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.

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.

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.

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.