New Multiobjective Proximal Bundle Method with Scaled Improvement Function

Improvement functions are used in nonsmooth optimization both for constraint handling and scalarization of multiple objectives. In the multiobjective case the improvement function possesses, for example the nice property that a descent direction for the improvement function improves all the objectives of the original problem. However, the numerical experiments have shown that the standard improvement function is rather sensitive for scaling. For this reason we present here a new scaled version of the improvement function capable not only for linear but also for polynomial, logarithmic, and exponential scaling for both objective and constraint functions. In order to be convinced about the usability of the scaled improvement function, we develop a new version of the multiobjective proximal bundle method utilizing the scaled improvement function. This new method can be proved to produce weakly Pareto stationary solutions. In addition, under some generalized convexity assumptions the solutions are guaranteed to be globally weakly Pareto optimal. Furthermore, we illustrate the affect of the scaling with some numerical examples.