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2026
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| Online Access: | https://doi.org/10.5281/zenodo.18765294 |
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| _version_ | 1866901720319131648 |
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| author | Zhang, Jincheng |
| author_facet | Zhang, Jincheng |
| contents | <p><span>This paper establishes a new optimization theoretical framework: the dual theory of stable polar families under uncertain cone structures. Unlike classical robust optimization, which only embeds uncertainty into the data layer (objective function or constraint coefficients), this paper embeds uncertainty into the geometric structure of the feasible region itself, constructing a parameterized closed convex cone family K(u). We propose the following structural model:</span></p> <p> </p> <p><span>min over x of sup over u in U of c(u)^T x</span></p> <p><span>subject to A(u)x = b(u)</span></p> <p><span>x in K(u)</span></p> <p> </p> <p><span>and systematically establish:</span></p> <p> </p> <p><span>(1) Theorem for the intersection closure representation of cone families</span></p> <p> </p> <p><span>(2) Theorem for the equivalence of union closure of polar cone families</span></p> <p> </p> <p><span>(3) Necessary and sufficient conditions for stable strong duality of cone families</span></p> <p> </p> <p><span>(4) Equivalent structure of geometric uncertainty and polar cone convexity</span></p> <p> </p> <p><span>This paper proves that: Worst-case cone constraints are geometrically equivalent to the closed convex hull structure of a polar cone family, and its dual cone is no longer a single polar cone, but a closed convex generating cone of the polar cone family. This result reveals that the essence of Worst-case optimization is the settling of polar structures</span>.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18765294 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Geometric Uncertainty in Conic Optimization: A Stable Duality Framework via Dual Cone Aggregation under Worst-Case Structures Zhang, Jincheng <p><span>This paper establishes a new optimization theoretical framework: the dual theory of stable polar families under uncertain cone structures. Unlike classical robust optimization, which only embeds uncertainty into the data layer (objective function or constraint coefficients), this paper embeds uncertainty into the geometric structure of the feasible region itself, constructing a parameterized closed convex cone family K(u). We propose the following structural model:</span></p> <p> </p> <p><span>min over x of sup over u in U of c(u)^T x</span></p> <p><span>subject to A(u)x = b(u)</span></p> <p><span>x in K(u)</span></p> <p> </p> <p><span>and systematically establish:</span></p> <p> </p> <p><span>(1) Theorem for the intersection closure representation of cone families</span></p> <p> </p> <p><span>(2) Theorem for the equivalence of union closure of polar cone families</span></p> <p> </p> <p><span>(3) Necessary and sufficient conditions for stable strong duality of cone families</span></p> <p> </p> <p><span>(4) Equivalent structure of geometric uncertainty and polar cone convexity</span></p> <p> </p> <p><span>This paper proves that: Worst-case cone constraints are geometrically equivalent to the closed convex hull structure of a polar cone family, and its dual cone is no longer a single polar cone, but a closed convex generating cone of the polar cone family. This result reveals that the essence of Worst-case optimization is the settling of polar structures</span>.</p> |
| title | Geometric Uncertainty in Conic Optimization: A Stable Duality Framework via Dual Cone Aggregation under Worst-Case Structures |
| url | https://doi.org/10.5281/zenodo.18765294 |