Second-order analysis of penalty function
The authors study global exact penalty properties for general nonlinear programming problems. Global exact penalty properties are conditions under which every global minimum of the original problem is also a global minimum of the penalized problem. The global second-order sufficient conditions are similar to those in [\textit{X. Q. Yang}, Math. Program. 81, 327--347 (1998; Zbl 0919.90125)]. More precisely, the authors use a weak second-order sufficient condition, together with a generalized representation condition and prove that these conditions ensure the existence of a global exact penalty function. The representation condition requires the objective function and constraint functions to have a specific kind of second-order representation, and it is satisfied by quadratic or linear fractional functions, as well as by functions with a surjective Hessian. In the last section, the authors apply their results to several specially structured penalty functions for a general nonlinear programming problem and in particular to a nonconvex quadratic program with a two-sided quadratic constraint. For such a quadratic problem, the authors show that a global exact penalty function exists if the Hessian of the Lagrangian is positive definite. For a class of linearly fractionally constrained linear fractional programs with the same denominator, the authors obtain the existence of a global exact penalty function by using a first-order condition.
scientific article; zbMATH DE number 5814304
scientific article