Conic quadratic optimization is the problem of minimizing a linear function subject to the intersection of an affine set and the product of quadratic cones. The problem is a convex optimization problem and has numerous applications in engineering, economics, and other areas of science. Indeed, linear and convex quadratic optimization is a special case. Conic quadratic optimization problems can in theory be solved efficiently using interior-point methods. In particular it has been shown by Nesterov and Todd that primal-dual interior-point methods developed for linear optimization can be generalized to the conic quadratic case while maintaining their efficiency. Therefore, based on the work of Nesterov and Todd, we discuss an implementation of a primal-dual interior-point method for solution of large-scale sparse conic quadratic optimization problems. The main features of the implementation are it is based on a homogeneous and self-dual model, handles the rotated quadratic cone directly, employs a Mehrotra type predictor-corrector

In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry.

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Abstract. We demonstrate that if A 1 , ..., A m are symmetric positive semidefinite n ×n matrices with positive definite sum and A is an arbitrary symmetric n × n matrix, then the relative accuracy, in terms of the optimal value, of the semidefinite relaxationof the optimization programis not worse than 1 − 1 2 ln(2m 2 ) . It is shown that this bound is sharp in order, as far as the dependence on m is concerned, and that a feasible solution x to (P) withcan be found efficiently. This somehow improves one of the results of Nesterov [4] where bound similar to ( * ) is established for the case when all A i are of rank 1. Key words. semidefinite relaxations -quadratic programming IntroductionLet A i , i = 1, ..., m, be positive semidefinite n × n matrices with positive definite sum, and A be a n × n symmetric matrix. Consider the optimization problemThis problem, in general, is NP-hard (take, e.g., m = n and A i = e i e T i , where e i are the standard basic orths in R n ; then (P) becomes the problem of maximizing a homogeneous quadratic form over the unit cube, which is known to be NP-hard even in the case of positive semidefinite A).

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