A Reduced Jacobian Scheme with Full Convergence for Multicriteria Optimization

In this paper, we propose a variant of the reduced Jacobian method (RJM) introduced by El Maghri and Elboulqe in [JOTA, 179 (2018) 917–943] for multicriteria optimization under linear constraints. Motivation is that, contrarily to RJM which has only global convergence to Pareto KKT-stationary points in the classical sense of accumulation points, this new variant … Read more

A quasi-Newton method with Wolfe line searches for multiobjective optimization

We propose a BFGS method with Wolfe line searches for unconstrained multiobjective optimization problems. The algorithm is well defined even for general nonconvex problems. Global and R-linear convergence to a Pareto optimal point are established for strongly convex problems. In the local convergence analysis, if the objective functions are locally strongly convex with Lipschitz continuous … Read more

Pareto Robust Optimization on Euclidean Vector Spaces

Pareto efficiency for robust linear programs was introduced by Iancu and Trichakis. We generalize their approach and theoretical results to robust optimization problems in Euclidean spaces with affine uncertainty. Additionally, we demonstrate the value of this approach in an exemplary manner in the area of robust semidefinite programming (SDP). In particular, we prove that computing … Read more

A study of Liu-Storey conjugate gradient methods for vector optimization

This work presents a study of Liu-Storey (LS) nonlinear conjugate gradient (CG) methods to solve vector optimization problems. Three variants of the LS-CG method originally designed to solve single-objective problems are extended to the vector setting. The first algorithm restricts the LS conjugate parameter to be nonnegative and use a sufficiently accurate line search satisfying … Read more

Pareto Adaptive Robust Optimality via a Fourier-Motzkin Elimination Lens

We formalize the concept of Pareto Adaptive Robust Optimality (PARO) for linear Adaptive Robust Optimization (ARO) problems. A worst-case optimal solution pair of here-and-now decisions and wait-and-see decisions is PARO if it cannot be Pareto dominated by another solution, i.e., there does not exist another such pair that performs at least as good in all … Read more

Nonmonotone line searches for unconstrained multiobjective optimization problems

In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the … Read more

A barrier-type method for multiobjective optimization

For solving constrained multicriteria problems, we introduce the multiobjective barrier method (MBM), which extends the scalar-valued internal penalty method. This multiobjective version of the classical method also requires a penalty barrier for the feasible set and a sequence of nonnegative penalty parameters. Differently from the single-valued procedure, MBM is implemented by means of an auxiliary … Read more

Inexact scalarization proximal methods for multiobjective quasiconvex minimization on Hadamard manifold

In this paper we extend naturally the scalarization proximal point method to solve multiobjective unconstrained minimization problems, proposed by Apolinario et al.(2016), from Euclidean spaces to Hadamard manifolds for locally Lipschitz and quasiconvex vector objective functions. Moreover, we present a convergence analysis, under some mild assumptions on the multiobjective function, for two inexact variants of … Read more

An external penalty-type method for multicriteria

We propose an extension of the classical real-valued external penalty method to the multicriteria optimization setting. As its single objective counterpart, it also requires an external penalty function for the constraint set, as well as an exogenous divergent sequence of nonnegative real numbers, the so-called penalty parameters, but, differently from the scalar procedure, the vector-valued … Read more

Minimum cost subset selection with two competing agents

We address an optimization problem in which two agents, each with a set of weighted items, compete in order to minimize the total weight of their solution sets. The latter are built according to a sequential game consisting in a fixed number of rounds. In every round each agent submits one item that may be … Read more