Rescaling and Asymptotic Acceleration in Unconstrained Quadratic Optimisation.

Area of Science:

• Optimization theory
• Numerical analysis
• Convex analysis

Background:

• Convex quadratic maps are fundamental in optimization.
• Existing methods may face convergence challenges.
• Rescaling techniques offer potential improvements.

Purpose of the Study:

• To investigate piecewise-fractional maps derived from rescaled convex quadratic maps.
• To analyze the optimization properties of these novel maps.
• To develop and evaluate accelerated minimization strategies.

Main Methods:

• Formulation of piecewise-fractional maps with rescaling invariance.
• Analysis of quasiconvexity and pseudoconvexity properties.
• Application of coordinate descent with gradient-based rules.
• Emphasis on maximum-alignment property in reproducing kernel Hilbert spaces.

Main Results:

• The derived maps exhibit quasiconvex and pseudoconvex properties.
• Exact line search is feasible along admissible directions.
• Coordinate descent with optimal rescaling demonstrates significant acceleration.
• Theoretical and empirical validation of the accelerated convergence.

Conclusions:

• Piecewise-fractional maps with rescaling invariance offer an effective optimization framework.
• Optimal rescaling of iterates is a key factor in accelerating unconstrained minimization.
• The proposed methods show promise for enhancing the efficiency of convex quadratic map optimization.