ETD

• bi-objective optimization
• bundle methods
• merit function
• nonmonotone algorithm
• nonsmooth convex optimization

This thesis introduces a bi-objective framework for bundle methods in nonsmooth convex optimization, based on the joint use of a lower and an upper model of the objective function. Classical stabilization strategies (proximal, trust-region, and level methods) are recovered through suitable choices of these models and appropriate rules for combining them into a unified objective. From this perspective, the framework strictly generalizes existing bundle approaches and clarifies their structural connections.
A key element is a merit function that measures model consistency and approximate optimality; its associated gap function provides an exact optimality certificate and enables a convergence analysis that avoids the traditional distinction between serious and null steps.
We further propose a new cut-aggregation mechanism derived from minimizing the merit function, yielding explicit aggregation parameters and reproducing classical step-acceptance logic through first-order optimality conditions. A complete convergence analysis is presented, proving monotonic decrease of the merit function and demonstrating that the generated iterates are minimizing.
Preliminary computational results indicate competitive performance compared to the standard Proximal Bundle Method, with the gap function closely tracking true relative accuracy and serving as a robust stopping criterion.