Error Bound and Reduced-Gradient Projection Algorithms for Convex Minimization over a Polyhedral Set

Consider the problem of minimizing, over a polyhedral set, the composition of an affine mapping with a strongly convex differentiable function. The polyhedral set is expressed as the intersection of an affine set with a (simpler) polyhedral set and a new local error bound for this problem, based on projecting the reduced gradient associated with the affine set onto the simpler polyhedral set, is studied. A class of reduced-gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence. Included in this class are the gradient projection algorithm of Goldstein and Levitin and Poljak, and an algorithm of Bertsekas. A new algorithm in this class, reminiscent of active set algorithms, is also proposed. Some of the results presented here extend to problems where the objective function is extended real valued and to variational inequality problems.

[1]  J. J. Moré.  Gradient projection techniques for large-scale optimization problems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[2]  Boris Polyak,et al.  Constrained minimization methods , 1966 .

[3]  D. Bertsekas,et al.  Projected Newton methods and optimization of multicommodity flows , 1982, 1982 21st IEEE Conference on Decision and Control.

[4]  S. M. Robinson.  Bounds for error in the solution set of a perturbed linear program , 1973 .

[5]  Dimitri P. Bertsekas,et al.  Data Networks , 1986 .

[6]  W. Tsai.  Convergence of gradient projection routing methods in an asynchronous stochastic quasi-static virtual circuit network , 1989 .

[7]  P. Tseng.  Descent methods for convex essentially smooth minimization , 1991 .

[8]  D. Bertsekas,et al.  Distributed asynchronous optimal routing in data networks , 1984, The 23rd IEEE Conference on Decision and Control.

[9]  J. Dunn.  Global and Asymptotic Convergence Rate Estimates for a Class of Projected Gradient Processes , 1981 .

[10]  Jong-Shi Pang,et al.  A Posteriori Error Bounds for the Linearly-Constrained Variational Inequality Problem , 1987, Math. Oper. Res..

[11]  J. Dunn.  On the convergence of projected gradient processes to singular critical points , 1987 .

[12]  D. Bertsekas,et al.  Projection methods for variational inequalities with application to the traffic assignment problem , 1982 .

[13]  S. M. Robinson.  Generalized equations and their solutions, part II: Applications to nonlinear programming , 1982 .

[14]  O. Mangasarian,et al.  Lipschitz continuity of solutions of linear inequalities, programs and complementarity problems , 1987 .

[15]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[16]  D. Bertsekas,et al.  TWO-METRIC PROJECTION METHODS FOR CONSTRAINED OPTIMIZATION* , 1984 .

[17]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[18]  M..  Projected Newton Methods and Optimization of Multicommodity Flows , 2022 .

[19]  D. Bertsekas.  Projected Newton methods for optimization problems with simple constraints , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[20]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[21]  A. Goldstein.  Convex programming in Hilbert space , 1964 .

[22]  Paul Tseng,et al.  Error Bound and Convergence Analysis of Matrix Splitting Algorithms for the Affine Variational Inequality Problem , 1992, SIAM J. Optim..

[23]  P. Tseng,et al.  On the linear convergence of descent methods for convex essentially smooth minimization , 1992 .

[24]  D. Bertsekas.  On the Goldstein-Levitin-Polyak gradient projection method , 1974, CDC 1974.

[25]  P. Tseng,et al.  On the convergence of the coordinate descent method for convex differentiable minimization , 1992 .

[26]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[27]  O. Mangasarian,et al.  Error bounds for strongly convex programs and (super)linearly convergent iterative schemes for the least 2-norm solution of linear programs , 1988 .

[28]  J. Dunn,et al.  Variable metric gradient projection processes in convex feasible sets defined by nonlinear inequalities , 1988 .