Creating Standard Load Profiles in Residential and Commercial Sectors in Germany for 2016, 2025 and 2040

Standard load profiles (SLPs) are used to calculate the natural gas demand of non-daily metered customers based on temperature forecasts. The most recent version of natural gas SLPs in Germany was published by the Federal Association of Energy and Water in June 2015. With the concept SigLinDE, a linearization of the old SLPs was carried … Read more

Perturbation Analysis of Singular Semidefinite Program and Its Application to a Control Problem

We consider the sensitivity of semidefinite programs (SDPs) under perturbations. It is well known that the optimal value changes continuously under perturbations on the right hand side in the case where the Slater condition holds in the primal problems. In this manuscript, we observe by investigating a concrete SDP that the optimal value can be … Read more

Doubly Nonnegative Relaxations for Quadratic and Polynomial Optimization Problems with Binary and Box Constraints

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems (QOPs) to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to … Read more

A bound on the Carathéodory number

The Carathéodory number k(K) of a pointed closed convex cone K is the minimum among all the k for which every element of K can be written as a nonnegative linear combination of at most k elements belonging to extreme rays. Carathéodory’s Theorem gives the bound k(K) <= dim (K). In this work we observe … Read more

Primal-dual potential reduction algorithm for symmetric programming problems with nonlinear objective functions

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

New error measures and methods for realizing protein graphs from distance data

The interval Distance Geometry Problem (iDGP) consists in finding a realization in R^K of a simple undirected graph G=(V,E) with nonnegative intervals assigned to the edges in such a way that, for each edge, the Euclidean distance between the realization of the adjacent vertices is within the edge interval bounds. Our aim is to determine … Read more

Application of Facial Reduction to \infty$ State Feedback Control Problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from $H_\infty$ control problems. We discuss the reason from the viewpoint of optimization, and provide necessary and sufficient conditions for LMI problem and its dual not to be strongly feasible. Moreover, we interpret them in terms of control system. In this analysis, … Read more

Kronecker Product Constraints for Semidefinite Optimization

We consider semidefinite optimization problems that include constraints that G(x) and H(x) are positive semidefinite (PSD), where the components of the symmetric matrices G(x) and H(x) are affine functions of an n-vector x. In such a case we obtain a new constraint that a matrix K(x,X) is PSD, where the components of K(x,X) are affine … Read more

Intrinsic Representation of Tangent Vectors and Vector transport on Matrix Manifolds

In Riemannian optimization problems, commonly encountered manifolds are $d$-dimensional matrix manifolds whose tangent spaces can be represented by $d$-dimensional linear subspaces of a $w$-dimensional Euclidean space, where $w > d$. Therefore, representing tangent vectors by $w$-dimensional vectors has been commonly used in practice. However, using $w$-dimensional vectors may be the most natural but may not … Read more

Solving PhaseLift by low-rank Riemannian optimization methods for complex semidefinite constraints

A framework, PhaseLift, was recently proposed to solve the phase retrieval problem. In this framework, the problem is solved by optimizing a cost function over the set of complex Hermitian positive semidefinite matrices. This approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite Hermitian matrices where the desired minimizers … Read more