Overview
This research introduces a methodology for addressing a class of fractional programs characterized by convex polynomial data. The proposed scheme integrates the Charnes–Cooper transformation with a high-degree perturbation method. This combination facilitates the construction of sums of squares (SOS) based relaxations for the specified fractional programs. The study examines properties including solution existence, strong duality, asymptotic convergence, and methods for solution extraction under defined conditions. Additionally, the framework is applied to fractional programs that feature SOS-convex polynomial data as a specific instance.
Research Context
The core of this work lies in fractional programming, a field of optimization concerned with the optimization of ratios of functions. The problem class specifically targets fractional programs where both the numerator and denominator are convex polynomials. The Charnes–Cooper transformation is a known technique for converting certain fractional programming problems into equivalent non-fractional forms, often linear or convex programs, which can simplify their solution. The introduction of the high-degree perturbation method further refines the approach, particularly in constructing sums of squares relaxations.
Approach
The proposed scheme operates on a foundational sequence involving the Charnes–Cooper transformation. This transformation is applied to the fractional programs under consideration. Following this, the high-degree perturbation method is employed. This method is instrumental in generating relaxations that are based on sums of squares. Such relaxations are a common technique in polynomial optimization, where a non-negative polynomial is represented as a sum of squares of other polynomials, converting a difficult non-convex problem into a more tractable semidefinite program.
Findings
Under specific, suitable conditions, the research establishes several key findings related to the class of fractional programs studied:
- Solution Existence: The methodology provides conditions under which solutions to the fractional programs are guaranteed to exist.
- Strong Duality: Strong duality results are demonstrated, indicating that the optimal value of the primal problem equals that of its dual counterpart. This is a significant finding for computational tractability and theoretical understanding.
- Asymptotic Convergence: The study indicates asymptotic convergence properties for the proposed scheme, implying that as parameters within the method are adjusted (e.g., degree of perturbation), the approximate solutions approach the true solutions.
- Solution Extraction: Mechanisms for extracting solutions from the constructed relaxations are detailed, which is crucial for practical implementation of the method.
A specialized application of this framework is also presented for fractional programs that utilize SOS-convex polynomial data. This suggests that the methodology is adaptable to specific structural properties of the polynomials involved, potentially simplifying analysis or improving performance in those cases.