August 17, 2025

Optimality Conditions at Infinity in Semialgebraic Vector Optimization

Key Points

Optimality conditions established at infinity indicate the behavior of solutions in semialgebraic vector optimization problems, with implications for theory and practice.
The study identifies cases where generalized nondominated points exist without being attained by weakly efficient solutions, emphasizing the complexity of vector optimization.
Exploration of cones linked to unbounded semialgebraic sets at infinity supports the development of optimality conditions, expanding existing frameworks in optimization theory.
Research contributes to a deeper understanding of the relationships between Newton polyhedra and constraint sets in vector optimization, suggesting avenues for future exploration.

Abstract

We are interested in optimality conditions for semialgebraic vector optimization problems with a situation in which generalized (weakly) nondominated points do exist but are not attained as image points of (weakly) efficient solutions. To this end, cones associated with unbounded semialgebraic sets at infinity are introduced and studied. Then, optimality conditions at infinity in terms of the Newton polyhedra of the objective mappings and of the cones associated with the constraint sets at infinity for the problems in question are proposed. Funding: L. Jiao was partially supported by the Chinese National Natural Science Foundation Grant 12371300. J. H. Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) Grant NRF-2021R1C1C2004488. T.-S. Pham was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) Grant 101.04-2023.06.

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