Vine copula is an important tool in modeling dependence structures of continuous-valued random variables. The maximum likelihood estimation (MLE) for vine copulas has long been considered computationally difficult in higher dimensions, even in 10 or 20 dimensions. Current computational practice, including the implementation in the state-of- the-art R package VineCopula, suffers from the bottleneck of … Read more
This paper considers the solution of tree-structured Quadratic Programs (QPs) as they may arise in multi- stage Model Predictive Control (MPC). In this context, sampling the uncertainty on prescribed decision points gives rise to different scenarios that are linked to each other via the so-called non-anticipativity constraints. Previous work suggests to dualize these constraints and … Read more
In this paper, we develop a Parameterized Proximal Point Algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case $O(1/t)$ convergence rate, where $t$ is the iteration number. By properly choosing the algorithm parameters, numerical experiments … Read more
This paper presents a mixed-integer quadratically constrained programming (MIQCP) formulation for B-spline constraints. The formulation can be used to obtain an exact MIQCP reformulation of any spline-constrained optimization problem, provided that the polynomial spline functions are continuous. This reformulation allows practitioners to use a general-purpose MIQCP solver, instead of a special-purpose spline solver, when solving … Read more
Motivated by applications in robotics and computer vision, we study problems related to spatial reasoning of a 3D environment using sublevel sets of polynomials. These include: tightly containing a cloud of points (e.g., representing an obstacle) with convex or nearly-convex basic semialgebraic sets, computation of Euclidean distance between two such sets, separation of two convex … Read more
We present a mixed-integer nonlinear programming (MINLP) formulation of a UAV path optimization problem in an attempt to find the globally optimum solution. As objective functions in UAV path optimization problems typically tend to be non-convex, traditional optimization solvers (typically local solvers) are prone to local optima, which lead to severely sub-optimal controls. For the … Read more
The design of hub-and-spoke transport networks is a strategic planning problem, as the choice of hub locations has to remain unchanged for long time periods. However, strikes, disasters or traffic breakdown can lead to the unavailability of a hub for a short period of time. Therefore it is important to consider such events already in … Read more
In this paper we investigate the family of functions representable by deep neural networks (DNN) with rectified linear units (ReLU). We give the first-ever polynomial time (in the size of data) algorithm to train a ReLU DNN with one hidden layer to {\em global optimality}. This follows from our complete characterization of the ReLU DNN … Read more
In this study, we develop heuristic methods to solve unimodular quadratic programming (UQP) approximately, which is known to be NP-hard. UQP-type problems appear naturally in radar waveform design and active sensing applications. In the UQP framework, we optimize a sequence of complex variables with unit modulus by maximizing a quadratic function. To solve the UQP … Read more
The uncertainty associated with renewable energy sources introduces significant challenges in optimal power flow (OPF) analysis. A variety of new approaches have been proposed that use chance constraints to limit line or bus overload risk in OPF models. Most existing formulations assume that the probability distributions associated with the uncertainty are known a priori or … Read more