On the behaviour of the (1, λ)-ES for conically constrained linear problems

Area of Science:

• Optimization
• Evolutionary Computation
• Mathematical Programming

Background:

• Constrained optimization problems are common in various scientific and engineering fields.
• Existing evolutionary strategies often struggle with conically constrained feasible regions.
• Generalizing analysis beyond specific cone orientations is crucial for broader applicability.

Purpose of the Study:

• To analyze the behavior of an evolutionary strategy (ES) that resamples infeasible solutions for linear optimization.
• To generalize the analysis of such strategies to arbitrary orientations of conical constraints.
• To determine the convergence speed and characterize step size adaptation in this context.

Main Methods:

• Derivation of expressions for the strategy's single-step behavior.
• Utilizing a zeroth-order model with scale-invariant mutation strength adaptation.
• Deriving approximate expressions for average step size and convergence rate with cumulative step size adaptation.

Main Results:

• The study provides a generalized analysis of an ES for conically constrained linear optimization.
• Expressions characterizing single-step behavior and convergence speed were derived.
• The performance of cumulative step size adaptation was analyzed and compared to optimal values.

Conclusions:

• The developed evolutionary strategy effectively handles constraints in linear optimization problems with conical feasible regions.
• The analysis offers a theoretical framework for understanding and improving the performance of such strategies.
• The findings contribute to the development of more robust and efficient optimization algorithms.