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We study the geometry of convex optimization problems given in aDomain-Driven form and categorize possible statuses of these problems usingduality theory. Our duality theory for the Domain-Driven form, which acceptsboth conic and non-conic constraints, lets us determine and certify statuses ofa problem as rigorously as the best approaches for conic formulations (whichhave been demonstrably very efficient in this context). We analyze theperformance of an infeasible-start primal-dual algorithm for the Domain-Drivenform in returning the certificates for the defined statuses. Our iterationcomplexity bounds for this more practical Domain-Driven form match the bestones available for conic formulations. At the end, we propose some stoppingcriteria for practical algorithms based on insights gained from our analyses.

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