Adaptive Weighted Sum Method for Multiobjective Optimization

This paper presents an adaptive weighted sum method for multiobjective optimization problems. The authors developed the bi-objective adaptive weighted sum method, which determines uniformly-spaced Pareto optimal solutions, finds solutions on non-convex regions, and neglects non-Pareto optimal solutions. However, the method could solve only problems with two objective functions. In this work, the bi-objective adaptive weighted sum method is extended to problems with more than two objective functions. In the first phase, the usual weighted sum method is performed to approximate Pareto surfaces quickly, and a mesh of Pareto front patches is identified. Each Pareto front patch is then refined by imposing additional equality constraints that connect the pseudo nadir point and the expected Pareto optimal solutions on a piecewise planar surface in the objective space. It is demonstrated that the method produces a well-distributed Pareto front mesh for effective visualization and finds solutions in non-convex regions. Two numerical examples and a simple structural optimization problem are solved as case studies.