A necessary condition for the guarantee of the superiorization method

Area of Science:

• Optimization Theory
• Applied Mathematics
• Numerical Analysis

Background:

• The superiorization method (SM) is an iterative technique combining feasibility-seeking with objective function value reduction.
• SM perturbs iterates of convergent feasibility algorithms using non-ascent steps to improve solution quality.
• Guarantees for SM's effectiveness in achieving superior objective values are not fully established.

Purpose of the Study:

• To investigate the conditions under which SM algorithms guarantee convergence to a superior feasible point.
• To identify specific circumstances where SM might fail to produce improved objective function values.
• To establish a foundational 'negative condition' crucial for future theoretical guarantees of SM.

Main Methods:

• Analysis of sequences generated by SM algorithms employing negative gradient descent for perturbations.
• Derivation of a specific condition that leads to a failure in achieving superior outcomes.
• Examination of the theoretical implications of this 'negative condition' for SM convergence proofs.

Main Results:

• A precise condition is identified under which an SM algorithm using negative gradient descent perturbations fails to yield a superior outcome.
• This 'negative condition' is significant as its inverse must hold for future SM guarantee results.
• The identified condition is practically relevant, as it can be avoided by practitioners to enhance SM's success rate.

Conclusions:

• The discovery of the 'negative condition' is a key step towards establishing robust theoretical guarantees for the superiorization method.
• Understanding and avoiding this condition is vital for practitioners aiming for successful real-world applications of SM.
• This research paves the way for more reliable and effective use of superiorization methods in optimization problems.