Numerical methods for nonlinearly constrained optimization

Technical Report · Fri Oct 01 04:00:00 EDT 1976

OSTI ID:7117668

Wright, M

A detailed description of a new feasible-point algorithm for nonlinearly constrained optimization is presented. The new method is based on the properties of the trajectory of minima obtained by varying the barrier parameter of the logarithmic barrier function. A nonfeasible-point method based on the trajectory of the quadratic penalty function is also described in detail. The search direction in the trajectory algorithms is determined by solving a quadratic programing sub-problem whose objective function is based on an approximation to the Lagrangian function. This sub-problem is well-posed whatever the value of the penalty or barrier parameter. The step taken along the search direction is obtained by considering an appropriate reduction in the respective penalty or barrier function. For the quadratic penalty function, this step can be efficiently determined by a regular safeguarded linear search; for a barrier function, however, it is crucial to use specially designed linear searches. Two such special linear searches are derived for the logarithmic barrier function. A description is included of penalty and barrier function methods, and of some other methods which are based on the Lagrangian function. A comprehensive set of programs has been developed, and a selection of typical numerical results is presented. A primary concern was with practical algorithms, and the need to consider methods that will converge from even a poor initial estimate of the solution. Many ''sophisticated'' algorithms often critically depend on properties that hold only in a close neighborhood of the solution and, consequently, may perform poorly or even fail for any initial point not in such a region. The experimentation with methods has, therefore, been directed toward analyzing their general behavior, as well as asymptotic convergence rates. 23 figures.

• Research Organization: Stanford Univ., CA (USA)
• DOE Contract Number: EY-76-S-03-0326; EY-76-C-03-0515
• OSTI ID: 7117668
• Report Number(s): SU-326P30-48
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
FUNCTIONS
LAGRANGIAN FUNCTION
MATHEMATICAL LOGIC
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
OPTIMIZATION
SERIES EXPANSION