While evolutionary computing inspired approaches to multi-objective optimization have many advantages over conventional approaches; they generally do not explicitly exploit directional/gradient information. This can be inefficient if the underlying objectives are reasonably smooth, and this may limit the application of such approaches to real-world problems. This paper develops a local framework for such problems by geometrically analyzing the multiobjective concepts of descent, diversity and convergence/optimality. It is shown that locally optimal, multi-objective descent direction can be calculated that maximally reduce all the objectives and a local sub-space also exists that is a basis for diversity updates. Local convergence of a point towards the optimal Pareto set is therefore assured. The concept of a population of points is also considered and it is shown that it can be used to locally increase the diversity of the population while still ensuring convergence and a method to extract the local directional information from the population is also described. The paper describes and introduces the basic theoretical concepts as well as demonstrating how they are used on simple test problems.
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