Group Counseling Optimization (GCO): Study Notes and Technical Guide

Overview of Group Counseling Optimization (GCO)

Group Counseling Optimization (GCO) is a novel population-based optimization algorithm developed by M.A. Eita and M.M. Fahmy at Tanta University, Egypt. Unlike traditional metaheuristic algorithms that mimic biological behaviors (e.g., birds, ants, bees), GCO emulates human problem-solving through group counseling.

• Motivation: Human thinking is perceived as the most reasonable and influential mechanism for problem-solving.
• Key Contribution: The algorithm establishes a formal analogy between the procedures of group counseling and computational population-based optimization.
• Performance: GCO has been validated against diverse benchmark functions (unrotated, rotated, and composition) and real-world spacecraft trajectory design problems, often outperforming established optimizers like CLPSO and NBIPOP-aCMA-ES.

Context: Existing Population-Based Optimizers

To understand the uniqueness of GCO, the paper reviews several biology-inspired and evolutionary algorithms:

• Genetic Algorithm (GA): Based on Darwinian evolution involving natural selection, survival of the fittest, and reproduction.
• Particle Swarm Optimization (PSO): Simulates social behavior of bird flocks or fish schools. It relies on individuals imitating the success of neighbors to maintain optimal separation and cooperation.
• Ant Colony Optimization (ACO): Inspired by the foraging behavior of ants. It uses pheromone concentrations to mark and select the shortest paths between nests and food sources.
• Artificial Bee Colony (ABC): Emulates honey bee foraging roles (employed, onlooker, and scout bees) and the use of the "waggle dance" to communicate food source quality.
• Differential Evolution (DE): An extension of GA that uses a weighted difference of randomly chosen vectors to perturb a third vector, followed by crossover and selection.

Analogy Between Group Counseling and Optimization

The paper identifies 20 striking items of analogy that justify group counseling as a metaphor for optimization:

1. Problem Solving: Life problems correspond to computational problems.
2. Purpose of Solution: Seeking the best way out of a difficulty corresponds to seeking the optimum of an objective function.
3. Method of Solution: Counseling approaches vary by situation; optimization approaches are problem-dependent.
4. Increments of Solution: Counseling occurs in sessions; optimization occurs in iterations.
5. Work-force: A minisociety of members corresponds to a population of candidate vectors.
6. Role of Participants: Members alternate between counselors (givers) and counselees (receivers). Vectors contribute to/receive improvements.
7. Best Possible Participation: Members use best experiences; vectors use best positions.
8. Equal Chances: All participants/vectors are eligible to help and be helped.
9. Active Members: Only verbally active members participate in specific discussions; only a subset of vectors contributes based on probability.
10. Brainstorming: Combining good ideas into better ones corresponds to adding vector contributions in specific proportions.
11. General Terms to Details: Moving from general discussion to specifics corresponds to the shift from exploration to exploitation.
12. Thorough Investigation: Avoiding immature solutions vs. avoiding local optima/premature convergence.
13. Problem Complication: Multi-dimensional life problems (compromise) vs. multi-objective optimization (conflicting solutions).
14. Inferior Solution: Support to contain distress vs. suboptimum solutions when global optima are unavailable.
15. Dependence on Other Members: Solving problems via peer help vs. vector improvement via other vectors.
16. Self-dependence: Self-counseling after discovery vs. self-improvement components.
17. Remaining Unchanged: Persisting in a state to avoid worsening vs. keeping a vector if a change yields a worse fitness value.
18. Standardization: Professional norms vs. benchmark/rotated test functions.
19. Judgment: Evaluating group progress vs. evaluating objective function convergence.
20. Termination: Ending sessions vs. stopping criteria (e.g., maximum iterations).

Mathematical Framework of GCO

Problem Statement

For a single-objective, multivariate, unconstrained continuous problem, the goal is to optimize a scalar objective function $f(X)$ where: $X = (x_1, x_2, \dots, x_D)$$X^* = (x^*_1, x^*_2, \dots, x^*_D)$ Each component $x_d$ has a search range defined as $[x_{d,min}, x_{d,max}]$ with a range length: $\text{range length}_d = x_{d,max} - x_{d,min}$

Population Dynamics

A population of size $m$ candidate vectors is maintained: $X_i = (x_{i,1}, x_{i,2}, \dots, x_{i,D}), \, i = 1, 2, \dots, m$

Algorithmic Steps and Procedures

Step 1: Initialization

Vectors are initialized randomly using a Beta probability distribution: $\beta(x) = \frac{x^{a-1}(1 - x)^{b-1}}{B(a, b)}, \, 0 < x < 1$$B(a, b) = \int_0^1 t^{a-1}(1 - t)^{b-1}\,dt$ Shaping parameters are set to $a = b < 1$ (typically $0.1$), creating a symmetric U-shape to emphasize search near boundaries.

Step 2: Fitness Evaluation

Calculate $f(X_i)$ for all members.

Step 3: Iterative Solution Production

For each solution $X_i$, an alternative $X'_i$ is produced component-wise using one of two strategies, determined by the counseling decisive coefficient (cdc) (a random number $[0, 1]$) relative to the counseling probability (cp).

Strategy 3a: Other-members Counseling (Brainstorming)

If $\text{cdc} \le \text{cp}$, the component is updated via a weighted sum of $c$ random counselors: $x'_{i,d} = \sum_{q=1}^c w_q x_{d}^{\text{int rand}_q}$ Where weights sum to unity: $\sum_{q=1}^c w_q = 1$

Strategy 3b: Self-counseling (Modification)

If $\text{cdc} > \text{cp}$, the component is modified within a reduced range: $x'_{i,d} = x_{i,d} + \text{rand}_{i,d}(-mdf_{max,itr,d}, mdf_{max,itr,d})$$mdf_{max,d} = 0.5 \times \text{red} \times \text{range length}_d$$mdf_{max,itr,d} = mdf_{max,d} \left( 1 - \frac{\text{itr}}{\text{itr}_{max}} \right)^{tr}$ Where:

• red = search range reduction coefficient ($0$ to $1$).
• tr = transition rate from exploration to exploitation.
• If the result exceeds bounds, it is clamped back into the range logic.

Step 4: Selection and Judgment

If $f(X'_i)$ is better than $f(X_i)$, $X'_i$ replaces $X_i$. Otherwise, the original solution is kept. Steps 3 and 4 repeat until $itr_{max}$ is reached.

Benchmark Test Functions and Results

Function Categories

1. Unrotated Benchmark Functions: Simple unimodal (Sphere, Rosenbrock) and multimodal (Ackley, Griewank, Weierstrass, Rastrigin, Schwefel).
2. Rotated Benchmark Functions: Vectors are multiplied by an orthogonal rotation matrix $R$ ($Y = RX$) to eliminate coordinate-axis biases. These include rotated Ackley ($f_8$), Griewank ($f_9$), Weierstrass ($f_{10}$), Rastrigin ($f_{11}$), and Schwefel ($f_{12}$).
3. Composition Functions: High-complexity functions ($f_{13}$ to $f_{20}$) combining multiple basic functions with specific weights and biases.

Algorithmic Settings

• Dimensions ($D$): $30$.
• Population Size ($m$): $40$.
• Max Evaluations (FEs): $200,000$.
• Beta Parameters ($a, b$): $0.1$.

Comparative Performance Highlights

• GCO vs. CLPSO (Unrotated): GCO achieved zero-value error for Griewank ($f_4$), Weierstrass ($f_5$), and Rastrigin ($f_6$). In most cases, GCO converged faster than CLPSO.
• GCO vs. CLPSO (Rotated): GCO showed significant superiority on the Schwefel function ($f_{12}$). While CLPSO error was $1700 \pm 186$, GCO was only $212 \pm 292$ (a reduction to $12.5\%$ of CLPSO's mean error).
• GCO vs. NBIPOP-aCMA-ES (Composition): In function $f_{14}$, GCO mean error was only $8\%$ of the winner of the IEEE CEC-2013 competition.

Real-World Application: Spacecraft Trajectory Design

Optimization in aerospace aims to minimize propellant consumption or maximize payload push. GCO is applied to Multiple Gravity Assist (MGA) models.

Case 1: Cassini1 Mission

• Goal: Reach Saturn from Earth via Venus, Venus, Earth, and Jupiter.
• Objective: Minimize total $\Delta V$ (velocity change).
• Variables ($D=6$): Launch date ($t_0$), and flight times along five legs ($T_1, \dots, T_5$).
• Search Range ($t_0$): $[-1000, 0]$ MJD2000.
• Results (200,000 FEs):
  • Best Known Solution: $4.9307\,km/s$.
  • GCO Best Found: $4.9466\,km/s$.
  • GCO Average: $5.2983\,km/s$ (best among nine compared optimizers).

Case 2: GTOC1 Mission (Asteroid Deflection)

• Goal: Impact Asteroid TW229 to maximize deflection.
• Objective: Maximize $f(X) = m_f u_{rel} v_{ast}$.
• Variables ($D=8$): Launch date and seven flight times.
• Results (200,000 FEs):
  • Best Known Solution: $1,580,599\,kg\,km^2/s^2$.
  • GCO Best Found: $1,480,000\,kg\,km^2/s^2$ (highest among compared optimizers).
  • GCO Average: $1,131,267\,kg\,km^2/s^2$.

Conclusion

GCO stands as a robust human-behavior-inspired alternative to biological metaphors in computational optimization. By utilizing simple mathematical operations without complex derivatives, it demonstrates efficient convergence and high accuracy across standard benchmarks and significant engineering problems in aerospace trajectory design. Its performance is characterized by an effective balance between global exploration and local exploitation facilitated by the transition rate ($tr$) and Beta-distribution initialization.