  We consider the minimization of a continuous function over the intersection
of a regular cone with an affine set via a new class of adaptive first- and
second-order optimization methods, building on the Hessian-barrier techniques
introduced in [Bomze, Mertikopoulos, Schachinger, and Staudigl, Hessian barrier
algorithms for linearly constrained optimization problems, SIAM Journal on
Optimization, 2019]. Our approach is based on a potential-reduction mechanism
and attains a suitably defined class of approximate first- or second-order KKT
points with the optimal worst-case iteration complexity $O(\varepsilon^{-2})$
(first-order) and $O(\varepsilon^{-3/2})$ (second-order), respectively. A key
feature of our methodology is the use of self-concordant barrier functions to
construct strictly feasible iterates via a disciplined decomposition approach
and without sacrificing on the iteration complexity of the method. To the best
of our knowledge, this work is the first which achieves these worst-case
complexity bounds under such weak conditions for general conic constrained
optimization problems.
