Name
Abstract | ||
|---|---|---|
Certain matrix relationships play an important role in optimality conditions and algorithms for nonlinear and semidefinite programming. Let H be an n × n symmetric matrix, A an m × n matrix, and Z a basis for the null space of A. (In a typical optimization context, H is the Hessian of a smooth function and A is the Jacobian of a set of constraints.) When the reduced Hessian ZTHZ is positive definite, augmented Lagrangian methods rely on the known existence of a finite $\bar\rho\ge 0$ such that, for all $\rho \bar\rho$, the augmented Hessian $H + \rho \ATA $ is positive definite. In this note we analyze the case when ZTHZ is positive semidefinite, i.e., singularity is allowed, and show that the situation is more complicated. In particular, we give a simple necessary and sufficient condition for the existence of a finite $\bar\rho$ so that $H + \rho \ATA$ is positive semidefinite for $\rho \ge \bar\rho$. A corollary of our result is that if H is nonsingular and indefinite while ZTHZ is positive semidefinite and singular, no such $\bar\rho$ exists. |
Year | DOI | Venue |
|---|---|---|
2000 | 10.1137/S1052623499351791 | SIAM Journal on Optimization |
Keywords | Field | DocType |
important role,augmented hessian,known existence,augmented lagrangian method,n matrix,reduced hessian zthz,reduced hessian,semidefinite programming,certain matrix relationship,n symmetric matrix,positive semidefinite,augmented lagrangian methods,inertia | Hessian equation,Mathematical optimization,Jacobian matrix and determinant,Matrix (mathematics),Positive-definite matrix,Hessian matrix,Symmetric matrix,Invertible matrix,Semidefinite programming,Mathematics | Journal |
Volume | Issue | ISSN |
11 | 1 | 1052-6234 |
Citations | PageRank | References |
11 | 1.48 | 3 |
Authors | ||
2 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Kurt M. Anstreicher | 1 | 633 | 86.40 |
| Margaret H. Wright | 2 | 1233 | 182.31 |