TY - JOUR

T1 - Worst-Case Complexity Bounds of Directional Direct-Search Methods for Multiobjective Optimization

AU - Custódio, Ana Luísa

AU - Diouane, Youssef

AU - Garmanjani, Rohollah

AU - Riccietti, Elisa

N1 - Funding support for Ana Luisa Custodio and Rohollah Garmanjani was provided by national funds through FCT-FundacAo para a Ciencia e a Tecnologia I. P., under the scope of Projects PTDC/MAT-APL/28400/2017 and UIDB/00297/2020.
Support for Elisa Riccietti was provided by TOTAL E&P.

PY - 2021/1

Y1 - 2021/1

N2 - 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.

AB - 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.

KW - Derivative-free optimization methods

KW - Directional direct-search

KW - Multiobjective unconstrained optimization

KW - Nonconvex smooth optimization

KW - Worst-case complexity

UR - http://www.scopus.com/inward/record.url?scp=85096451584&partnerID=8YFLogxK

U2 - 10.1007/s10957-020-01781-z

DO - 10.1007/s10957-020-01781-z

M3 - Article

AN - SCOPUS:85096451584

VL - 188

SP - 73

EP - 93

JO - Journal Of Optimization Theory And Applications

JF - Journal Of Optimization Theory And Applications

SN - 0022-3239

IS - 1

ER -