On the Dynamic Stability of Electricity Markets

In this work, we present new insights into the dynamic stability of electricity markets. In particular, we discuss how short forecast horizons, incomplete gaming, and physical ramping constraints can give rise to stability issues. Using basic concepts of market efficiency, Lyapunov stability, and predictive control, we construct a new stabilizing market design. A numerical case … Read more

Real-Time Optimization as a Generalized Equation

We establish results for the problem of tracking a time-dependent manifold arising in online nonlinear programming by casting this as a generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a linear complementarity problem (LCP) at each time step. … Read more

The Effects of Adding Objectives to an Optimization Problem on the Solution Set

Suppose that for a given optimisation problem (which might be multicriteria problem or a single-criteron problem), an additional objective function is introduced. How does the the set of solutions, i.~e.\ the set of efficient points change when instead of the old problem the new multicriteria problem is considered? How does the set of properly efficient … Read more

Prox-Regularity and Stability of the Proximal Mapping

Fundamental insights into the properties of a function come from the study of its Moreau envelopes and Proximal point mappings. In this paper we examine the stability of these two objects under several types of perturbations. In the simplest case, we consider tilt-perturbations, i.e. perturbations which correspond to adding a linear term to the objective … Read more

The mathematics of eigenvalue optimization

Optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge. Such problems arise often in theory and practice, particularly in engineering design, and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory. This essay presents a personal choice of some central mathematical ideas, outlined for … Read more

Solving second order cone programming via a reduced augmented system approach

The standard Schur complement equation based implementation of interior-point methods for second order cone programming may encounter stability problems in the computation of search directions, and as a consequence, accurate approximate optimal solutions are sometimes not attainable. Based on the eigenvalue decomposition of the $(1,1)$ block of the augmented equation, a reduced augmented equation approach … Read more

Variational Analysis of the Abscissa Mapping for Polynomials

The abscissa mapping on the affine variety $M_n$ of monic polynomials of degree $n$ is the mapping that takes a monic polynomial to the maximum of the real parts of its roots. This mapping plays a central role in the stability theory of matrices and dynamical systems. It is well known that the abscissa mapping … Read more