# No-gap second-order conditions under $n$-polyhedric constraints and finitely many nonlinear constraints

## TL;DR

This paper establishes second-order optimality conditions for constrained optimization problems using the concept of n-polyhedricity, under weak regularity assumptions, and minimizes the gap between necessary and sufficient conditions.

## Contribution

It introduces second-order optimality conditions under weak regularity assumptions using n-polyhedricity, reducing the gap between necessary and sufficient conditions.

## Key findings

- Derived necessary first and second order optimality conditions.
- Established sufficient optimality conditions with minimal gap.
- Applied the concept of n-polyhedricity to nonlinear constraints.

## Abstract

We consider an optimization problem subject to an abstract constraint and finitely many nonlinear constraints. Using the recently introduced concept of $n$-polyhedricity, we are able to provide second-order optimality conditions under weak regularity assumptions. In particular, we prove necessary optimality conditions of first and second order under the constraint qualification of Robinson, Zowe and Kurcyusz. Similarly, sufficient optimality conditions are stated. The gap between both conditions is as small as possible.