Computing modified Newton directions using a partial Cholesky factorization

Technical Report · Mon Mar 01 04:00:00 EST 1993

DOI:https://doi.org/10.2172/10138165· OSTI ID:10138165

Forsgren, A ^[1]; Gill, P E ^[2]; Murray, W ^[3]

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1. Royal Inst. of Tech., Stockholm (Sweden). Dept. of Mathematics
2. California Univ., San Diego, La Jolla, CA (United States)
3. Stanford Univ., CA (United States). Systems Optimization Lab.

The effectiveness of Newton`s method for finding an unconstrained minimizer of a strictly convex twice continuously differentiable function has prompted the proposal of various modified Newton inetliods for the nonconvex case. Linesearch modified Newton methods utilize a linear combination of a descent direction and a direction of negative curvature. If these directions are sufficient in a certain sense, and a suitable linesearch is used, the resulting method will generate limit points that satisfy the second-order necessary conditions for optimality. We propose an efficient method for computing a descent direction and a direction of negative curvature that is based on a partial Cholesky factorization of the Hessian. This factorization not only gives theoretically satisfactory directions, but also requires only a partial pivoting strategy, i.e., the equivalent of only two rows of the Schur complement need be examined at each step.

• Research Organization: Stanford Univ., CA (United States). Systems Optimization Lab.
• Sponsoring Organization: USDOE, Washington, DC (United States); Department of Defense, Washington, DC (United States); National Science Foundation, Washington, DC (United States)
• DOE Contract Number: FG03-92ER25117
• OSTI ID: 10138165
• Report Number(s): SOL--93-1; ON: DE93009584; CNN: Grant DDM-9204208; Grant DDM-9204547; Grant N00014-90-J-1242
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200
FACTORIZATION
FUNCTIONS
MATHEMATICS AND COMPUTERS
MATRICES
NEWTON METHOD