# A numerically stable dual method for solving strictly convex quadratic programs

@article{Goldfarb1983ANS, title={A numerically stable dual method for solving strictly convex quadratic programs}, author={Donald Goldfarb and Ashok U. Idnani}, journal={Mathematical Programming}, year={1983}, volume={27}, pages={1-33} }

An efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point. Its implementation utilizes the Cholesky and QR factorizations and procedures for updating them. The performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated…

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## References

SHOWING 1-10 OF 51 REFERENCES

Numerically stable dual projection methods for solving positive definite quadratic programs

- Computer Science
- 1980

Tests on the use of this numerically stable DP method in Powell's algorithm on six published NLP problems produced a 45 to 70 percent reduction in the computational effort over that required by Fletcher's QP algorithm.

A Parametric Method for Semidefinite Quadratic Programs

- Mathematics
- 1969

A parametric method for solving semidefinite quadratic programs with a large number of constraints is described. All computations are performed by pivotal operations on a tableau, or more efficiently…

A Method of Solution for Quadratic Programs

- Mathematics
- 1962

This paper describes a method of minimizing a strictly convex quadratic functional of several variables constrained by a system of linear inequalities. The method takes advantage of strict convexity…

The Simplex and the Dual Method for Quadratic Programming

- Computer Science, Mathematics
- 1964

The paper presents a straightforward generalization of the Simplex and the dual method for linear programming to the case of convex quadratic programming. The two algorithms, called the Simplex and…

Numerically stable methods for quadratic programming

- Computer Science, MathematicsMath. Program.
- 1978

A new algorithm is described for indefinite quadratic programming which utilizes methods for updating positivedefinite factorizations only and can be used for the positive-definite case without loss of efficiency.

A stable method for solving certain constrained least squares problems

- Mathematics, Computer ScienceMath. Program.
- 1979

The algorithm generates a finite sequence of subproblems that are solved using the numerically stable technique of orthogonal factorization with reorthogonalization and Given's transformation updating.

The Gradient Projection Method for Nonlinear Programming. Part I. Linear Constraints

- Mathematics
- 1960

more constraints or equations, with either a linear or nonlinear objective function. This distinction is made primarily on the basis of the difficulty of solving these two types of nonlinear…

ON MINIMIZING A CONVEX FUNCTION SUBJECT TO LINEAR INEQUALITIES

- Mathematics
- 1955

SUMMARY THE minimization of a convex function of variables subject to linear inequalities is discussed briefly in general terms. Dantzig's Simplex Method is extended to yield finite algorithms for…