P-Factor-Approach to Degenerate Optimization Problems
The paper describes and analyzes an application of the p-regularity theory to nonregular, (irregular, degenerate) nonlinear optimization problems. The p-regularity theory, also known as the factor-analysis of nonlinear mappings, has been developing successfully for the last twenty years. The p-factor-approach is based on the construction of a p-factor-operator, which allows us to describe and analyze nonlinear problems in the degenerate case.
First, we illustrate how to use the p-factor-approach to solve degenerate optimization problems with equality constraints, in which the Lagrange multiplier associated with the objective function might be equal to zero. We then present necessary and sufficient optimality conditions for a degenerate optimization problem with inequality constraints. The p-factor-approach is also used for solving mathematical programs with equilibrium constraints (MPECs). We show that the constraints are 2-regular at the solution of the MPEC. This property allows us to localize the minimizer independently of the objective function. The same idea is applied to some other nonregular nonlinear programming problems and allows us to reduce these problems to a regular system of equations without an objective function.
keywordsLagrange optimality conditions degeneracy p-regularity
- A. F. Izmailov, and A. A. Tret’yakov, Factor-analysis of nonlinear mappings, Nauka, Moscow, 1994 (in Russian).Google Scholar