On the characterization of Pareto-optimal solutions in bicriterion optimization

Convexity and Differentiability of Controlled Risk

“…In fact, as noted by Geahart (1979), his algorithm will work even if we assume a substantially weaker property than convexity. Indeed, it is sufficient that for one of the criterion components every local minimum is a global minimum, whereas the other component is upper semicontinuous (see Theorem 3.3 in Geahart, 1979).…”

“…Indeed, it is sufficient that for one of the criterion components every local minimum is a global minimum, whereas the other component is upper semicontinuous (see Theorem 3.3 in Geahart, 1979).…”

“…We mention two algorithms that have been specially developed for problems with a two-component criterion (Geahart, 1979;Paine et al, 1975). The algorithm of Paine et al (1975) is designed for problems with continuously differentiable, but not necessarily convex functions; the algorithm of Geahart (1979) is designed for problems that have convexity properties, but are not necessarily differentiable.…”

“…We focus on two algorithms especially developed for problems with a two-dimensional criterion (Geahart, 1979;Paine et al, 1975). Geahart's (1979) algorithm utilizes the convexity properties of the problem, whereas the algorithm of Paine et al (1975) assumes continuous differentiability of the functions. Both algorithms produce Pareto-optimal solutions.…”

On the Structure of the Non-dominated Set for Bicriteria Programmes

“…Theoretical research on bicriteria optimization problems, initiated perhaps by Geoffrion (1967), was conducted by Pasternak and Passy (1 973), Benson (1979), Gearhart (1979) and others. Payne et al (1975) were among the first researchers to propose algorithms for solving bicriteria problems.…”