Functional Analysis and Optimization

In this monograph we develop the function space method for optimization problems and operator equations in Banach spaces. Optimization is the one of key components for mathematical modeling of real world problems and the solution method provides an accurate and essential description and validation of the mathematical model. Optimization problems are encountered frequently in engineering and sciences and have widespread practical applications.In general we optimize an appropriately chosen cost functional subject to constraints. For example, the constraints are in the form of equality constraints and inequality constraints. The problem is casted in a function space and it is important to formulate the problem in a proper function space framework in order to develop the accurate theory and the effective algorithm. Many of the constraints for the optimization are governed by partial differential and functional equations. In order to discuss the existence of optimizers it is essential to develop a comprehensive treatment of the constraints equations. In general the necessary optimality condition is in the form of operator equations. Non-smooth optimization becomes a very basic modeling toll and enlarges and enhances the applications of the optimization method in general. For example, in the classical variational formulation we analyze the non Newtonian energy functional and nonsmooth friction penalty. For the imaging/signal analysis the sparsity optimization is used by means of L1 and TV regularization. In order to develop an efficient solution method for large scale optimizations it is essential to develop a set of necessary conditions in the equation form rather than the variational inequality. The Lagrange multiplier theory for the constrained minimizations and nonsmooth optimization problems is developed and analyzed The theory derives the complementarity condition for the multiplier and the solution. Consequently, one can derive the necessary optimality system for the solution and the multiplier for a general class of nonsmooth and noncovex optimizations. In the light of the theory we derive and analyze numerical algorithms based the primal-dual active set method and the semi-smooth Newton method. The monograph also covers the basic materials for real analysis, functional analysis, Banach space theory, convex analysis, operator theory and PDE theory, which makes the book self-contained, comprehensive and complete. The analysis component is naturally connected to the optimization theory. The necessary optimality condition is in general written as nonlinear operator equations for the primal variable and Lagrange multiplier. The Lagrange multiplier theory of a general class of nonsmooth and ∗Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA