The Hypervolume Indicator Hessian Matrix: Analytical Expression, Computational Time Complexity, and Sparsity

The problem of approximating the Pareto front a multiobjective optimization can be reformulated as finding set that maximizes hypervolume indicator. This paper establishes analytical expression Hessian matrix mapping from (fixed size) collection n points in d-dimensional decision space (or m dimensional objective space) to scalar indicator value. To define matrix, input is vectorized, and derived by differentiation vectorized plays crucial role second-order methods, such Newton-Raphson method, it used for verification local optimal sets. So far, full was only established analyzed relatively simple bi-objective case. will derive arbitrary dimensions ( $$m\ge 2$$ functions). For practically important three-dimensional case, we also provide an asymptotically efficient algorithm with time complexity $$O(n\log n)$$ exact computation Matrix’ non-zero entries. We establish sharp bound $$12m-6$$ number Also, general m-dimensional compact recursive established, its algorithmic implementation discussed. some sparsity results established; these are implied expression. validate illustrate analytically algorithms results, few numerical examples using Python Mathematica implementations. Open-source implementations testing data made available supplement this paper.