A Large-update Primal-dual Interior-point Algorithm for Convex Quadratic Optimization Based on a New Bi-parameterized Bi-hyperbolic Kernel Function

Abstract

We present a polynomial-time primal-dual interior-point algorithm (IPA) for solving convex quadratic optimization (CQO) problems, based on a bi-parameterized bi-hyperbolic kernel function (KF). The growth term is a combination of the classical quadratic term and a hyperbolic one depending on a parameter \(p\in[0,1],\) while the barrier term is hyperbolic and depends on a parameter \(q\geq\frac{1}{2}\sinh 2.\) Using some simple analysis tools, we prove with a special choice of the parameter \(q,\) that the worst-case iteration bound for the new corresponding algorithm is \(\textbf{O}\big{(}\sqrt{n}\log n\log\frac{n}{\epsilon}\big{)}\) iterations for large-update methods. This improves the result obtained in (Optimization 70 (8), 1703–1724 (2021)) for CQO problems and matches the currently best-known iteration bound for large-update primal-dual interior-point methods (IPMs). Numerical tests show that the parameter \(p\) influences also the computational behavior of the algorithm although the theoretical iteration bound does not depends on this parameter. To our knowledge, this is the first bi-parameterized bi-hyperbolic KF-based IPM introduced for CQO problems, and the first KF that incorporates a hyperbolic function in its growth term while all KFs existing in the literature have a polynomial growth term exepct the KFs proposed in (Optimization 67 (10), 1605–1630 (2018)) and (J. Optim. Theory Appl. 178, 935–949 (2018)) which have a trigonometric growth term.