A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization
Yanqin Bai Lipu Zhang
In this paper, we present a full-Newton step primal-dual interior-point algorithm for solving symmetric cone convex quadratic optimization problem, where the objective function is a convex quadratic function and the feasible set is the intersection of an affine subspace and a symmetric cone lies in Euclidean Jordan algebra. The search directions of the algorithm are obtained from the modification of NT-search direction in terms of the quadratic representation in Euclidean Jordan algebra. We prove that the algorithm has a quadratical convergence result. Furthermore, we present the complexity analysis for the algorithm and obtain the complexity bound as $\left\lceil 2\sqrt{r}\log\frac{\mu^0 r}{\epsilon}\right\rceil$, where $r$ is the rank of Euclidean Jordan algebras where the symmetric cone lies in.
keywords: complexity analysis. Euclidean Jordan algebras full-Newton step NT-search direction symmetric cone
An efficient algorithm for convex quadratic semi-definite optimization
Lipu Zhang Yinghong Xu Zhengjing Jin
We present a full-step interior-point algorithm for convex quadratic semi-definite optimization based on a simple univariate function. The algorithm uses the simple function to determine the search direction and define the neighborhood of central path. The full-step used in the algorithm has local quadratic convergence property according to the proximity function which is also constructed by the simple function. We derive the iteration complexity for the algorithm and obtain the best-known iteration bounds for convex quadratic semi-definite optimization.
keywords: complexity analysis. Convex quadratic semi-definite optimization interior point algorithm
A full-modified-Newton step infeasible interior-point algorithm for linear optimization
Yinghong Xu Lipu Zhang Jing Zhang
Based on an equivalent reformulation of the central path, we obtain a modified-Newton step for linear optimization. Using this step, we propose an infeasible interior-point algorithm. The algorithm uses only one full-modified-Newton step search in each iteration. The complexity bound of the algorithm is the best known for infeasible interior-point algorithm.
keywords: modified-Newton-direction linear optimization complexity analysis. Infeasible interior-point algorithm full-Newton step
The stable duality of DC programs for composite convex functions
Gang Li Lipu Zhang Zhe Liu

In this paper, we consider a composite DC optimization problem with a cone-convex system in locally convex Hausdorff topological vector spaces. By using the properties of the epigraph of the conjugate functions, some necessary and sufficient conditions which characterize the strong Fenchel-Lagrange duality and the stable strong Fenchel-Lagrange duality are given. We apply the results obtained to study the minmax optimization problem and $l_1$ penalty problem.

keywords: Strong Fenchel-Lagrange duality DC programming stable duality

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