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Ellipsoidal approximations of convex sets based on the volumetric barrier

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  • ANSTREICHER, Kurt M.

    (Department of Management Sciences, University of Iowa)

Abstract

Let C [included] R[exp.n] be a convex set. We assume that [norm.x][infinity]≤1for all x [belong] C; and that C contains a ball of radius 1/R. For x [belong] R[exp.n] , r [belong] R, and B an nxn positive definite matrix, let E (x,B,r) ={y|(y - x) [exp.t] B(y - x) ≤ r[exp.2]}. A [bêta] -rounding of C is an ellipsoid E(x, B, r/[bêta][included]C[included]E(x, B, r).In the case that C is characterized by a separation oracle, it is well known that an O(n[exp.3/2] ) -rounding of C can be obtained using the shallow cut ellipsoid method in O(n[exp.3] ln(nR)) oracle calls. We show that a modification of the volumetric cutting plane method obtains an O(n[exp.3/2]) -rounding of C in O(n2 ln(nR)) oracle calls. We also consider the problem of obtaining an O(n) -rounding of C when C has an explicit polyhedral description. Our analysis uses a new characterization of circumscribing ellipsoids centered at, or near, the volumetric center of a polyhedral set.

Suggested Citation

  • ANSTREICHER, Kurt M., 1997. "Ellipsoidal approximations of convex sets based on the volumetric barrier," LIDAM Discussion Papers CORE 1997045, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:1997045
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    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp1997.html
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