SOLVING CONVEX PROBLEMS INVOLVING POWERS USING CONIC OPTIMIZATION AND A NEW SELF-CONCORDANT BARRIER The first part of this talk will introduce a common framework unifying several classes of convex optimization problems involving powers, including linear programming, second-order cone programming, quadratically constrained convex quadratic programming, $l_p$ programming, minimization of sums of Euclidean or $p$-norms, geometric programming and entropy programming. Any of these problems can be modelled as a conic optimization problem, where all the cones used in the formulation belong to a single family of three-dimensional self-dual convex cones, defined as \[ K_\alpha = \left\{ (x, y, z) \in \mathbb{R}_+ \times \mathbb{R}_+ \times \mathbb{R} \mid x^\alpha y^{1-\alpha} \ge |z| \right\} \] where $\alpha$ is a real parameter between $0$ and $1$. The second part will deal with algorithms to solve these problems. We rely on interior-point methods based on the theory of self-concordant barriers. In particular, we propose a new self-concordant barrier for the above-mentioned cone, featuring a complexity parameter equal to $3$ (better than what was known before), and study its behaviour when used in a practical implementation of a path-following algorithm. Preliminary computational results on some problem classes will be reported.