Study On Nonsmooth Bundle Method For Semi-infinite Programming Problems

Semi-infinite programming (SIP) not only has been applied in economic balance, optimal control, information technology, engineering design, but also plays an important role in Cheby-shev approximation theory, robust optimization, fuzzy set and so on. Thus, the study of nu-merical algorithms on SIP have strong application value, and have aroused great enthusiasm of Scholars at home and abroad.It is well known that bundle methods are considered as one of the most efficient and promis-ing methods for solving nonsmooth optimization problems. Bundle methods have been devel-oped into various variants for special properties of different problems, and have been applied to many classic optimization problems, such as bilevel programming, chance constrained problem, min-max problem, equilibrium problem and so on. They are also applied to many practical areas for example, economy, mechanics, engineering and optimal control.This dissertation is devoted to study on nonsmooth numerical algorithms for SIP problem-s, including incremental bundle method with partial inexact oracle for nonsmooth convex SIP, an infeasible bundle method for nonsmooth nonconvex constrained optimization with applica-tion to SIP, a special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization. The main results of this dissertation are summarized as follows:1. Chapter 3 firstly presents a bundle method for solving the nonsmooth convex SIP prob-lems. The proposed algorithm is mainly based on a so-called "improvement function", "in-cremental idea" and "incomplete knowledge" of the constraints. It is well known that the main difficulty for solving SIPs lies in that there are infinitely many constraints. By using the improve-ment function, the constrained optimization can be rewritten as a unconstrained one. Besides, by using the incremental idea, it does not require all information of the constraints, but only one function value and one corresponding subgradient to update the bundle information and generate the search direction. The proposed algorithm, whenever a new stabilized center is obtained, re-quires an evaluation within some accuracy for the value of constraints. Thus the computational cost is significantly reduced. Under the EMFCQ, the global convergence of the method can be proved. Numerical experiments also show that our method is efficient for solving nonsmooth convex SIP problems.2. Chapter 4 presents an infeasible bundle method for nonsmooth nonconvex constrained optimization (NNCO) with application to to semi-infinite programming (SIP) problems. By us-ing a maximum function, the SIP problems can be rewritten as a class of nonconvex nonsmooth constrained optimization problems. Both functions in the problem are a class of special non-convex function, which is called the " lower-C2" function. The proposed method is based on the "improvement function" and "redistributed idea". The cutting-planes models in this method, are not of the objective function or its local convexification, but a local convexification of the improvement function. All the parameters are computed adaptively, and will be stabilized even-tually. Under the MFCQ, this method converges to the KKT points of the NNCO problems. Under the EMFCQ, this method converges to the KKT points of the SIP problems. Numerical results show that our method is robust and efficient for nonconvex nonsmooth examples and SIP problems.3. Chapter 5 study a special SIP problem, i.e., the nonconvex maximum eigenvalue opti-mization, and presents a special backtracking proximal bundle method for minimizing the prob-lem. The maximum eigenvalue optimization can be rewritten as a unconstrained SIP problem. Based on a so-called "Euler formula", we defined a better approximation of the composite ob-jective function, which is called the "conceptual model". In each iteration, we solve a certain quadratic programming problem in which the smooth inner mapping is replaced by its Taylor-series linearization around the current serious step. By using the backtracking test, we can make a better approximation of the objective function. The global convergence of the special bun-dle method has been proved without additional assumption. Numerical results demonstrate the efficiency of our algorithm on linear matrix problems, bilinear matrix problems and feedback control syntheses.