Northern Goshawk Optimization Algorithm

• Northern Goshawk Optimization (NGO) is a metaheuristic that mimics goshawk predatory behavior with distinct global exploration and local exploitation phases.
• It combines biologically-inspired operators with a two-phase update mechanism—'Prey Identification and Rapid Strike' and 'Chase and Escape'—to effectively traverse the search space.
• Enhanced versions like INGO integrate diverse chaotic initialization (DCMIS) and bidirectional population evolution (BPED) to improve convergence and optimize Wireless Sensor Network coverage.

Northern Goshawk Optimization (NGO) is a population-based metaheuristic that emulates the predatory tactics of the northern goshawk, characterized by two principal phases: global exploration (“Prey Identification and Rapid Strike”) and local exploitation (“Chase and Escape”). NGO and its advanced multi-strategy version, INGO, have demonstrated high efficacy in combinatorial and continuous optimization, with notable success in Wireless Sensor Network (WSN) coverage problems by leveraging novel strategies for initialization and population evolution (Tian et al., 5 Jan 2026).

1. Algorithmic Structure of Standard Northern Goshawk Optimization (NGO)

NGO employs a population of candidate solutions, denoted as $X_i \in \mathbb{R}^D$, and iteratively applies biologically inspired operators for search space traversal. The process consists of two key phases per iteration:

Phase I: Prey Identification & Strike (Exploration)

• Each agent $i$ randomly selects agent $k \ne i$ as the prey $P_i = X_k$.
• For each dimension $j$, generate a candidate:

$$x_{i,j}^{new,P_1} = \begin{cases} x_{i,j} + r (p_{i,j} - I x_{i,j}), & F(P_i) < F(X_i) \ x_{i,j} + r (x_{i,j} - p_{i,j}), & F(P_i) \geq F(X_i) \end{cases}$$

with $r \sim U(0,1)$, $I \in \{1,2\}$.

• Greedy selection replaces $X_i$ if the new candidate is superior.

Phase II: Chase & Escape (Exploitation)

• Each agent locally refines its position:

$R = 0.02 (1 - t/T)$

$$x_{i,j}^{new,P_2} = x_{i,j} + R (2r - 1) x_{i,j}, \quad r \sim U(0,1)$$

• Greedy update is performed as in Phase I.

Pseudocode (algorithm summary):

Input: population size N, dimensions D, bounds lb, ub, max iterations T Output: best solution X_best // 1. Random initialization for i = 1 to N X_i ← lb + rand(1,D) ∘ (ub - lb) F_i ← F(X_i) end for X_best ← argmin_i F_i // 2. Main loop for t = 1 to T for i = 1 to N // Phase I: Exploration pick k ≠ i at random ... apply greedy selection to update X_i // Phase II: Exploitation ... apply greedy selection to update X_i if F(X_i) < F(X_best) X_best ← X_i end if end for end for return X_best

2. Multi-Strategy Enhancements: DCMIS and BPED

INGO introduces two principal improvements over baseline NGO to address premature convergence and diversity loss:

2.1 Diverse Chaotic Map Initialization Strategy (DCMIS):

• Seeding via a coupled Logistic–Sine chaotic sequence:

$z_{\text{seed}} = \text{rand}(N, D)$

$$z = \sin\left( \pi z_{\text{seed}} (1 - z_{\text{seed}}) + \sin( \pi z_{\text{seed}} ) \right)$$

$X = \text{lb} + |z| (\text{ub} - \text{lb})$

• This yields a high-diversity, low-correlation population, enhancing global search capability.

2.2 Bidirectional Population Evolutionary Dynamics (BPED):

• Population is split:
  • Top 20% (elite): refined using the current best.
  • Bottom 20% (non-elite): re-explored via local refinement (shrinking bounds) or large mutation.
• Elite refinement:

$$w = \frac{1}{2} \left( \sin \left( 2\pi\,\text{freq}\,t + \pi \right) \pi \frac{t}{T_{\max}+1} \right), \quad \text{freq} = \frac{1}{D}$$

$$X_q^{new} = X_q + w [x_{\rm best} - \text{round}(1+z) X_k]$$

• Non-elite re-exploration:
  • If $r_0 < 0.5$:

  $$X^{new} = x_{\rm best} + \text{sign}(r_0-0.5)[lb_{\rm ap} + \text{rand}(ub_{\rm ap} - lb_{\rm ap})]$$ - Else:

  $$X^{new} = X_z - 2\,\text{sign}(r_0-0.5)[lb + \text{rand}(ub-lb)]$$

3. Integrated Algorithm Workflow

The combination of DCMIS (for initialization) and BPED (for dynamic refinement) is formalized in the consolidated INGO pseudocode:

Input: Objective F, bounds lb/ub, population N, dimensions D, max iterations T Output: x_best, Convergence_curve // 1) Initialization via DCMIS z_seed ← rand(N,D) z ← sin( π*z_seed*(1−z_seed) + sin(π*z_seed) ) X ← lb + |z| ∘ (ub - lb) Evaluate F(X_i), set x_best ← best of X // 2) Main Loop for t = 1 to T // Standard NGO phases ... // BPED enhancement rank X by fitness X_elite ← top 20% X_non-elite ← bottom 20% // (a) Elite Evolution for each X_q in X_elite pick random X_k in X_elite X_q ← X_q + w*(x_best - round(1+z)*X_k) end for // (b) Non-elite Re-exploration for each X_z in X_non-elite if rand < 0.5 X_z ← x_best + sign(rand-0.5)*(lb_ap + rand*(ub_ap-lb_ap)) else X_z ← X_z - 2*sign(rand-0.5)*(lb + rand*(ub-lb)) end if end for end for return x_best, Convergence_curve

4. Parameterization and Experimental Setup

WSN coverage enhancement experiments employed the following protocol:

• Monitoring region: $50 \times 50$ m²
• Number of sensors: 35
• Sensing radius: 5 m; Communication radius: 10 m
• Grid discretization: $\Delta = 0.8$
• Population size: 30
• Maximum iterations: 500
• Replications: 30 (coverage), 20 (benchmark functions) No further parameter sensitivity analyses beyond ablation of DCMIS and BPED were performed. All remaining hyperparameters followed the original NGO formulation (Tian et al., 5 Jan 2026).

5. Empirical Performance and Comparative Results

INGO demonstrated substantial improvements to WSN deployment, achieving the following statistical metrics across 30 runs:

Algorithm	Avg Coverage	Std Dev	Avg Connectivity
INGO	91.90%	0.01223	100.00%
FA	88.30%	0.01231	96–98%
IWHO	82.66%	0.01420	92–95%
NGO (std)	82.51%	0.01088	90–93%
ABC	72.54%	0.01261	75–80%

• Coverage ($Cov$): maximized as primary objective
• Connectivity ($\eta_{\rm conn}$): percentage nodes in largest connected component, consistently 100% with INGO

Convergence analysis identified rapid ascent above 85% coverage (within 100 iterations) due to DCMIS, continued refinement by BPED, and ultimate outperformance versus all comparison algorithms. INGO achieved the narrowest interquartile range and lowest medians across benchmark functions; ablation studies indicate complementary strengths for DCMIS and BPED alone.

6. Algorithmic Properties and Implications

The distinct features of INGO—namely multivariate chaotic initialization (DCMIS) and bidirectional evolutionary dynamics (BPED)—allow robust rebalancing between exploration and exploitation. DCMIS provides improved initial population dispersion, accelerating early search and reducing clustering. BPED implements dynamic dual strategies: elite exploitation near the global best, and forced re-exploration for poor solutions, mitigating stagnation in local optima. Mathematically, these steps replace the standard random initialization and single update rule, overlaying nonlinear mapping and rank-based population partitioning.

A plausible implication is that such hybrid strategies may generalize well to other population-based metaheuristics susceptible to diversity loss. The 9.4% increase in average coverage and universal connectivity was achieved without significant parameter overhead. This suggests that multi-strategy metaheuristics may be particularly effective in highly multimodal, constrained deployment problems.

7. Context, Applicability, and Outlook

Northern Goshawk Optimization and its multi-strategy improvement (INGO) have proven effective as primary methods for WSN deployment, with the potential for adaptation to other coverage, clustering, and combinatorial optimization tasks. Current developments focus on initialization dynamics and adaptive population strategies; further research may explore sensitivity analysis, hybridization with other evolutionary mechanisms, or expansion to high-dimensional and real-time domains.

The methodology and empirical outcomes illustrate the integration of biological inspiration with rigorous population-based optimization, producing state-of-the-art results in practical deployment scenarios (Tian et al., 5 Jan 2026).