Projected gradient methods for linearly constrained problems

Technical Report · Thu May 01 04:00:00 EDT 1986

OSTI ID:5424690

Calamai, P H; More, J J

The aim of this paper is to study the convergence properties of the projected gradient method and to apply these results to algorithms for linearly constrained problems. The main convergence result is obtained by defining a projected gradient, and proving that the gradient projection method forces the sequence of projected gradients to zero. A consequence of this result is that if the projected gradient method converges to a nondegenerate point of a linearly constrained problem, then the active and binding constraints are identified in a finite number of iterations. As an application to the theory, quadratic programming algorithms are developed that iteratively explore a subspace defined by the active constraints. These algorithms are able to drop and add many constraints from the active set, and can either compute an accurate minimizer by a direct method, or an approximate minimizer by an iterative method of the conjugate gradient type. Thus, these algorithms are attractive for large scale problems. It is shown that it is possible to develop a finitely terminating quadratic programming algorithm without non-degeneracy assumptions. 14 refs.

• Research Organization: Argonne National Lab., IL (USA). Mathematics and Computer Science Div.; Waterloo Univ., Ontario (Canada). Dept. of Systems Design Engineering
• DOE Contract Number: W-31109-ENG-38
• OSTI ID: 5424690
• Report Number(s): ANL/MCS-TM-73; ON: DE86014084
• Country of Publication: United States
• Language: English

Related Subjects

658000 -- Mathematical Physics-- (-1987)
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
CONVERGENCE
ITERATIVE METHODS
MATHEMATICAL LOGIC
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
OPTIMIZATION
PROGRAMMING