Leonid Faybusovich Let $V$ be an Euclidean Jordan algebra and $\Omega$ be a cone of invertible squares in $V$. Suppose that $g:\mathbb{R}_{+} \to \mathbb{R}$ is a matrix monotone function on the positive semiaxis which naturally induces a function $\tilde{g}: \Omega \to V$. We show that $-\tilde{g}$ is compatible (in the sense of Nesterov-Nemirovski) with the standard self-concordant … Read more
We describe a long-step path-following algorithm for a class of symmetric programming problems with nonlinear convex objective functions. The complexity estimates similar to the case of a linear-quadratic objective function are established. The results of numerical experiments for the class of optimization problems involving quantum entropy are presented. Citation Preprint, University of Notre Dame, December … Read more
We consider a primal-dual potential reduction algorithm for nonlinear convex optimization problems over symmetric cones. The same complexity estimates as in the case of linear objective function are obtained provided a certain nonlinear system of equations can be solved with a given accuracy. This generalizes the result of K. Kortanek, F. Potra and Y.Ye. We … Read more
We consider multi-target,robust linear-quadratic control problem on semi-infinite interval. Using functional-analytic approach developed in [2], we reduce this problem to a convex optimization problem on the simplex. Explicit procedure for the reduced optimization problem is described. Citation Preprint, University of Notre Dame, August,2015 Article Download View Remark on multi-target,robust linear-quadratic control problem on semi-infinite interval
Let $g$ be a continuously differentiable function whose derivative is matrix monotone on positive semi-axis. Such a function induces a function $\phi (x)=tr(g(x))$ on the cone of squares of an arbitrary Euclidean Jordan algebra. We show that $\phi (x) -\ln \det(x)$ is a self-concordant function on the interior of the cone. We also show that … Read more
We describe a Jordan-algebraic version of E. Lieb convexity inequalities. A joint convexity of Jordan-algebraic version of quantum entropy is proven. SA spectral theory on semi-simple complex Jordan algebras is used as atool to prove the convexity results. Possible applications to optimization and statistics are indicated Citation Preprint, University of Notre Dame, August 2014 Article … Read more
We describe the generalization of Hazan’s algorithm for symmetric programming problems Citation Notre Dame Report, December, 2012 Article Download View On Hazan's algorithm for symmetric programming problems
We relate a deterministic Kalman filter on semi-infinite interval to linear-quadratic tracking control model with unfixed initial condition Citation Technical report, University of Notre Dame, October, 2011 Article Download View Deterministic Kalman Filtering on Semi-infinite Interval
We consider multi-target linear-quadratic control problem on semi-infinite interval. We show that the problem can be reduced to a simple convex optimization problem on the simplex. Citation To appear in Mathematical Problems in Engineering 2012 Article Download View Multi-target Linear-quadratic control problem: semi-infinite interval
We describe a classof measures on second-order cones as a push-forward of the Cartesian product of a probabilistic measure on positive semi-line corresponding to Gamma distribution and the uniform measure on the sphere Citation report, Department of Mathematics, University of Notre Dame, July, 2011 Article Download View Sampling with respect to a class of measures … Read more