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1. CJM 2014 (vol 67 pp. 90)

Bousch, Thierry
Une propriété 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)

Marshall, M.
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)

Penot, Jean-Paul
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)

Bauschke, Heinz H.; Güler, Osman; Lewis, Adrian S.; Sendov, Hristo S.
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)

Combari, C.; Poliquin, R.; Thibault, L.
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, Painlevé-Kuratowski convergence, primal-lower-nice functions, constraint qualification, slice convergence, graph convergence of subdifferentials, convexly composite functions
Categories:49A52, 58C06, 58C20, 90C30

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