We show that efficient points can be obtained with the weighted norm scalarization under assumptions less restrictive than in the existing literature. For properly efficient points with a given trade-off bound, we provide an easy-to-compute lower bound on the norm parameter that allows the approximation of the desired points within a specified tolerance. We apply … Read more
This paper studies geometric and structural properties of quadratic multiobjective optimization problems. Thereby, a multiobjective optimization problem is called convexlike if the upper image, i.e., the image set plus the nonnegative orthant, is a convex set. Moreover, we say that a feasible point is supported in case it is a minimal solution of a weighted … Read more
Direct Multisearch (DMS) is a class of direct-search algorithms designed for multiobjective derivative-free optimization. Its framework consists of an optional search step and a poll step, the latter ensuring the corresponding theoretical convergence properties. Recently, a search strategy based on the minimization of quadratic polynomial models, constructed from previously evaluated points, was proposed to improve … Read more
In this paper, we develop a unified majorization-minimization scheme and convergence analysis with first-order surrogate functions for unconstrained vector optimization problems (VOPs). By selecting different surrogate functions, the unified method can be reduced to various existing first-order methods. The unified convergence analysis reveals that the slow convergence of the steepest descent method is primarily attributed … Read more
We propose an active-set algorithm for smooth multiobjective optimization problems subject to box constraints. The method works on one face of the feasible set at a time, treating it as a lower-dimensional region on which the problem simplifies. At each iteration, the algorithm decides whether to remain on the current face or to move to … Read more
We study the supportedness of nondominated points of multiobjective optimization problems, that is, whether they can be obtained via weighted sum scalarization. One key question is how supported points behave under an efficiency-preserving transformation of the original problem. Under a differentiability assumption, we characterize the transformations that preserve both efficiency and supportedness as the component-wise … Read more
A widely used approximation concept in multiobjective optimization is the concept of enclosures. These are unions of boxes defined by lower and upper bound sets that are used to cover optimal sets of multiobjective optimization problems in the image space. The width of an enclosure is taken as a quality measure. In this paper, we … Read more
Optimization problems that involve multiple, conflicting criteria lead to a set of efficient solutions, and when there are discrete decisions, some solutions may be unsupported. Applications where it is difficult to estimate the parameters for criteria motivate inverse optimization techniques. We provide a theoretical analysis of the set of (unknown) objective parameters which lead to … Read more
Achievement Scalarizing Functions (ASFs) are a class of scalarizing functions for multiobjective optimization problems that have been successfully implemented in many applications due to their mathematical elegance and decision making utility. However, no formal proofs of the fundamental properties of ASFs have been presented in the literature. Furthermore, developments of ASFs, including the construction of … Read more