# Status Determination by Interior-Point Methods for Convex Optimization Problems in Domain-Driven Form

## TL;DR

This paper develops a duality theory for convex optimization problems in Domain-Driven form, enabling rigorous status determination and certification, and analyzes an infeasible-start primal-dual algorithm with optimal complexity bounds.

## Contribution

It introduces a duality framework for Domain-Driven convex problems, extending conic methods to non-conic constraints, and analyzes an efficient primal-dual algorithm for these problems.

## Key findings

- Duality theory for Domain-Driven form established
- Iteration complexity bounds match conic formulations
- Practical stopping criteria proposed

## Abstract

We study the geometry of convex optimization problems given in a Domain-Driven form and categorize possible statuses of these problems using duality theory. Our duality theory for the Domain-Driven form, which accepts both conic and non-conic constraints, lets us determine and certify statuses of a problem as rigorously as the best approaches for conic formulations (which have been demonstrably very efficient in this context). We analyze the performance of an infeasible-start primal-dual algorithm for the Domain-Driven form in returning the certificates for the defined statuses. Our iteration complexity bounds for this more practical Domain-Driven form match the best ones available for conic formulations. At the end, we propose some stopping criteria for practical algorithms based on insights gained from our analyses.