Superlinear and Quadratic Convergence of Riemannian Interior Point Methods

We extend the classical primal-dual interior point algorithms from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point (RIP) method, is for solving Riemannian constrained optimization problems. Under the standard assumptions in the Riemannian setting, we establish locally superlinear, quadratic convergence for the Newton version of RIP and locally linear, … Read more

A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor C. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following … Read more

A barrier Lagrangian dual method for multi-stage stochastic convex semidefinite optimization

In this paper, we present a polynomial-time barrier algorithm for solving multi-stage stochastic convex semidefinite optimization based on the Lagrangian dual method which relaxes the nonanticipativity constraints. We show that the barrier Lagrangian dual functions for our setting form self-concordant families with respect to barrier parameters. We also use the barrier function method to improve … Read more

On solving large-scale multistage stochastic problems with a new specialized interior-point approach

A novel approach based on a specialized interior-point method (IPM) is presented for solving large-scale stochastic multistage continuous optimization problems, which represent the uncertainty in strategic multistage and operational two-stage scenario trees, the latter being rooted at the strategic nodes. This new solution approach considers a split-variable formulation of the strategic and operational structures, for … Read more

New interior-point approach for one- and two-class linear support vector machines using multiple variable splitting

Multiple variable splitting is a general technique for decomposing problems by using copies of variables and additional linking constraints that equate their values. The resulting large optimization problem can be solved with a specialized interior-point method that exploits the problem structure and computes the Newton direction with a combination of direct and iterative solvers (i.e., … Read more

Stochastic infinity norm optimization and self-concordant primal interior-point algorithms

We study the two-stage stochastic infinity norm optimization problem with recourse. First, we study and analyze the algebraic structure of the infinity norm cone, and use its algebra to compute the derivatives of the barrier recourse functions. Then, we show that the barrier recourse functions and the composite barrier functions for this optimization problem are … Read more

An Improved Penalty Algorithm using Model Order Reduction for MIPDECO problems with partial observations

This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra … Read more

A new stopping criterion for Krylov solvers applied in Interior Point Methods

A surprising result is presented in this paper with possible far reaching consequences for any optimization technique which relies on Krylov subspace methods employed to solve the underlying linear equation systems. In this paper the advantages of the new technique are illustrated in the context of Interior Point Methods (IPMs). When an iterative method is … Read more

A Homogeneous Predictor-Corrector Algorithm for Stochastic Nonsymmetric Convex Conic Optimization With Discrete Support

We consider a stochastic convex optimization problem over nonsymmetric cones with discrete support. This class of optimization problems has not been studied yet. By using a logarithmically homogeneous self-concordant barrier function, we present a homogeneous predictor-corrector interior-point algorithm for solving stochastic nonsymmetric conic optimization problems. We also derive an iteration bound for the proposed algorithm. … Read more

Barrier Methods Based on Jordan-Hilbert Algebras for Stochastic Optimization in Spin Factors

We present decomposition logarithmic-barrier interior-point methods based on unital Jordan-Hilbert algebras for infinite-dimensional stochastic second-order cone programming problems in spin factors. The results show that the iteration complexity of the proposed algorithms is independent on the choice of Hilbert spaces from which the underlying spin factors are formed, and so it coincides with the best … Read more