On the convergence of critical points on real algebraic sets and applications to optimization
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| author | Basu, Saugata Mohammad-Nezhad, Ali |
| author_facet | Basu, Saugata Mohammad-Nezhad, Ali |
| contents | Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an infinitesimal perturbation $V_ξ = \zero(\{P_1-ξ_1,\ldots,P_s-ξ_s\},\R^n)$. Our main motivation is to understand the limiting behavior of local minimizers of the log-barrier function (and central paths) in polynomial optimization, whose existence plays a fundamental role, in theory and practice, for modern interior point methods. We establish different sets of conditions that ensure existence, finiteness, boundedness, and non-degeneracy of critical points of $F$ on $V_ξ$, respectively. These lead to new conditions for the existence, convergence, and smoothness of central paths of polynomial optimization and its extension to non-linear optimization problems involving definable sets and functions in an o-minimal structure. In particular, for non-linear programs defined by real globally analytic functions, our extension provides a stronger form of the convergence result obtained by Drummond and Peterzil. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the convergence of critical points on real algebraic sets and applications to optimization Basu, Saugata Mohammad-Nezhad, Ali Algebraic Geometry Optimization and Control 14P10, 90C23, 90C51 Let $F \in \R[X_1,\ldots,X_n]$ and the zero set $V=\zero(\mathcal{P},\R^n)$, where $\mathcal{P}:=\{P_1,\ldots,P_s\} \subset \R[X_1,\ldots,X_n]$ is a finite set of polynomials. We investigate existence of critical points of $F$ on an infinitesimal perturbation $V_ξ = \zero(\{P_1-ξ_1,\ldots,P_s-ξ_s\},\R^n)$. Our main motivation is to understand the limiting behavior of local minimizers of the log-barrier function (and central paths) in polynomial optimization, whose existence plays a fundamental role, in theory and practice, for modern interior point methods. We establish different sets of conditions that ensure existence, finiteness, boundedness, and non-degeneracy of critical points of $F$ on $V_ξ$, respectively. These lead to new conditions for the existence, convergence, and smoothness of central paths of polynomial optimization and its extension to non-linear optimization problems involving definable sets and functions in an o-minimal structure. In particular, for non-linear programs defined by real globally analytic functions, our extension provides a stronger form of the convergence result obtained by Drummond and Peterzil. |
| title | On the convergence of critical points on real algebraic sets and applications to optimization |
| topic | Algebraic Geometry Optimization and Control 14P10, 90C23, 90C51 |
| url | https://arxiv.org/abs/2506.20565 |