Abstract
Various Monte Carlo techniques for random point generation over sets of interest are widely used in many areas of computational mathematics, optimization, data processing, etc. Whereas for regularly shaped sets such sampling is immediate to arrange, for nontrivial, implicitly specified domains these techniques are not easy to implement. We consider the so-called Hit-and-Run algorithm, a representative of the class of Markov chain Monte Carlo methods, which became popular in recent years. To perform random sampling over a set, this method requires only the knowledge of the intersection of a line through a point inside the set with the boundary of this set. This component of the Hit-and-Run procedure, known as boundary oracle, has to be performed quickly when applied to economy point representation of many-dimensional sets within the randomized approach to data mining, image reconstruction, control, optimization, etc. In this paper, we consider several vector and matrix sets typically encountered in control and specified by linear matrix inequalities. Closed-form solutions are proposed for finding the respective points of intersection, leading to efficient boundary oracles; they are generalized to robust formulations where the system matrices contain norm-bounded uncertainty.
Highlights
One of the first issues in data mining and pattern recognition is an economy representation of implicitly specified massive data arrays with the subsequent extraction of specific features and classification [1,2,3]
We propose efficient computations of several BOs for various vector and matrix sets defined by linear matrix inequalities (LMIs)
For a wide range of problems, a boundary oracle can be formulated in closed form, and below we present particular boundary oracles for matrix sets specified by linear matrix inequalities
Summary
One of the first issues in data mining and pattern recognition is an economy representation of implicitly specified massive data arrays with the subsequent extraction of specific features and classification [1,2,3]. One of the efficient sampling procedures that are based on the availability of boundary oracles is the Hit-and-Run (HR) algorithm, which was proposed in [10] with the primary goal to facilitate numerical calculation of multi-dimensional integrals over convex domains. In [17], a comparison tool was proposed to quantify the performance of several commonly used Markov chain sampling techniques, and it turned out that the basic HR procedure outperforms other approaches and is less intensive computationally Those interested in the history of the HR algorithm are referred to [10,11,12]; later developments can be found in [18] and in [19,20], where a promising modification using barrier functions was proposed; of the most recent papers we mention [17,21,22]. We use the following acronyms: BO—Boundary Oracle; HR—Hit-and-Run; LMI—Linear Matrix inequality; LQR—Linear Quadratic Regulation; SDP—SemiDefinite Programming
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