Numerical certification of Pareto optimality for biobjective nonlinear problems

The solution to a biobjective optimization problem is composed of a collection of trade-off solution called the Pareto set. Based on a computer assisted proof methodology, the present work studies the question of certifying numerically that a conjectured set is close to the Pareto set. Two situations are considered. First, we analyze the case where the conjectured set is directly provided: one objective is explicitly given as a function of the other. Second, we analyze the situation where the conjectured set is parameterized: both objectives are explicitly given as functions of a parameter. In both cases, we formulate the question of verifying that the conjectured set is close to the Pareto set as a global optimization problem. These situations are illustrated on a new class of extremal problems over convex polygons in the plane. The objectives are to maximize the area and perimeter of a polygon with a fixed diameter, for a given number of sides.