Abstract
The kernel functions play an important role in the amelioration of the computational complexity of algorithms. In this paper, we present a primal-dual interior-point algorithm for solving convex quadratic programming based on a new parametric kernel function. The proposed kernel function is not logarithmic and not self-regular. We analysis a large and small-update versions which are based on a new kernel function. We obtain the best known iteration bound for large-update methods, which improves signicantly the so far obtained complexity results. Thisresult is the rst to reach this goal.
Highlights
Primal-dual interior point methods based on a kernel function were studied extensively by many authors for linear optimization (LO)
We propose a primal-dual interior-point method for solving CQP based on a new parametric kernel function, this function is used for determining the new search directions and for measuring the distance between the given iterate and the center
We proposed a new kernel function consisting of a polynomial function in its barrier term defined by (6.1) for the primal-dual interior point methods for convex quadratic programming
Summary
Throughout the paper, we make the following assumptions:. (H1) The matrix A has full row rank (rank(A) = m < n). (H2) (P ) and (D) satisfy the interior-point condition (IPC ), i.e., there exist x0, y0, z0 such that: Ax0 = b, x0 > 0, Aty0 + z0 − Qx0 = c, z0 > 0. (H2) (P ) and (D) satisfy the interior-point condition (IPC ), i.e., there exist x0, y0, z0 such that: Ax0 = b, x0 > 0, Aty0 + z0 − Qx0 = c, z0 > 0. The basic idea of primal-dual IPMs is to replace the complementarity condition xz = 0 in (2.2) by the parameterized equation xz = μe, one obtains the following perturbed system: Ax = b, x ≥ 0, Aty + z − Qx = c, z ≥ 0, xz = μe,. The set of all μ-centers forms the so called central path for (P ) and (D). The principal idea of IPMs is to follow this central path and approach the optimal set of CQP as μ goes to zero. From a theoretical point of view, the IPC can be assumed without loss of generality.
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