This paper formalizes a demand response task as an optimization problem featuring a known time-varying engineering cost and an unknown (dis)comfort function. Based on this model, this paper develops a feedback-based projected gradient method to solve the demand response problem in an online fashion, where: i) feedback from the user is leveraged to learn the (dis)comfort function concurrently with the execution of the algorithm; and, ii) measurements of electrical quantities are used to estimate the gradient of the known engineering cost. To learn the unknown function, a shape-constrained Gaussian Process is leveraged; this approach allows one to obtain an estimated function that is strongly convex and smooth. The performance of the online algorithm is analyzed by using metrics such as the tracking error and the dynamic regret. A numerical example is illustrated to corroborate the technical findings.
√x x. A vector of zeros is represented by 0 and a vector of ones by 1, with the corresponding dimensions. O refers to the big O notation; that is, given two positive sequences {a k } ∞ k=0 and {b k } ∞ k=0 , we say that
This paper considers a time-varying optimization problem associated with a network of systems, with each of the systems shared by (and affecting) a number of individuals. The objective is to minimize cost functions associated with the individuals' preferences, which are unknown, subject to timevarying constraints that capture physical or operational limits of the network. To this end, the paper develops a distributed online optimization algorithm with concurrent learning of the cost functions. The cost functions are learned on-the-fly based on the users' feedback (provided at irregular intervals) by leveraging tools from shape-constrained Gaussian Processes. The online algorithm is based on a primal-dual method, and acts effectively in a closed-loop fashion where: i) users' feedback is utilized to estimate the cost, and ii) measurements from the network are utilized in the algorithmic steps to bypass the need for sensing of (unknown) exogenous inputs of the network. The performance of the algorithm is analyzed in terms of dynamic network regret and constraint violation. Numerical examples are presented in the context of real-time optimization of distributed energy resources.
The main contribution of this study is to offer a general hierarchy of the most important financial variables associated with determining the presence of innovation. The variables' importance is computed based on an ensemble neural network model, which predicts the presence of innovation in a sample of the small-and medium-sized enterprises (SMEs) from the emerging market of Colombia. The results suggest that to innovate, the variables associated with the sources and uses of financing, and not the variables associated with the characteristics or credit of the company, predominate. The variables related to managers are secondary.
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