The Lack of Positive Definiteness in the Hessian in Constrained Optimization

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Abstract:

The use of the DFP or the BFGS secant updates requires the Hessian at the solution to be positive-definite. The second order sufficiency conditions insure the positive definiteness only in a sub-space of Rexp n. Conditions are given so the author can safely update with either update. The author proposes a new class of algorithms that generate a sequence converging 2-step q-superlinearly. He also proposes two specific algorithms. The first one converges q-superlinearly if the Hessian is positive-definite in Rexp n, and it converges 2-step q-superlinearly if the Hessian is positive-definite only in a subspace. The second one generates a sequence converging 1-step q-superlinearly. While the former costs one extra gradient evaluation, the latter costs one extra gradient evaluation and one extra function evaluation on the constraints.

Author(s):
Fontecilla, Rodrigo
Author Organization(s):
RICE UNIV HOUSTON TX DEPT OF MATHEMATICAL SCIENCES
Descriptive Note:
Technical rept.
Supplementary Note:
Presented at the SIAM National Meeting held in Denver, CO, in Jun 1983.
Pagination:
0036

Security Markings

DOCUMENT & CONTEXTUAL SUMMARY

Distribution:
Approved For Public Release
Distribution Statement:
Approved For Public Release; Distribution Is Unlimited.

RECORD

Collection: TR
Identifying Numbers
Report Number(s):
TR-83-17
Contract/Grant Number(s):
DE-AS05-82ERI-3016, DAAG29-83-K-0035
Monitor Series:
ARO
 Subject Terms
Joint Capability Areas:
JCA_8_Building Partnerships
Communities of Interest:
No COI(s) Identified
Descriptor(s):
*ALGORITHMS, *OPTIMIZATION, *MATRICES(MATHEMATICS), *CONVERGENCE, *ITERATIONS, *SEQUENCES(MATHEMATICS), *NONLINEAR PROGRAMMING, APPROXIMATION(MATHEMATICS), SYMMETRY, LAGRANGIAN FUNCTIONS, BANACH SPACE, NONLINEAR ALGEBRAIC EQUATIONS, PERTURBATIONS
Field(s)/Group(s):
Numerical Mathematics, Operations Research
Keyword(s):
*CONSTRAINED OPTIMIZATION, *Q SUPERLINEAR CONVERGENCE, *2 STEP ALGORITHMS, *HESSIANS, BFGS SECANT UPDATE, DFP SECANT UPDATE, QUASI NEWTON METHODS, NEWTON MULTIPLIER UPDATE, BROYDEN SECANT UPDATE, DQMM(DIAGONALIZED QUASI NEWTON MULTIPLIER METHOD), POSITIVE DEFINITE SECANT UPDATES, BANACH PERTURBATION LEMMA, MDQN(MODIFIED DIAGONALIZED QUASI NEWTON METHOD), IDQN(IMPROVED DIAGONALIZED QUASI NEWTON METHOD)
Report Date:
1983 Oct 01
Creation Date:
2006 Sep 27