Convexification of a Separable Function over a Polyhedral Ground Set
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2025
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| author | Dey, Santanu S. Kocuk, Burak |
| author_facet | Dey, Santanu S. Kocuk, Burak |
| contents | In this paper, we study the set $\mathcal{S}^κ= \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^κ, j=1,\dots,n\}$, where $κ> 1$ and the ground set $\mathcal{G}$ is a nonempty polytope contained in $[0,1]^n$. This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems from gas and water networks. Our aim is to obtain the convex hull of $\mathcal{S}^κ$ or its tight outer-approximation for the special case when the ground set $\mathcal{G}$ is the standard simplex. We propose power cone, second-order cone and semidefinite programming relaxations for this purpose, which are further strengthened by the Reformulation-Linearization Technique and the Reformulation-Perspectification Technique. For $κ=2$, we obtain the convex hull of $\mathcal{S}^κ$ in the low-dimensional setting. For general $κ$, we give approximation guarantees for the power cone representable relaxation, the weakest relaxation we consider. We prove that this weakest relaxation is tight with probability one as $n\to\infty$ when a uniformly generated linear objective is optimized over it. Finally, we provide the results of our extensive computational experiments comparing the empirical strength of several conic programming relaxations that we propose. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16595 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convexification of a Separable Function over a Polyhedral Ground Set Dey, Santanu S. Kocuk, Burak Optimization and Control In this paper, we study the set $\mathcal{S}^κ= \{ (x,y)\in\mathcal{G}\times\mathbb{R}^n : y_j = x_j^κ, j=1,\dots,n\}$, where $κ> 1$ and the ground set $\mathcal{G}$ is a nonempty polytope contained in $[0,1]^n$. This nonconvex set is closely related to separable standard quadratic programming and appears as a substructure in potential-based network flow problems from gas and water networks. Our aim is to obtain the convex hull of $\mathcal{S}^κ$ or its tight outer-approximation for the special case when the ground set $\mathcal{G}$ is the standard simplex. We propose power cone, second-order cone and semidefinite programming relaxations for this purpose, which are further strengthened by the Reformulation-Linearization Technique and the Reformulation-Perspectification Technique. For $κ=2$, we obtain the convex hull of $\mathcal{S}^κ$ in the low-dimensional setting. For general $κ$, we give approximation guarantees for the power cone representable relaxation, the weakest relaxation we consider. We prove that this weakest relaxation is tight with probability one as $n\to\infty$ when a uniformly generated linear objective is optimized over it. Finally, we provide the results of our extensive computational experiments comparing the empirical strength of several conic programming relaxations that we propose. |
| title | Convexification of a Separable Function over a Polyhedral Ground Set |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2510.16595 |