In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming ( SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e. g., ...
详细信息
In this paper, an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming ( SDP) problems is described and analyzed. Such nonlinear and nonconvex programs arise, e. g., in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional Hilbert space. The algorithm treats the abstract states and controls as independent variables. In particular, an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear SDP-condition, a positive definiteness condition, and a nonlinear matrix equation is considered. The algorithm is designed to take advantage of the structure of the problem. It is an extension of an interior point trust region method to nonlinear and nonconvex SDPs, with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration. Some convergence results are given, and, finally, several numerical examples demonstrate the applicability of the considered algorithm.
The Tikhonov regularization method for non-linear ill-posed problems requires us to globally solve non-convex optimization problem which have been very little studied in the inverse problems community. In this paper w...
详细信息
The Tikhonov regularization method for non-linear ill-posed problems requires us to globally solve non-convex optimization problem which have been very little studied in the inverse problems community. In this paper we suggest a method which is applicable to the Tikhonov method for a wide class of non-linear ill-posed problems. This is a class of problems when the Tikhonov functional for them can be represented by the difference of two convex functionals. Our method for these problems is a combination of the recently developed algorithm DCA in dc programming with the branch-and-bound techniques. (C) 2002 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.
In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certa...
详细信息
In this paper, new classes of generalized convex functions are introduced, extending the concepts of quasi-convexity, pseudoconvexity, and their associate subclasses. Functions belonging to these classes satisfy certain local-global minimum properties. Conversely, it is shown that, under some mild regularity conditions, functions for which the local-global minimum properties hold must belong to one of the classes of functions introduced.
We study the local convergence of a proximal point method in a metric space under the presence of computational errors. We show that the proximal point method generates a good approximate solution if the sequence of c...
详细信息
We study the local convergence of a proximal point method in a metric space under the presence of computational errors. We show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by some constant. The principle assumption is a local error bound condition which relates the growth of an objective function to the distance to the set of minimizers introduced by Hager and Zhang (SIAM J Control Optim 46:1683-1704, 2007).
This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447-3475, 2010), a method tha...
详细信息
This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447-3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhancements, such as functionality to avoid the computation of a normal step during every iteration. The implementation is included in the IPOPT software package paired with an iterative linear system solver and preconditioner provided in PARDISO. Numerical results on a large nonlinear optimization test set and two PDE-constrained optimization problems with control and state constraints are presented to illustrate that the implementation is robust and efficient for large-scale applications.
We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper a...
详细信息
We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper and the lower objectives are linear fractional. Our solution method adopts the vertex-searching approach on the polyhedron, and the search space is determined by the set of candidates of optimal base to the lower level problem. In order to search base, a modified enumerative scheme, that is a new upper bound filter scheme inserted into the classical enumerative scheme, is proposed. The main solution procedure is designed on solving a sequence of upper and lower level mathematical programs;instead of a single-level problem reformulation approach, which is popularly and widely used in literature. An extension on general upper level objective functions such as quasiconvex/quasiconcave for the proposed vertex-searching approach is discussed. Numerical experiments show that our algorithm leads us to a global optimum. We conclude that our proposed algorithm framework has the simplest solution procedure and has potential efficiency advantages, which may reduce the complexity of enumerative schemes for medium or large-scale problems, while comparing with existing global algorithms such as theKth-best algorithm (a two-level vertex-searching algorithm) and the single-level duality-based reformulation algorithm (a single-level vertex-searching algorithm).
We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on appl...
详细信息
We present a branch and bound algorithm for the global optimization of a twice differentiable nonconvex objective function with a Lipschitz continuous Hessian over a compact, convex set. The algorithm is based on applying cubic regularisation techniques to the objective function within an overlapping branch and bound algorithm for convex constrained global optimization. Unlike other branch and bound algorithms, lower bounds are obtained via nonconvex underestimators of the function. For a numerical example, we apply the proposed branch and bound algorithm to radial basis function approximations.
Determination of the Nash-equilibria in the bimatrix game was considered by reducing it to its equivalent nonconvex problem of optimization solved by the algorithm of global search based on the theory of global extrem...
详细信息
Determination of the Nash-equilibria in the bimatrix game was considered by reducing it to its equivalent nonconvex problem of optimization solved by the algorithm of global search based on the theory of global extremum for this problem. Efficiency of this method was illustrated by numerical solution of bimatrix games of sufficiently high dimensions.
Problems of control are considered, for which the maximum principle releases extremal modes and points of the set of accessibility without optimality guaranty. In the framework of technology of the confidence region a...
详细信息
Problems of control are considered, for which the maximum principle releases extremal modes and points of the set of accessibility without optimality guaranty. In the framework of technology of the confidence region and on the basis of modified optimality criteria, some approaches to the problem of search for and improvement of extremal controls are presented.
The problem of finding an extremum for a nonconvex function under convex constraints is considered. The original nonconvex function is replaced by an auxiliary one, called a smoothed function, which possesses some nic...
详细信息
The problem of finding an extremum for a nonconvex function under convex constraints is considered. The original nonconvex function is replaced by an auxiliary one, called a smoothed function, which possesses some nice properties. Operating with the smoothed function and the given convex constraints the global extremum of the original problem is found.
暂无评论