First-Order Algorithms for Optimization Problems with a Maximum Eigenvalue/Singular Value Cost and or Constraints
Optimization problems with maximum eigenvalue or singular eigenvalue cost or constraints occur in the design of linear feedback systems, signal processing, and polynomial interpolation on a sphere. Since the maximum eigenvalue of a positive definite matrix Q(x) is given by max‖y‖=1〈(y, Q(x)y〉, we see that such problems are, in fact, semi-infinite optimization problems. We will show that the quadratic structure of these problems can be exploited in constructing specialized first-order algorithms for their solution that do not require the discretization of the unit sphere or the use of outer approximations techniques.
KeywordsAccumulation Point Support Point Maximum Eigenvalue Unique Minimizer Outer Approximation
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