An Interactive Interior Point Method for Multiobjective Nonlinear Programming Problems

Abstract

An interactive interior point algorithm for solving a
multiobjective nonlinear programming problem has been
proposed in this paper. The algorithm uses a single-objective
nonlinear variant based on both logarithmic barrier function and
Newton’s method in order to generate, at each iterate, interior
search directions. New feasible points are found along these
directions which will be later used for deriving bestapproximation
to the gradient of the implicitly-known utility
function at the current iterate. Using this approximate gradient,
a single feasible interior direction for the implicitly-utility
function could be found by solving a set of linear equations. It
may be taken an interior step from the current iterate to the next
one along this feasible direction. During the execution of the
algorithm, a sequence of interior points will be generated. It has
been proved that this sequence converges to an ε − optimal
solution, whereε is a predetermined error tolerance known a
priori. A numerical multiobjective example is illustrated using
this algorithm