# Optimization with Semidefinite, Quadratic and Linear Constraints

@inproceedings{Alizadeh1997OptimizationWS, title={Optimization with Semidefinite, Quadratic and Linear Constraints}, author={Farid Alizadeh and Stefan Schmieta}, year={1997} }

We consider optimization problems where variables have either linear, or convex quadratic or semideenite constraints. First, we deene and characterize primal and dual nondegeneracy and strict complementarity conditions. Next, we develop primal-dual interior point methods for such problems and show that in the absence of degeneracy these algorithms are numerically stable. Finally we describe an implementation of our method and present numerical experiments with both degenerate and nondegenerate…

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## 46 Citations

Optimization Over Symmetric Cones

- Mathematics
- 1999

We consider the problem of optimizing a linear function over the intersection of an affine space and a special class of closed, convex cones, namely the symmetric cones over the reals. This problem…

A polynomial primal-dual affine scaling algorithm for symmetric conic optimization

- MathematicsComput. Optim. Appl.
- 2017

The primal-dual Dikin-type affine scaling method is generalized to symmetric conic optimization using the notion of Euclidean Jordan algebras and is shown to be viable and robust when compared to SeDuMi, MOSEK and SDPT3.

Quadratic convergence to the optimal solution of second-order conic optimization without strict complementarity

- MathematicsOptim. Methods Softw.
- 2019

Under primal and dual nondegeneracy conditions, the quadratic convergence of Newton's method is established to the unique optimal solution of second-order conic optimization, which depends on the optimal partition of the problem, which can be identified from a bounded sequence of interior solutions.

A Continuation Method for the Linear Second-Order Cone Complementarity Problem

- MathematicsICCSA
- 2005

It is proved that the algorithm approximates an optimum of the linear second-order cone complementarity problem in finite steps under certain conditions and it is shown that the system of nonlinear equations of the reformulation is nonsingular at optimum undercertain conditions.

Quadratic convergence of Newton ’ s method to the optimal solution of second-order conic optimization

- Mathematics
- 2017

Under strict complementarity and primal and dual nondegeneracy conditions we establish the quadratic convergence of Newton’s method to the unique strictly complementary optimal solution of…

Handbook of Semidefinite Programming

- Computer Science
- 2000

Conditions and an accurate semidefinite programming solver are described in The Journal of the SDPA family for solving large-scale SDPs and in Handbook on Semidefinitely Programming.

Unifying Optimal Partition Approach to Sensitivity Analysis in Conic Optimization

- Computer Science
- 2004

A unifying geometric framework is presented that subsumes the concept of the optimal partition in linear programming and semidefinite programming and extends it to conic optimization and discusses briefly the properties of the optimize value function under such perturbations.

On implementing a primal-dual interior-point method for conic quadratic optimization

- Computer Science, MathematicsMath. Program.
- 2003

The main features of the implementation are it is based on a homogeneous and self-dual model, it handles rotated quadratic cones directly, it employs a Mehrotra type predictor-corrector extension and sparse linear algebra to improve the computational efficiency and it exploits fixed variables which naturally occurs in many conic Quadratic optimization problems.

Second-order cone programming

- Computer ScienceMath. Program.
- 2003

SOCP formulations are given for four examples: the convex quadratically constrained quadratic programming (QCQP) problem, problems involving fractional quadRatic functions, and many of the problems presented in the survey paper of Vandenberghe and Boyd as examples of SDPs can in fact be formulated as SOCPs and should be solved as such.

Solving symmetric indefinite systems in an interior-point method for second order cone programming

- Computer Science
- 2002

An augmented system approach is proposed to overcome the stability problems of the standard normal equation based implementation of IPM and numerical experiments show that the new approach can improve the stability.

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