Sparse principal component analysis (PCA) is a popular dimensionality reduction technique for obtaining principal components which are linear combinations of a small subset of the original features. Existing approaches cannot supply certifiably optimal principal components with more than $p=100s$ of variables. By reformulating sparse PCA as a convex mixed-integer semidefinite optimization problem, we design a … Read more
In this paper, we consider binary quadratically constrained quadratic problems and propose a new approach to generate stronger bounds than the ones obtained using the Semidefinite Programming relaxation. The new relaxation is based on the Boolean Quadric Polytope and is solved via a Dantzig-Wolfe Reformulation in matrix space. For block-decomposable problems, we extend the relaxation … Read more
Potential-based flows constitute a basic model to represent physical behavior in networks. Under natural assumptions, the flow in such networks must be acyclic. The goal of this paper is to exploit this property for the solution of corresponding optimization problems. To this end, we introduce several combinatorial models for acyclic flows, based on binary variables … Read more
We present a mixed-integer nonlinear optimization model for computing the optimal expansion of an existing tree-shaped district heating network given a number of potential new consumers. To this end, we state a stationary and nonlinear model of all hydraulic and thermal effects in the pipeline network as well as nonlinear models for consumers and the … Read more
The thermal unit commitment (UC) problem often can be formulated as a mixed integer quadratic programming (MIQP), which is difficult to solve efficiently, especially for large-scale instances. The tighter characteristic re-duces the search space, therefore, as a natural conse-quence, significantly reduces the computational burden. In the literature, many tightened formulations for single units with parts … Read more
In this paper, we study the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive … Read more
We consider the least squares regression problem, penalized with a combination of the L0 and L2 norms (a.k.a. L0 L2 regularization). Recent work presents strong evidence that the resulting L0-based estimators can outperform popular sparse learning methods, under many important high-dimensional settings. However, exact computation of L0-based estimators remains a major challenge. Indeed, state-of-the-art mixed … Read more
Traffic jams cause economical damage which has been estimated between 10 and 100 billion Euros per year in Germany, also due to inefficient urban traffic. It is currently open how the situation will change with upcoming technological advances in autonomous and electric mobility. On the one hand, autonomous cars may lead to an increased number … Read more
Matrix copositivity has an important theoretical background. Over the last decades, the use of algorithms to check copositivity has made a big progress. Methods are based on spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the … Read more
In this letter, we present an alternative mixed-integer non-liner programming formulation of the reactive optimal power flow (ROPF) problem. We utilize a mixed-integer second-order cone programming (MISOCP) based approach to find global optimal solutions of the proposed ROPF problem formulation. We strengthen the MISOCP relaxation via the addition of convex envelopes and cutting planes. Computational … Read more