Quasi-Newton Method for Set Optimization Problems with Set-Valued Mapping Given by Finitely Many Vector-Valued Functions
Debdas Ghosh, Anshika, Jen-Chih Yao, Xiaopeng Zhao
TL;DR
This work addresses unconstrained set optimization with a set-valued objective defined by a finite collection of vector-valued functions, ordered via a lower set less relation induced by a cone $K$. It introduces a quasi-Newton method that uses Hessian approximations via BFGS and a partition-based family of vector optimization subproblems to characterize and reach weakly minimal points, proving global convergence to stationary points and local superlinear convergence under stronger regularity. The algorithm computes a descent direction by solving a minimization of a Gerstewitz-based scalarization, uses an Armijo-type line search, and updates Hessian approximations for each component function; a key theoretical contribution is linking weakly minimal points to a stationary condition expressed through a continuity-preserving $\Phi$ functional. Numerical experiments on diverse test problems show that the proposed quasi-Newton method outperforms the existing steepest-descent approach in terms of iterations and CPU time, highlighting its practical impact for set-valued optimization tasks with finite vector-valued objectives.
Abstract
In this article, we propose a quasi-Newton method for unconstrained set optimization problems to find its weakly minimal solutions with respect to lower set-less ordering. The set-valued objective mapping under consideration is given by a finite number of vector-valued functions that are twice continuously differentiable. To find the necessary optimality condition for weak minimal points with the help of the proposed quasi-Newton method, we use the concept of partition and formulate a family of vector optimization problems. The evaluation of necessary optimality condition for finding the weakly minimal points involves the computation of the approximate Hessian of every objective function, which is done by a quasi-Newton scheme for vector optimization problems. In the proposed quasi-Newton method, we derive a sequence of iterative points that exhibits convergence to a point which satisfies the derived necessary optimality condition for weakly minimal points. After that, we find a descent direction for a suitably chosen vector optimization problem from this family of vector optimization problems and update from the current iterate to the next iterate. The proposed quasi-Newton method for set optimization problems is not a direct extension of that for vector optimization problems, as the selected vector optimization problem varies across the iterates. The well-definedness and convergence of the proposed method are analyzed. The convergence of the proposed algorithm under some regularity condition of the stationary points, a condition on nonstationary points, the boundedness of the norm of quasi-Newton direction, and the existence of step length that satisfies the Armijo condition are derived. We obtain a local superlinear convergence of the proposed method under uniform continuity of the Hessian approximation function.