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1. CJM 2014 (vol 67 pp. 90)
Une propriÃ©tÃ© de domination convexe pour les orbites sturmiennes Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$-periodic sequence of integers
($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter
(and also $N$-periodic, consequently). It is shown that, for affine
functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively
to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$-periodic
sequences), the average of $|\alpha|$ on the orbit of ${\bf x}$ is
greater than its average on the orbit of ${\bf s}$.
Keywords:suite sturmienne, domination convexe, optimisation ergodique Categories:37D35, 49N20, 90C27 |
2. CJM 2009 (vol 61 pp. 205)
Representations of Non-Negative Polynomials, Degree Bounds and Applications to Optimization Natural sufficient conditions for a polynomial to have a local minimum
at a point are considered. These conditions tend to hold with
probability $1$. It is shown that polynomials satisfying these
conditions at each minimum point have nice presentations in terms of
sums of squares. Applications are given to optimization on a compact
set and also to global optimization. In many cases, there are degree
bounds for such presentations. These bounds are of theoretical
interest, but they appear to be too large to be of much practical use
at present. In the final section, other more concrete degree bounds
are obtained which ensure at least that the feasible set of solutions
is not empty.
Categories:13J30, 12Y05, 13P99, 14P10, 90C22 |
3. CJM 2004 (vol 56 pp. 825)
Differentiability Properties of Optimal Value Functions Differentiability properties of optimal value functions associated with
perturbed optimization problems require strong assumptions. We consider such
a set of assumptions which does not use compactness hypothesis but which
involves a kind of coherence property. Moreover, a strict differentiability
property is obtained by using techniques of Ekeland and Lebourg and a result
of Preiss. Such a strengthening is required in order to obtain genericity
results.
Keywords:differentiability, generic, marginal, performance function, subdifferential Categories:26B05, 65K10, 54C60, 90C26, 90C48 |
4. CJM 2001 (vol 53 pp. 470)
Hyperbolic Polynomials and Convex Analysis A homogeneous real polynomial $p$ is {\em hyperbolic} with respect to
a given vector $d$ if the univariate polynomial $t \mapsto p(x-td)$
has all real roots for all vectors $x$. Motivated by partial
differential equations, G{\aa}rding proved in 1951 that the largest
such root is a convex function of $x$, and showed various ways of
constructing new hyperbolic polynomials. We present a powerful new
such construction, and use it to generalize G{\aa}rding's result to
arbitrary symmetric functions of the roots. Many classical and recent
inequalities follow easily. We develop various convex-analytic tools
for such symmetric functions, of interest in interior-point methods
for optimization problems over related cones.
Keywords:convex analysis, eigenvalue, G{\aa}rding's inequality, hyperbolic barrier function, hyperbolic polynomial, hyperbolicity cone, interior-point method, semidefinite program, singular value, symmetric function Categories:90C25, 15A45, 52A41 |
5. CJM 1999 (vol 51 pp. 250)
Convergence of Subdifferentials of Convexly Composite Functions In this paper we establish conditions that guarantee, in the
setting of a general Banach space, the Painlev\'e-Kuratowski
convergence of the graphs of the subdifferentials of convexly
composite functions. We also provide applications to the
convergence of multipliers of families of constrained optimization
problems and to the generalized second-order derivability of
convexly composite functions.
Keywords:epi-convergence, Mosco convergence, PainlevÃ©-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions Categories:49A52, 58C06, 58C20, 90C30 |