Optimization and Engineering, 4, 231–261, 2003 c  2003 Kluwer Academic Publishers. Manufactured in The Netherlands Minimum Effort Approximation of the Pareto Space of Convex Bi-Criteria Problems YUSHENG LI AND GEORGES M. FADEL Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA email: yli16@ford.com email: gfadel@ces.clemson.edu MARGARET WIECEK ∗ Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA email: wmalgor@clemson.edu VINCENT Y. BLOUIN Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA email: vblouin@clemson.edu Received October 10, 2000; Revised September 23, 2002 Abstract. One of the major issues facing design practitioners using multiple-criteria optimization problems is how to decide on the trade-off between the various objectives. Since the Pareto set of those problems gives the ability to visualize the trade-off in the objective space, based on the Pareto set, a decision maker could conceivably make trade-off decisions without repeatedly solving the problem. The Pareto set of bi-criteria problems is a curve. To generate that Pareto curve, the problem needs to be solved many times. This may not be feasible for some computationally intensive applications. One way to avoid this issue is to approximate the Pareto curve using partial information from the Pareto set. In this paper, the hyper- ellipse is used to approximate the entire Pareto set of convex bi-criteria optimization problems. The approximation is achieved by means of fitting a hyper-ellipse to a minimum number of Pareto points and the equation of the hyper-ellipse yields an explicit analytical description of the Pareto set. The method is progressively applied to unconstrained and constrained problems to understand its behavior, and illustrated on a structural example to show its efficiency. The paper identifies the limits of applicability of the approach and proposes further extensions. Keywords: bi-criteria optimization, Pareto set, approximation, hyper-ellipse 1. Introduction Many researchers have recognized that engineering design involves multiple and often conflicting criteria or objectives (e.g., mass, stiffness, stress, deformation, stability in a structural problem) and should be treated in a multi-criteria framework (Eschenauer, 1992; Stadler, 1995). The solution set of a multi-criteria design problem usually includes a large number of points, which are referred to as Pareto (efficient, non-dominated) decisions (solutions). Trading one Pareto decision for another results in improvement of at least one ∗ Corresponding author.