Iterative Algorithm for Triple-Hierarchical Constrained Nonconvex Optimization Problem and Its Application to Network Bandwidth Allocation

In this paper, we discuss a variational inequality problem with the gradient of a nonconvex objective function in which the constraint set composed of the absolute set and the subsidiary sets is not feasible. We formulate a compromise solution of the problem by using a solution to a variational inequality for the gradient over a subset of the absolute set with the elements closest to the subsidiary sets in terms of the norm. The objective of this paper is to enable us to discuss a network bandwidth allocation problem with a nonconcave utility function in which the compoundable constraints about the preferable transmission rate fall in the infeasible region. For such a bandwidth allocation, the ideal bandwidth must be quickly shared among traffic sources so as to satisfy the capacity constraints (absolute constraints) and all sources' demands regarding the transmission rate (subsidiary constraints) as much as possible. Iterative algorithms for the bandwidth allocation should thus not involve auxiliary opti...

[1]  Isao Yamada,et al.  Convex Feasibility Problem with Prioritized Hard Constraints : Double Layered Projected Gradient Method(Numerical Analysis and Optimization) , 2004 .

[2]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[3]  J. Borwein,et al.  Convex Analysis And Nonlinear Optimization , 2000 .

[4]  Z. Opial.  Weak convergence of the sequence of successive approximations for nonexpansive mappings , 1967 .

[5]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[6]  Jean C. Walrand,et al.  Fair end-to-end window-based congestion control , 2000, TNET.

[7]  R. Tyrrell Rockafellar,et al.  Variational Analysis , 1998, Grundlehren der mathematischen Wissenschaften.

[8]  Hideaki Iiduka,et al.  Fixed Point Optimization Algorithms for Network Bandwidth Allocation Problems with Compoundable Constraints , 2011, IEEE Communications Letters.

[9]  Rayadurgam Srikant,et al.  The Mathematics of Internet Congestion Control , 2003 .

[10]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[11]  Heinz H. Bauschke,et al.  Convex Analysis and Monotone Operator Theory in Hilbert Spaces , 2011, CMS Books in Mathematics.

[12]  Hideaki Iiduka,et al.  Iterative Algorithm for Solving Triple-Hierarchical Constrained Optimization Problem , 2011, J. Optim. Theory Appl..

[13]  Bethany L. Nicholson,et al.  Mathematical Programs with Equilibrium Constraints , 2021, Pyomo — Optimization Modeling in Python.

[14]  Isao Yamada,et al.  A Use of Conjugate Gradient Direction for the Convex Optimization Problem over the Fixed Point Set of a Nonexpansive Mapping , 2008, SIAM J. Optim..

[15]  Heinz H. Bauschke,et al.  On Projection Algorithms for Solving Convex Feasibility Problems , 1996, SIAM Rev..

[16]  Patrick L. Combettes,et al.  Hard-constrained inconsistent signal feasibility problems , 1999, IEEE Trans. Signal Process..

[17]  J. Hiriart-Urruty,et al.  Convex analysis and minimization algorithms , 1993 .

[18]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[19]  Patrick L. Combettes,et al.  A block-iterative surrogate constraint splitting method for quadratic signal recovery , 2003, IEEE Trans. Signal Process..

[20]  Frank Kelly,et al.  Charging and rate control for elastic traffic , 1997, Eur. Trans. Telecommun..

[21]  I. Yamada.  A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems , 2002 .

[22]  I. Yamada.  The Hybrid Steepest Descent Method for the Variational Inequality Problem over the Intersection of Fixed Point Sets of Nonexpansive Mappings , 2001 .