Interior-point algorithms for convex optimization based on primal-dual metrics

Abstract

We propose and analyse primal-dual interior-point algorithms for convex optimization problems in conic form. The families of algorithms we analyse are so-called shortstep algorithms and they match the current best iteration complexity bounds for primal-dual symmetric interior-point algorithm of Nesterov and Todd, for symmetric cone programming problems with given self-scaled barriers. Our results apply to any self-concordant barrier for any convex cone. We also prove that certain specializations of our algorithms to hyperbolic cone programming problems (which lie strictly between symmetric cone programming and general convex optimization problems in terms of generality) can take advantage of the favourable special structure of hyperbolic barriers. We make new connections to Riemannian geometry, integrals over operator spaces, Gaussian quadrature, and strengthen the connection of our algorithms to quasi-Newton updates and hence first-order methods in general.

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

“…Our approach (and in general interior-point methods) returns more robust certificates in provably stronger (polynomial) iteration complexity bounds compared to first-order methods such as Douglas-Rachford splitting [14], at the price of higher computational cost per iteration. However, as explained in [9], the quasi-Newton type ideas for deriving suitable primal-dual local metrics in [40,24] can be used to make our algorithm scalable, while preserving some primal-dual symmetry. The rest of the article covers:…”

“…Based on the insights we have gained by our performance analyses of the PtPCA algorithm in detecting the possible statuses for a given problem, we can discuss the stopping criteria and returned certificates by this algorithm in a practical setup. Even though applications of interiorpoint methods beyond the scope of symmetric cones have been studied [40,28,36,24,1], there is no well-stablished software close to optimization in the Domain-Driven from. Let us review the existing stopping criteria for some well-known optimization over symmetric cones solvers (using the formulation in ( 5)).…”

Certifying Polynomial Nonnegativity via Hyperbolic Optimization

“…As such, hyperbolic optimization problems can be solved using interior point methods as long as the polynomial p can be evaluated efficiently. More recently, other algorithmic approaches to solving hyperbolic optimization problems have been developed, including primal-dual interior point methods [MT14], affine scaling methods [RS14], firstorder methods based on applying a subgradient method to a transformation of the problem [Ren16] and accelerated modifications tailored for hyperbolic programs [Ren19].…”

Primal-Dual Interior-Point Methods for Domain-Driven Formulations

“…However, the applications and software for conic optimization itself have not gone much beyond optimization over symmetric (self-scaled) cones; more specifically linear programming (LP), second-order cone programming (SOCP), and semidefinite programming (SDP). Some of the desired properties of optimization over symmetric cones have been extended to general conic optimization [49,38,48,33]. While the conic reformulation implies that, under reasonable assumptions, all convex optimization problems enjoy the same iteration complexity bounds, there is a gap (remained unchanged for many years) between the efficiency and robustness of the software we have for optimization over symmetric cones and many other classes of problems.…”