Nonlinear programming and variational inequality problems. A unified approach
The book proposes a unified approach of variational inequality problems and of nonlinear programming through the utilization of some algorithms from the same class which allow the author formulates a framework algorithm named ``cost approximation algorithm'' (shortly ``CA algorithm''). Let \(X\subseteq \mathbb{R}^n\) be a nonempty closed convex set, \(u:\mathbb{R}^n\to \mathbb{R}\cup \{+\infty\}\) be a lower semicontinuous proper and convex function and \(F: \text{dom }u\times X\to \mathbb{R}^n\) be a continuous vector-valued mapping. The main problem approached in this book is: ``\([\text{GVIP}(F, u,X)]\): Find a vector \(x^*\in \mathbb{R}^n\) such that \(F(x^*)+\partial u(x^*)+ N_X(x^*)\ni 0^n\)''. Here \(\partial u(x^*)\) is the Moreau subdifferential of \(u\) at \(x^*\), \(N_X(x^*)\) is the normal cone at \(X\) in \(x^*\) and \(0^n\) is the null element of \(\mathbb{R}^n\). This problem is known as a ``generalized variational inequality'', or as a ``nonlinear variational inequality'' or as a ``generalized equation''. This problem and its various special cases have a large variety of applications in the mathematical and engineering science. The book is divided into nine chapters. 1. Introduction (p. 1-38). Various specially cases of \([\text{GVIP}(F, u,X)]\) problem are presented, as: saddle point problem, nonlinear programming problem, penalty function problem, Nash equilibrium problem in games etc. It is described also the ``CA algorithm''. 2. Technical preliminaries (p. 39-56). The cases when the problem \([\text{GVIP}(F,u,X)]\) admits solutions are presented. The CA algorithm is used in the resolution of a variational subproblem, and lower and upper bounds of the arising error are estimated. 3. Instances of the cost approximation algorithm (p. 57-94). Some algorithms of CA type are presented, as: interior point algorithm, Jacobi and Gauss-Seidel methods, regularization and proximal point methods, decomposition-coordination methods, decomposition of optimization problems. 4. Merit functions for variational inequality problems (p. 95-134). A class of the merit functions is presented, properties and examples of merit functions are given, some cost approximating mapping and primal-dual variational inequalities are studied. 5. Convergence of the CA algorithm for nonlinear programs (p. 135-168). The convergence of the CA algorithm is studied under an exact line search, under the Armijo and modified Armijo rules, and under various step length rules. A non-monotone CA algorithm and linear convergence results are given, too. 6. Convergence of the CA algorithm for variational inequality problems (p. 169-190). Independent and dependent iterations of the cost approximation are studied. Non-descendent ACA methods, averaging schemes and ergodic sequences are given. 7. Finite identification of active constraints and of solutions (p. 191-210). 8. Parallel and sequential decomposition CA algorithms (p. 211-252). Synchronous and partially asynchronous aspects and the convergence of some algorithms of CA type as well as variational inequality problems over Cartesian product sets are studied. 9. A column generation/simplicial decomposition algorithm (p. 253-275). This subject approached here are the following ones: column generation approaches, the column generation CA algorithm, convergence for nonlinear programs and for variational inequalities. References (p. 283-324): over 800 titles.
scientific article; zbMATH DE number 1250530