Formulating the alternating current optimal power flow (ACOPF) as a polynomial optimization problem makes it possible to solve large instances in practice and to guarantee asymptotic convergence in theory.We formulate the ACOPF as a degree-two polynomial program and study two approaches to solving it via convexifications. In the first approach, we tighten the first order relaxation of the non-convex quadratic program by adding valid inequalities. In the second approach, we exploit the structure of the polynomial program by using a sparse variant of Lasserre's hierarchy. This allows us to solve instances of up to 39 buses to global optimality and to provide strong bounds for the Polish network within an hour.

This article outlines our point of view regarding the applicability, state-of-the-art, and potential of quantum computing for problems in finance. We provide an introduction to quantum computing as well as a survey on problem classes in finance that are computationally challenging classically and for which quantum computing algorithms are promising. In the main part, we describe in detail quantum algorithms for specific applications arising in financial services, such as those involving simulation, optimization, and machine learning problems. In addition, we include demonstrations of quantum algorithms on IBM Quantum back-ends and discuss the potential benefits of quantum algorithms for problems in financial services. We conclude with a summary of technical challenges and future prospects.

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We present a simple model and algorithm for predicting driver destinations and routes, based on the input of the latest road links visited as part of an ongoing trip. The algorithm may be used to predict any clusters previously observed in a driver's trip history. It assumes that the driver's historical trips are grouped into clusters sharing similar patterns. Given a new trip, the algorithm attempts to predict the cluster in which the trip belongs. The proposed algorithm has low temporal complexity. In addition, it does not require the transition and emission matrices of the Markov chain to be computed. Rather it relies on the frequencies of co-occurrences of road links and trip clusters. We validate the proposed algorithm against an experimental dataset. We discuss the success and convergence of the algorithm and show that our algorithm has a high prediction success rate.

We consider the problem of approximating the unknown density u ∈ L 2 (Ω, λ) of a measure µ on Ω ⊂ R n , absolutely continuous with respect to some given reference measure λ, only from the knowledge of finitely many moments of µ. Given d ∈ N and moments of order d, we provide a polynomial p d which minimizes the mean square error (u − p) 2 dλ over all polynomials p of degree at most d. If there is no additional requirement, p d is obtained as solution of a linear system. In addition, if p d is expressed in the basis of polynomials that are orthonormal with respect to λ, its vector of coefficients is just the vector of given moments and no computation is needed.In general nonnegativity of p d is not guaranteed even though u is nonnegative. However, with this additional nonnegativity requirement one obtains analogous results but computing p d ≥ 0 that minimizes (u − p) 2 dλ now requires solving an appropriate semidefinite program. We have tested the approach on some applications arising from the reconstruction of geometrical objects and the approximation of solutions of nonlinear differential equations. In all cases our results are significantly better than those obtained with the maximum entropy technique for estimating u.