## Journals

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### Open Access Journals

JIMO

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.

NACO

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.

JIMO

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.

JIMO

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.

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