We present a framework for designing and analyzing primal-dual interior-point methods for convex optimization. We assume that a self-concordant barrier for the convex domain of interest and the Legendre transformation of the barrier are both available to us. We directly apply the theory and techniques of interior-point methods to the given good formulation of the problem (as is, without a conic...

Polynomially structured low-rank approximation problems occur in • algebraic curve fitting, e.g., conic section fitting, • subspace clustering (generalized principal component analysis), and • nonlinear and parameter-varying system identification. The maximum likelihood estimation principle applied to these nonlinear models leads to nonconvex optimization problems and yields inconsistent estima...

We study solution approaches to a class of mixed-integer nonlinear programming problems that arise from recent developments in risk-averse stochastic optimization and contain second-order and p-order cone programming as special cases. We explore possible applications of some of the solution techniques that have been successfully used in mixed-integer conic programming and show how they can be g...

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for ...

The main goal of this paper is to develop applications of advanced tools of first-order and second-order variational analysis and generalized differentiation to the fundamental notion of full stability of local minimizers of general classes of constrained optimization and minimax problems. In particular, we derive second-order characterizations of full stability and investigate its relationship...

In this paper we develop several polynomial-time interior-point methods (IPM) for solving nonlinear primal-dual conic optimization problem. We assume that the barriers for the primal and the dual cone are not conjugate. This broken symmetry does not allow to apply the standard primal-dual IPM. However, we show that in this situation it is also possible to develop very efficient optimization met...

We present minimax and stochastic formulations of some linear approximation problems with uncertain data in R n equipped with the Euclidean (l2), Absolute-sum (l1) or Chebyshev (l1) norms. We then show that these problems can be solved using convex optimization. Our results parallel and extend the work of El-Ghaoui and Lebret on robust least squares 3], and the work of Ben-Tal and Nemirovski on...