Projected Lagrangian algorithms for nonlinear minimax and l/sub 1/ optimization

Technical Report · Wed Aug 01 04:00:00 EDT 1979

OSTI ID:5784880

Overton, M L

This thesis presents new algorithms for solving the nonlinear minimax and l/sub 1/ optimization problems. The minimax problem often arises in circuit design and game theory, while solving the l/sub 1/ problem is useful in the robust analysis of experimental data. Both the minimax and l/sub 1/ problems are unconstrained optimization problems whose objective functions are not differentiable everywhere, and hence cannot be solved efficiently by standard techniques for unconstrained optimization. The minimax problem can be transformed into a nonlinearly constrained optimization problem with one extra variable, where the objective and constraint functions are continuously differentiable. The algorithm presented exploits the special structure of this equivalent problem. A direction of search is obtained at each iteration of the algorithm by solving an equality-constrained quadratic programing problem, related to one a projected Lagrangian method might use to solve the equivalent constrained optimization problem. Special Lagrange multiplier estimates are used to form an approximation to the Hessian of the Lagrangian function, which appears in the quadratic program. The direction of search is guaranteed to be a descent direction for the minimax objective function. Under mild conditions the algorithms are locally quadratically convergent if analytical Hessians are used. The l/sub 1/ problem can also be transformed into a nonlinearly constrained optimization problem, but it involves many extra variables. A method closely related to that for the minimax problem which requires successively solving quadratic programs in the same number of variables as that of the original problem is constructed. Several flexible steplength algorithms are presented, along with algorithms for univariate minimization of the minimax, l/sub 1/ and related functions.

• Research Organization: Stanford Univ., CA (USA). Dept. of Computer Science
• OSTI ID: 5784880
• Report Number(s): SU-326-P3067
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
990200* -- Mathematics & Computers
ALGORITHMS
ECONOMETRICS
ECONOMICS
FUNCTIONS
LAGRANGIAN FUNCTION
MATHEMATICAL LOGIC
MATHEMATICAL OPERATORS
NONLINEAR PROBLEMS
NONLINEAR PROGRAMMING
OPTIMIZATION
PROJECTION OPERATORS
SERIES EXPANSION