70,178 research outputs found
A comparative note on the relaxation algorithms for the linear semi-infinite feasibility problem
The problem (LFP) of finding a feasible solution to a given linear
semi-infinite system arises in different contexts. This paper provides
an empirical comparative study of relaxation algorithms for (LFP).
In this study we consider, together with the classical algorithm, imple-
mented with different values of the fixed parameter (the step size), a
new relaxation algorithm with random parameter which outperforms
the classical one in most test problems whatever fixed parameter is
taken. This new algorithm converges geometrically to a feasible so-
lution under mild conditions. The relaxation algorithms under com-
parison have been implemented using the Extended Cutting Angle
Method (ECAM) for solving the global optimization subproblems.Peer ReviewedPreprin
A comparative note on the relaxation algorithms for the linear semi-infinite feasibility problem
The problem (LFP) of finding a feasible solution to a given linear semi-infinite system arises in different contexts. This paper provides an empirical comparative study of relaxation algorithms for (LFP). In this study we consider, together with the classical algorithm, implemented with different values of the fixed parameter (the step size), a new relaxation algorithm with random parameter which outperforms the classical one in most test problems whatever fixed parameter is taken. This new algorithm converges geometrically to a feasible solution under mild conditions. The relaxation algorithms under comparison have been implemented using the extended cutting angle method for solving the global optimization subproblems.This research was partially supported by MICINN of Spain, Grant MTM2014-59179-C2-1-P and Sistema Nacional de Investigadores, Mexico
Nonnegative Matrix Inequalities and their Application to Nonconvex Power Control Optimization
Maximizing the sum rates in a multiuser Gaussian channel by power control is a nonconvex NP-hard problem that finds engineering application in code division multiple access (CDMA) wireless communication network. In this paper, we extend and apply several fundamental nonnegative matrix inequalities initiated by Friedland and Karlin in a 1975 paper to solve this nonconvex power control optimization problem. Leveraging tools such as the PerronâFrobenius theorem in nonnegative matrix theory, we (1) show that this problem in the power domain can be reformulated as an equivalent convex maximization problem over a closed unbounded convex set in the logarithmic signal-to-interference-noise ratio domain, (2) propose two relaxation techniques that utilize the reformulation problem structure and convexification by Lagrange dual relaxation to compute progressively tight bounds, and (3) propose a global optimization algorithm with Ï”-suboptimality to compute the optimal power control allocation. A byproduct of our analysis is the application of FriedlandâKarlin inequalities to inverse problems in nonnegative matrix theory
Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design
We describe an elementary algorithm to build convex inner approximations of
nonconvex sets. Both input and output sets are basic semialgebraic sets given
as lists of defining multivariate polynomials. Even though no optimality
guarantees can be given (e.g. in terms of volume maximization for bounded
sets), the algorithm is designed to preserve convex boundaries as much as
possible, while removing regions with concave boundaries. In particular, the
algorithm leaves invariant a given convex set. The algorithm is based on
Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial
optimization problems with the help of convex semidefinite programming
(optimization over linear matrix inequalities, or LMIs). We illustrate how the
algorithm can be used to design fixed-order controllers for linear systems,
following a polynomial approach
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