TRACING LOCALLY PARETO-OPTIMAL POINTS BY NUMERICAL INTEGRATION

We suggest a novel approach for the efficient and reliable approximation of the Pareto front of sufficiently smooth unconstrained biobjective optimization problems. Optimality conditions formulated for weighted sum scalarizations of the problem yield a description of (parts of) the Pareto front as a parametric curve, parameterized by the scalarization parameter (i.e., the weight in the weighted sum scalarization). Its sensitivity w.r.t. parameter variations can be described by an ordinary differential equation (ODE). Starting from an arbitrary initial Pareto-optimal solution, the Pareto front can then be traced by numerical integration. We provide an error analysis based on Lipschitz properties and suggest an explicit Runge-Kutta method for the numerical solution of the ODE. The method is validated and compared with a predictor-corrector method on biobjective convex quadratic programming problems and the biobjective test function ZDT3, for which the exact solution is explicitly known and numerically tested on complex biobjective shape optimization problems involving finite element discretizations of the state equation.

Automated summary

A novel approach is suggested for approximating the Pareto front of biobjective optimization problems by scalarizing the problem and tracing the front through numerical integration using an ODE. Error analysis based on Lipschitz properties and an explicit Runge-Kutta method are provided for numerical solution. The method is validated on convex quadratic programming and tested on a known biobjective function ZDT3 involving complex shape optimization problems.