Studies On The Augmented Lagrangian Methods And Convexfication Methods For Constrained Global Optimization

Constrained global optimization is concerned with the following problem: Given a compact set S(?)R~n, and continuous function f : S→R, find a x~*(?)S such that f(x~*)≤f(x), x∈S, where S or f may be nonconvex. Due to the presence of nonconvexity in the problem, multiple local optimal solutions may exist and thus the traditional solution methods in nonlinear programming do not guarantee to find a global solution to the problem. The constrained global optimization has a wide range of applications in many important fields, including engineering, economics, finance, military and management science. It is an important and challenging research topic in modern optimization theory and methods.This thesis investigates augmented Lagrangian methods and convexification methods for constrained global optimization problems. We propose several new primal-dual method based on augmented Lagrangrian functions, develop and unify the convergence theory for augmented Lagrangian methods in the literature, and establish new convexification results for a class of nonsmooth monotone optimization problems. The results obtained in this thesis provide theoretical foundation for the use of augmented Lagrangian methods and convexification methods in constrained global optimization. The main contributions of the thesis are as follows:(1) We study the properties of saddle points of several types of augmented Lagrangian functions for constrained global optimization. First, we prove that under second-order sufficiency conditions, five types of augmented Lagrangian functions admit a local saddle point, without requiring the strictly complementary conditions. Second, we show that under the conditions that the set X is compact and the global solution of problem (P) is unique, the local saddle point of five augmented Lagrangian functions is also a global saddle point. We further prove the existence of global saddle points without assuming the uniqueness of the global solution of the primal problem. These results are then extended to a more general augmented Lagrangian function, which includes five important classes of augmented Lagrangian functions as special cases.(2) We present new convergence results of primal-dual methods based on ten types of augmented Lagrangian functions in the context of constrained global optimization. We show that under standard assumptions, every limit point of the sequences generated by the augmented Lagrangian methods is a global solution of the primal problem. We further show that under weaker conditions, the convergence to a global solution can be still achieved by modifying either the basic primal-dual scheme or the augmented Lagrangian functions.(3) We further investigate the global convergence properties of primal-dual methods based on five classes of augmented Lagrangians for inequality constrained optimization problems. Our main result is that under certain approximate conditions for the unconstrained augmented Lagrangian relaxation problems, the global convergence of the augmented Lagrangian methods can be obtained without requiring the boundedness of the multiplier sequence, namely, every limit point of the sequences generated by the augmented Lagrangian methods is a KKT point of the primal problem for any initial multiplier vector.(4) We present new results on global exact penalization of Di Pillo-Grippo's three types of augmented Lagrangian functions for constrained nonconvex optimization. First, we show that under second-order sufficiency conditions and uniqueness of the global solution, the global optimal solution of the original problem is also a global optimal solution of the three exact augmented Lagrangian functions. We then establish new convergence results of the exact augmented Lagrangian methods based on the three types of augmented Lagrangian functions.We show that under certain assumptions, every limit point of the sequences generated by the exact augmented Lagrangian methods is a global optimal solution of the original problem. Furthermore, we prove the exact penalization of the three augmented Lagrangian functions without appealing to the uniqueness of the global solution of the original problem.(5) We study the convexification method for a special class of constrained global optimization problems: nonsmooth monotone optimization problems. We prove that a semismooth monotone function can be converted into a convex function via certain convexification transformations. This result essentially generalizes the convexification schemes for smooth functions to nonsmooth functions, and extends the reach of convexification methods to nonsmooth situations. Finally, as an application of the generalized convexification result, we investigate the global saddle points of constrained global optimization by means of perturbation analysis.