{"ID":2884975,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.04991","arxiv_id":"2508.04991","title":"Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems","abstract":"In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative regularity conditions for vector optimization problems whose objective functions are a vector polynomial and investigate their properties and characterizations. Moreover, we establish relationships between the relative regularity conditions, Palais-Smale condition, weak Palais-Smale condition, M-tameness and properness with respect to some index set. Under the relative regularity and non-regularity conditions, we establish nonemptiness of the efficient solution sets of the polynomial vector optimization problems respectively. As a by-product, we infer Frank-Wolfe type theorems for a non-convex polynomial vector optimization problem. Finally, we study the local properties and genericity characteristics of the relative regularity conditions.","short_abstract":"In this paper, we establish the existence of the efficient solutions for polynomial vector optimization problems on a nonempty closed constraint set without any convexity and compactness assumptions. We first introduce the relative regularity conditions for vector optimization problems whose objective functions are a v...","url_abs":"https://arxiv.org/abs/2508.04991","url_pdf":"https://arxiv.org/pdf/2508.04991v1","authors":"[\"Danyang Liu\"]","published":"2025-08-07T02:54:13Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}