78 research outputs found
Analytic Formulas for Alternating Projection Sequences for the Positive Semidefinite Cone and an Application to Convergence Analysis
We find analytic formulas for the alternating projection method for the cone
of positive semidefinite matrices and an affine subspace. More
precisely, we find recursive relations on parameters representing a sequence
constructed by the alternating projection method. By applying the formulas, we
analyze the alternating projection method in detail and show that the upper
bound given by the singularity degree is actually tight when the alternating
projection method is applied to and a -plane whose
intersection is a singleton and has singularity degree .Comment: 26 page
Exact Convergence Rates of Alternating Projections for Nontransversal Intersections
We consider the convergence rate of the alternating projection method for the
nontransversal intersection of a semialgebraic set and a linear subspace. For
such an intersection, the convergence rate is known as sublinear in the worst
case. We study the exact convergence rate for a given semialgebraic set and an
initial point, and investigate when the convergence rate is linear or
sublinear. As a consequence, we show that the exact rates are expressed by
multiplicities of the defining polynomials of the semialgebraic set, or related
power series in the case that the linear subspace is a line, and we also decide
the convergence rate for given data by using elimination theory. Our methods
are also applied to give upper bounds for the case that the linear subspace has
the dimension more than one. The upper bounds are shown to be tight by
obtaining exact convergence rates for a specific semialgebraic set, which
depend on the initial points.Comment: 22 pages, 1 figur
Induced Norm Analysis of Linear Systems for Nonnegative Input Signals
This paper is concerned with the analysis of the induced norms of continuous-time linear systems where input signals
are restricted to be nonnegative. This norm is referred to as the
induced norm in this paper. It has been shown recently that the
induced norm is effective for the stability analysis of nonlinear feedback
systems where the nonlinearity returns only nonnegative signals. However, the
exact computation of the induced norm is essentially difficult. To get
around this difficulty, in the first part of this paper, we provide a
copositive-programming-based method for the upper bound computation by
capturing the nonnegativity of the input signals by copositive multipliers.
Then, in the second part of the paper, we derive uniform lower bounds of the
induced norms with respect to the standard
induced norms that are valid for all linear systems including
infinite-dimensional ones. For each linear system, we finally derive a
computation method of the lower bounds of the induced norm that are
larger than (or equal to) the uniform one. The effectiveness of the upper/lower
bound computation methods are fully illustrated by numerical examples.Comment: 12 pages, 3 figures. A preliminary version of this paper was
presented at ECC 2022 (arXiv:2401.03242) and IFAC WC 202
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