Optimization of High-Dimensional Functions through Hypercube Evaluation

Area of Science:

• Computational Mathematics
• Optimization Theory

Background:

• Global numerical optimization presents significant computational challenges, especially for high-dimensional problems.
• Existing stochastic search methods often struggle with efficiency and scalability.

Purpose of the Study:

• To introduce a novel learning algorithm, the hypercube optimization (HO) algorithm, for solving global numerical optimization problems.
• To evaluate the performance of the HO algorithm on benchmark functions across a wide range of dimensions.

Main Methods:

• The hypercube optimization (HO) algorithm employs a stochastic search strategy based on hypercube evaluation and optimization.
• Key processes include initialization and evaluation, displacement-shrink, and searching space determination.
• Algorithms for each process were designed and implemented for testing.

Main Results:

• The HO algorithm was tested on benchmark functions with dimensions up to 10,000.
• Simulations demonstrated the algorithm's capability in handling both low and high-dimensional optimization tasks.
• Comparative analyses indicated competitive performance against other optimization approaches.

Conclusions:

• The proposed hypercube optimization (HO) algorithm is a viable and effective method for global numerical optimization.
• Its performance across diverse dimensionalities suggests broad applicability in computational mathematics and engineering.