In this paper, we are concerned with a worst-case complexity analysis of a-posteriori algorithms for unconstrained multiobjective optimization. Specifically, we propose an algorithmic framework that generates sets of points by means of $p$th-order models regularized with a power $p+1$ of the norm of the step. Through a tailored search procedure, several trial points are generated … Read more
Multiobjective binary quadratic programming refers to optimization problems involving multiple quadratic – potentially non-convex – objective functions and a feasible set that includes binary constraints on the variables. In this paper, we extend the well-established Quadratic Convex Reformulation technique, originally developed for single-objective binary quadratic programs, to the multiobjective setting. We propose a branch-and-bound algorithm … Read more
Many real-life optimization problems need to make decisions with discrete variables and multiple, conflicting objectives. Due to this need, the ability to solve such problems is an important and active area of research. We present a new algorithm, called Pareto Leap, for identifying the (weak) Pareto slices of biobjective mixed-integer programs (BOMIPs), even when Pareto … Read more
The concept of local upper bounds plays an important role for numerical algorithms in nonconvex, integer, and mixed-integer multiobjective optimization with respect to the componentwise partial ordering, that is, where the ordering cone is the nonnegative orthant. In this paper, we answer the question on whether and how this concept can be extended to arbitrary … Read more
Over the past two decades, descent methods have received substantial attention within the multiobjective optimization field. Nonetheless, both theoretical analyses and empirical evidence reveal that existing first-order methods for multiobjective optimization converge slowly, even for well-conditioned problems, due to the objective imbalances. To address this limitation, we incorporate curvature information to scale each objective within … Read more
Taking inspiration from what is commonly done in single-objective optimization, most local algorithms proposed for multiobjective optimization extend the classical iterative scalar methods and produce sequences of points able to converge to single efficient points. Recently, a growing number of local algorithms that build sequences of sets has been devised, following the real nature of … Read more
Nonlinear conjugate gradient methods have recently garnered significant attention within the multiobjective optimization community. These methods aim to maintain consistency in conjugate parameters with their single-objective optimization counterparts. However, the preservation of the attractive conjugate property of search directions remains uncertain, even for quadratic cases, in multiobjective conjugate gradient methods. This loss of interpretability of … Read more
Preconditioning is a powerful approach for solving ill-conditioned problems in optimization, where a preconditioning matrix is used to reduce the condition number and speed up the convergence of first-order method. Unfortunately, it is impossible to capture the curvature of all objective functions with a single preconditioning matrix in multiobjective optimization. Instead, second-order methods for multiobjective … Read more
Direct Multisearch (DMS) is a well-established class of methods for multiobjective derivative-free optimization, where constraints are addressed by an extreme barrier approach, only evaluating feasible points. In this work, we propose a filter approach, combined with an inexact feasibility restoration step, to address constraints in the DMS framework. The filter approach treats feasibility as an … Read more
Sandwiching algorithms, also known as Benson-type algorithms, approximate the nondominated set of convex bounded multiobjective optimization problems by constructing and iteratively improving polyhedral inner and outer approximations. Using a set-valued metric, an estimate of the approximation quality is determined as the distance between the inner and outer approximation. The convergence of the algorithm is evaluated … Read more