Asymmetric Long-Step Primal-Dual Interior-Point Methods with Dual Centering
Yurii Nesterov
TL;DR
This paper tackles conic optimization by introducing an asymmetric, dual-centric interior-point framework that prioritizes the dual barrier when it is simpler or smaller, enabling efficient long-step predictor-corrector methods. The core idea is to compute a dual-centered correction (Dual Gambit) and then generate a feasible primal-dual pair using a simple dual-to-primal scaling rule, with the Affine-Scaling Direction computed in a way that relies on Cholesky factorizations rather than matrix square roots. The approach yields iteration bounds that depend on the minimum barrier parameter between the primal and dual feasible sets, with particularly favorable behavior for symmetric cones and semidefinite problems where the dual barrier is substantially cheaper. Numerical experiments on Low-Rank Quadratic Interpolation indicate small predictor counts and practical performance approaching linear-optimization scales for certain SDP-like problems, supported by the theoretical framework that leverages self-concordant barriers and dual scaling. Overall, the work provides a principled, dual-aware pathway to scalable primal-dual IPMs with automatic adaptation to problem structure and barrier parameters, enabling efficient handling of SDO instances that are traditionally more challenging than LO problems.
Abstract
In this paper, we develop a new asymmetric framework for solving primal-dual problems of Conic Optimization by Interior-Point Methods (IPMs). It allows development of efficient methods for problems, where the dual formulation is simpler than the primal one. The problems of this type arise, in particular, in Semidefinite Optimization (SDO), for which we propose a new method with very attractive computational cost. Our long-step predictor-corrector scheme is based on centering in the dual space. It computes the affine-scaling predicting direction by the use of the dual barrier function, controlling the tangent step size by a functional proximity measure. We show that for symmetric cones, the search procedure at the predictor step is very cheap. In general, we do not need sophisticated Linear Algebra, restricting ourselves only by Cholesky factorization. However, our complexity bounds correspond to the best known polynomial-time results. Moreover, for symmetric cones the bounds automatically depend on the minimal barrier parameter between the primal or the dual feasible sets. We show by SDO-examples that the corresponding gain can be very big. We argue that the dual framework is more suitable for adjustment to the actual complexity of the problem. As an example, we discuss some classes of SDO-problems, where the number of iterations is proportional to the square root of the number of linear equality constraints. Moreover, the computational cost of one iteration there is similar to that one for Linear Optimization. We support our theoretical developments by preliminary but encouraging numerical results with randomly generated SDO-problems of different size.