Rank-one convex functions on 2×2 symmetric matrices and laminates on rank-three lines

Conti, S.; Faraco, D.; Maggi, F.; Müller, S.
December 2005
Calculus of Variations & Partial Differential Equations;Dec2005, Vol. 24 Issue 4, p479
Academic Journal
We construct a function on the space of symmetric 2× 2 matrices in such a way that it is convex on rank-one directions and its distributional Hessian is not a locally bounded measure. This paper is also an illustration of a recently proposed technique to disprove L 1 estimates by the construction of suitable probability measures (laminates) in matrix space. From this point of view the novelty is that the support of the laminate, besides satisfying a convex constraint, needs to be contained on a rank-three line, up to arbitrarily small errors.


Related Articles

  • The space decomposition theory for a class of eigenvalue optimizations. Huang, Ming; Pang, Li-Ping; Xia, Zun-Quan // Computational Optimization & Applications;Jun2014, Vol. 58 Issue 2, p423 

    In this paper we study optimization problems involving eigenvalues of symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, considered as functions of a symmetric matrix, are not differentiable at those points where they coalesce. Here we...

  • Testing copositivity with the help of difference-of-convex optimization. Dür, Mirjam; Hiriart-Urruty, Jean-Baptiste // Mathematical Programming;Aug2013, Vol. 140 Issue 1, p31 

    We consider the problem of minimizing an indefinite quadratic form over the nonnegative orthant, or equivalently, the problem of deciding whether a symmetric matrix is copositive. We formulate the problem as a difference of convex functions problem. Using conjugate duality, we show that there is...

  • Characterization of quadratic growth of extended-real-valued functions. Wang, Jin; Song, Wen // Journal of Inequalities & Applications;2/1/2016, Vol. 2016 Issue 1, p1 

    This paper shows that the sharpest possible bound in the second-order growth condition of a proper lower semicontinuous function can be attained under some assumptions. We also establish a relationship among strong metric subregularity, quadratic growth, the positive-definiteness property of the...

  • One case of analytic integrability of a nonstationary matrix ordinary differential equation. Fokin, L.; Shchipitsyn, A. // Differential Equations;Sep2008, Vol. 44 Issue 9, p1343 

    We find the general solution of the equation for the error in the matrix of direction cosines of the form $$ \dot x $$ = αx + β, where α( t) is a skew-symmetric matrix with zero main diagonal, β( t) is the product of a skew-symmetric matrix by a matrix exponential, and α( t) and...

  • Global error bound for convex inclusion problems. Yiran He // Journal of Global Optimization;Nov2007, Vol. 39 Issue 3, p419 

    Abstract  The existence of global error bound for convex inclusion problems is discussed in this paper, including pointwise global error bound and uniform global error bound. The existence of uniform global error bound has been carefully studied in Burke and Tseng (SIAM J. Optim. 6(2),...

  • Large Deviations of Max-Weight Scheduling Policies on Convex Rate Regions. Subramanian, Vijay G. // Mathematics of Operations Research;Nov2010, Vol. 35 Issue 4, p881 

    We consider a single server discrete-time system with a fixed number of users where the server picks operating points from a compact, convex, and coordinate convex set. For this system, we analyse the performance of a stabilising policy that at any given time picks operating points from the...

  • SYMMETRIC DERIVATIVES AND MONOTONICITY. Szyszkowski, Marcin // Real Analysis Exchange;June2003 Conference Reports, Vol. 29, p183 

    Presents an analysis on symmetric derivatives and monotonicity.

  • Preface. Frangioni, A.; Overton, M.; Sagastizábal, C. // Mathematical Programming;Aug2013, Vol. 140 Issue 1, p1 

    An introduction is presented in which the editor discusses various reports within the issue on convex analysis.

  • ON THE LEAST SQUARES PROBLEM OF A MATRIX EQUATION. An-ping Liao // Journal of Computational Mathematics;Nov99, Vol. 17 Issue 6, p589 

    Focuses on the least squares problem of the matrix equation F=PG with respect to positive semidefinite symmetric P. Solvability condition for (P[sub1]); Expression of solution for (P[sub1]); Expression of solution for (P[sub2]).


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics