Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization

• Published in: Journal of optimization theory and applications Vol. 188; no. 1; pp. 73 - 93
• Main Authors: Custódio, Ana Luísa, Diouane, Youssef, Garmanjani, Rohollah, Riccietti, Elisa
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-01-2021; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Applications of Mathematics; Calculus of Variations and Optimal Control; Optimization; Complexity; Computer Science; Engineering; Engineering Sciences; Mathematics; Mathematics and Statistics; Multiple objective analysis; Operations Research/Decision Theory; Optimization; Other; Theory of Computation; Multiobjective unconstrained optimization; Nonconvex smooth optimization; 90C30; 90C56; Derivative-free optimization methods; 90C29; Worst-case complexity; 90C60; Directional direct-search
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1007/s10957-020-01781-z

Direct Multisearch is a well-established class of algorithms, suited for multiobjective derivative-free optimization. In this work, we analyze the worst-case complexity of this class of methods in its most general formulation for unconstrained optimization. Considering nonconvex smooth functions, we show that to drive a given criticality measure below a specific positive threshold, Direct Multisearch takes at most a number of iterations proportional to the square of the inverse of the threshold, raised to the number of components of the objective function. This number is also proportional to the size of the set of linked sequences between the first unsuccessful iteration and the iteration immediately before the one where the criticality condition is satisfied. We then focus on a particular instance of Direct Multisearch, which considers a more strict criterion for accepting new nondominated points. In this case, we can establish a better worst-case complexity bound, simply proportional to the square of the inverse of the threshold, for driving the same criticality measure below the considered threshold.