Optimization and Control (math.OC)

Abstract: We extend the polynomial Pell's equation satisfied by univariate Chebyshev polynomials on [--1, 1] from one variable to several variables, using orthogonal polynomials on regular domains that include cubes, balls, and simplexes of arbitrary dimension. Moreover, we show that such an equation is strongly connected (i) to a certificate of positivity (from real algebraic geometry) on the domain, as well as (ii) to the Christoffel functions of the equilibrium measure on the domain. In addition, the solution to Pell's equation reflects an extremal property of orthonormal polynomials associated with an entropy-like criterion.

Abstract: Industrial decision-makers often base decisions on mathematical optimization models to achieve cost-efficient design solutions in energy transitions. However, since a model can only approximate reality, the optimal solution is not necessarily the best real-world energy system. Exploring near-optimal design spaces, e.g., by the Modeling All Alternatives (MAA) method, provides a more holistic view of decision alternatives beyond the cost-optimal solution. However, the MAA method misses out on discrete in-vestment decisions. Incorporating such discrete investment decisions is crucial when modeling industrial energy systems. Our work extends the MAA method by integrating discrete design decisions. We optimize the design and operation of an industrial energy system transformation using a mixed-integer linear program. First, we explore the continuous, near-optimal design space by applying the MAA method. Thereafter, we sample all discrete design alternatives from the continuous, near-optimal design space. In a case study, we apply our method to identify all near-optimal design alternatives of an industrial energy system. We find 128 near-optimal design alternatives where costs are allowed to increase to a maximum of one percent offering decision-makers more flexibility in their investment decisions. Our work enables the analysis of discrete design alternatives for industrial energy transitions and supports the decision-making process for investments in energy infrastructure.

Abstract: A final-state observability result in the Banach space setting for non-autonomous observation problems is obtained that covers and extends all previously known results in this context, while providing a streamlined proof that follows the established Lebeau-Robbiano strategy.

Abstract: In this paper, we present a method to certify the approximation quality of a low rank tensor to a given third order symmetric tensor. Under mild assumptions, best low rank approximation is attained if a control parameter is zero or quantified quasi-optimal low rank approximation is obtained if the control parameter is positive.This is based on a primal-dual method for computing a low rank approximation for a given tensor. The certification is derived from the global optimality of the primal and dual problems, and is characterized by easily checkable relations between the primal and the dual solutions together with another rank condition. The theory is verified theoretically for orthogonally decomposable tensors as well as numerically through examples in the general case.

Abstract: In this paper we study a generalization of distributed conditional gradient method to time-varying network architectures. We theoretically analyze convergence properties of the algorithm and provide numerical experiments. The time-varying network is modeled as a deterministic of a stochastic sequence of graphs.