# Complementarity and nondegeneracy in semidefinite programming

@article{Alizadeh1997ComplementarityAN, title={Complementarity and nondegeneracy in semidefinite programming}, author={Farid Alizadeh and Jean Pierre Haeberly and Michael L. Overton}, journal={Mathematical Programming}, year={1997}, volume={77}, pages={111-128} }

Primal and dual nondegeneracy conditions are defined for semidefinite programming. Given the existence of primal and dual solutions, it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution. The converses hold if strict complementarity is assumed. Primal and dual nondegeneracy assumptions do not imply strict complementarity, as they do in LP. The primal and dual nondegeneracy assumptions imply a range of possible ranks for…

## 272 Citations

Constraint Nondegeneracy, Strong Regularity, and Nonsingularity in Semidefinite Programming

- MathematicsSIAM J. Optim.
- 2008

This paper proves the equivalence between each of these conditions and the nonsingularity of Clarke's generalized Jacobian of the smoothed counterpart of this nonsmooth system used in several globally convergent smoothing Newton methods.

Equivalence of Two Nondegeneracy Conditions for Semidefinite Programs

- Mathematics
- 2007

Abstract
Nondegeneracy assumptions are often needed in order to prove the local fast convergence of suitable algorithms as well as in the sensitivity analysis for semidefinite programs. One of the…

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- MathematicsMath. Oper. Res.
- 1999

It is shown that the primal-dual affine-scaling algorithm using the NT direction for the same semidefinite programming problem always generates a sequence converging to the optimal solution.

Analyticity of weighted central paths and error bounds for semidefinite programming

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- 2008

It is shown that every Cholesky-based weighted central path for semidefinite programming is analytic under strict complementarity, and this result is applied to homogeneous cone programming to show that the central paths defined by the known class of optimal self-concordant barriers are analytic in the presence of strictly complementary solutions.

Primal-dual Aane-scaling Algorithms Fail for Semideenite Programming

- Mathematics
- 1997

In this paper, we give an example of a semide nite programming problem in which primaldual a ne-scaling algorithms using the HRVW/KSH/M, MT, and AHO directions fail. We prove that each of these…

Sensitivity of Solutions to Semidefinite Programs

- Mathematics
- 1997

Differential sensitivity of solutions to semidefinite programs are obtained by applying the implicit function theorem to a system of equations satisfied by the solutions. A certain Jacobian matrix…

Universal duality in conic convex optimization

- Mathematics, EconomicsMath. Program.
- 2007

The fact that the feasible sets of a primal convex program and its dual cannot both be bounded, unless they are both empty is related to universal duality.

A COMPARISON OF THREE NONDEGENERACY CONDITIONS FOR SEMIDEFINITE PROGRAMS

- Mathematics
- 2005

Nondegeneracy assumptions are often needed in order to prove local fast convergence of suitable algorithms as well as in the sensitivity analysis for semidefinite programs. Here we investigate the…

Local Duality of Nonlinear Semidefinite Programming

- MathematicsMath. Oper. Res.
- 2009

This paper introduces the dual SSOSC at a Karush-Kuhn-Tucker triple of NSDP and study its various characterizations and relationships to the primal nondegeneracy, and reveals that the nearest correlation matrix problem satisfies not only the primal and dualSSOSC but also the Primal and dual nondEGeneracy at its solution, suggesting that it is a well-conditioned QSDP.

Optimization with Semidefinite, Quadratic and Linear Constraints

- Mathematics
- 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…

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