8,915 research outputs found
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
Multidimensional bisection: a dual viewpoint
AbstractThis paper provides an alternative viewpoint of multidimensional bisection global optimization methods of Wood. A dual coordinate representation of convex bodies is introduced which leads to an easy implementation and eliminates the need to see the geometry of intersecting simplexes. Although developed in the context of global optimization, the techniques deal more generally with regions represented as the union of convex bodies. With this dual framework the algorithm can be implemented efficiently using any multiattribute index data structure that allows for quick range queries. A C version using a “multi-key double linked skip list” based on Pugh's skip list has been implemented
A Modified Levenberg-Marquardt Method for the Bidirectional Relay Channel
This paper presents an optimization approach for a system consisting of
multiple bidirectional links over a two-way amplify-and-forward relay. It is
desired to improve the fairness of the system. All user pairs exchange
information over one relay station with multiple antennas. Due to the joint
transmission to all users, the users are subject to mutual interference. A
mitigation of the interference can be achieved by max-min fair precoding
optimization where the relay is subject to a sum power constraint. The
resulting optimization problem is non-convex. This paper proposes a novel
iterative and low complexity approach based on a modified Levenberg-Marquardt
method to find near optimal solutions. The presented method finds solutions
close to the standard convex-solver based relaxation approach.Comment: submitted to IEEE Transactions on Vehicular Technology We corrected
small mistakes in the proof of Lemma 2 and Proposition
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
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