Inexact Projection Research Articles

Convergence properties of inexact projected gradient methods

In this article, we give a more comprehensive theoretical analysis of the inexact spectral projected gradient (ISPG) method introduced by Birgin et al. (Birgin, E.G., Martínez, J.M. and Raydan, M., 2003, Inexact spectral projected gradient methods on convex set. IMA Journal of Numerical Analysis, 23, 539–559) for the minimization of differentiable functions on closed convex sets. In doing so, we remove the boundedness of level sets of the objective function. Under weaker conditions, we establish the convergence theory. Moreover, we present a new inexact hybrid projection method based on the ISPG method. This new method has an encouraging convergence property which is that the whole sequence of iterates converges to a solution of the problem under no assumptions other than pseudoconvexity and continuous differentiability of f(·).

Inexact spectral projected gradient methods on convex sets

A new method is introduced for large-scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

Generalized Bregman Projections in Convex Feasibility Problems

We present a method for finding common points of finitely many closed convex sets in Euclidean space. The Bregman extension of the classical method of cyclic orthogonal projections employs nonorthogonal projections induced by a convex Bregman function, whereas the Bauschke and Borwein method uses Bregman/Legendre functions. Our method works with generalized Bregman functions (B-functions) and inexact projections, which are easier to compute than the exact ones employed in other methods. We also discuss subgradient algorithms with Bregman projections.

A Class of Trust Region Methods for Nonlinear Network Optimization Problems

We describe the results of a series of tests upon a class of new methods of trust region type for solving the nonlinear network optimization problem. The trust region technique considered is characterized by the use of the infinity norm and of inexact projections on the network constraints. The results are encouraging and show that this approach is particularly useful in solving large-scale nonlinear network optimization problems, especially when many bound constraints are expected to be active at the solution.

Global Convergence of a Class of Trust Region Algorithms for Optimization Using Inexact Projections on Convex Constraints

A class of trust region-based algorithms is presented for the solution of nonlinear optimization problems with a convex feasible set. At variance with previously published analyses of this type, the theory presented allows for the use of general norms. Furthermore, the proposed algorithms do not require the explicit computation of the projected gradient, and can therefore be adapted to cases where the projection onto the feasible domain may be expensive to calculate. Strong global convergence results are derived for the class. It is also shown that the set of linear and nonlinear constraints that are binding at the solution are identified by the algorithms of the class in a finite number of iterations.

A review of methods to forecast restorative treatment needs.

Decision makers in the areas of health policy, resource allocation, and manpower requirements rely implicitly on estimations of treatment needs on which to base their forecasts. The less specific the treatment-need estimate, the less precise the forecast. In previous decades, high caries rates were so prevalent that the dental profession could risk having inexact projections because overwhelming need and demand existed. However, rampant decay is no longer a common occurrence. Decay levels are declining in our nation's children and adults have fewer missing teeth. Therefore, restorative treatment needs and patterns in adult populations are transforming at a time when health care costs are spiraling and budget analysts at all levels of government are questioning the priority of continued support of dental care, dental education, and dental research at current levels. The purpose of this review is to present the existing methods of forecasting restorative treatment needs and to postulate the development of a new method based on the collective experiences of practicing dentists nationwide, an empirical method, to convert surface-specific oral health status data to restorative treatment need information. Need estimations based on empirical data would more accurately reflect the actual distribution of services that practicing dentists provide.

A relaxed version of Karmarkar's method

A relaxed version of Karmarkar's method is developed. This method is proved to have the same polynomial time complexity as Karmarkar's method and its efficient implementation using inexact projections is discussed. Computational results obtained using a preliminary implementation of the method are presented which indicate that the method is practicable.