in\mathbb{R}^M} 1^T e \quad \text{s.t.} \sum_{m=1}^M e_m F_{a,m}(s) \succeq \lambda I_2,\, 0 \le e \le e_b 1$$

• OW-S: Boolean constraints are relaxed using a Shor-type SDP lift:

$$\min 1^T w \quad \text{s.t.} \sum_{m=1}^M w_m F_{s,m}(s) \succeq \lambda I_2,\, \begin{bmatrix} W & w \ w^T & 1 \ \end{bmatrix} \succeq 0,\, \text{diag}(W) = w,\, w \ge 0$$

This convex relaxation allows polynomial-time approximation to the original combinatorial problem.

5. Sparsity-Promoting Iterative Reweighted $\ell_1$

To induce higher sparsity, iterative reweighted $\ell_1$ optimization is applied. At iteration $k$, selection weights are updated as $u_i^{(k+1)} = \frac{1}{\epsilon + e_i^{(k)}}$ and the current SDP is resolved with a weighted objective $u^{(k)T} e$. This process typically converges within 3–6 iterations to a solution with substantially fewer nonzero entries compared to plain $\ell_1$ relaxation. After convergence, all $e_i$ below a set threshold are discarded to finalize the sparse support.

In the OW-S mode, post-processing via randomization or simple thresholding of the continuous relaxation solution ($w^* \in [0,1]^M$) yields a binary selection. One standard approach is to sample Gaussian vectors from $\mathcal{N}(0, W^*)$, set $w_i^{(\text{trial})} = 1$ if $\xi_i \geq \theta$, and select the best cardinality-constrained trial that satisfies the LMIs (Chepuri et al., 2013).

6. Algorithmic Complexity and Numerical Behavior

Each $\ell_1$-relaxed problem forms an SDP of size $M \times M$ with $|\mathcal{S}|$ LMIs (each $2\times2$). Standard interior-point SDP solvers address problems of this scale efficiently (e.g., CVX+SeDuMi), with worst-case per-iteration complexity $O((M + |\mathcal{S}|)^3)$. For $M$ up to a few hundred and $|\mathcal{S}|$ up to a few thousand, run times are practical.

Iterative reweighted SDPs converge rapidly, and while global optimality to the original $\ell_0$ combinatorial problem is not ensured, empirical cardinalities are near-optimal. For example, compared to an intractable exhaustive search involving $\sim 10^{17}$ feasibility checks for moderate problem sizes, the algorithm executes efficiently and to high sparsity (Chepuri et al., 2013).

7. Numerical Results and Empirical Evaluation

Selected numerical evaluations with $M=80$ anchor candidates (on a circle), accuracy specs $R_e = 4$ cm, $P_e = 0.95$, path-loss $\beta = 2,\, \alpha = 1$, and $\mathrm{SNR} \approx 10$ dB at $10$ m, substantiate the algorithm’s performance:

Mode	Plain $\ell_1$	Iterative/Reweighted $\ell_1$	Final Support	Total Energy [J]
OW-A	$\sim$9 anchors	5 anchors ($\sim$45% fewer)	5	6
OW-S	$\sim$20 anchors (soft)	4 anchors (binary)	4	N/A

For all tested grid points, the critical smallest FIM eigenvalue ($\lambda_{\min}$) constraint is satisfied. These results highlight that iterative reweighting and proper relaxation can yield extremely sparse and energy-efficient anchor placements—orders of magnitude more efficiently than combinatorial search (Chepuri et al., 2013).

8. Summary of Methodological Innovations

The Binary Anchor Optimization Algorithm unifies several principles:

1. Performance constraint handling: CRB requirements are encoded as small-eigenvalue LMIs for guaranteed localization accuracy.
2. Sparsity-exploiting setup: Anchor deployment is naturally cast as an $\ell_0$-minimization problem.
3. Convex tractable surrogates: Relaxation to $\ell_1$ norm or SDP renders the selection problem solvable in polynomial time.
4. Sparsity enhancement: Iterative reweighted schemes and randomized rounding bridge the gap from convex relaxation to near-binary support, reconciling tractability and optimality.

A plausible implication is that these mathematical programming principles are extensible to other sensor deployment and energy allocation problems where sparsity and geometric coverage under statistical error constraints are critical (Chepuri et al., 2013).