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

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 is an element of [0, 1], while the barrier term is hyperbolic and depends on a parameter q >= 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 O(root n log n log n/epsilon) 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.