Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908480904888320 |
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| author | Liu, Danyang |
| author_facet | Liu, Danyang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems Liu, Danyang Optimization and Control 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. |
| title | Existence of Solutions and Relative Regularity Conditions for Polynomial Vector Optimization Problems |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2508.04991 |