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Optimization and Control (math.OC)

Thu, 27 Apr 2023

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1.Solving Data-Driven Newsvendor Pricing Problems with Decision-Dependent Effect

Authors:Wenxuan Liu, Zhihai Zhang

Abstract: This paper investigates the data-driven pricing newsvendor problem, which focuses on maximizing expected profit by deciding on inventory and pricing levels based on historical demand and feature data. We first build an approximate model by assigning weights to historical samples. However, due to decision-dependent effects, the resulting approximate model is complicated and unable to solve directly. To address this issue, we introduce the concept of approximate gradients and design an Approximate Gradient Descent (AGD) algorithm. We analyze the convergence of the proposed algorithm in both convex and non-convex settings, which correspond to the newsvendor pricing model and its variants respectively. Finally, we perform numerical experiment on both simulated and real-world dataset to demonstrate the efficiency and effectiveness of the AGD algorithm. We find that the AGD algorithm can converge to the local maximum provided that the approximation is effective. We also illustrate the significance of two characteristics: distribution-free and decision-dependent of our model. Consideration of the decision-dependent effect is necessary for approximation, and the distribution-free model is preferred when there is little information on the demand distribution and how demand reacts to the pricing decision. Moreover, the proposed model and algorithm are not limited to the newsvendor problem, but can also be used for a wide range of decision-dependent problems.

2.Optimal Transmission Switching with Uncertainties from both Renewable Energy and N-k Contingencies

Authors:Tong Han, David J. Hill, Yue Song

Abstract: This paper focuses on the N-k security-constrained optimal transmission switching (OTS) problem for variable renewable energy (VRE) penetrated power grids. A new three-stage stochastic and distributionally robust OTS model is proposed. The first stage has the primary purpose to schedule the power generation and network topology based on the forecast of VRE. The second stage controls the power generation and voltage magnitudes of voltage-controlled buses in response to VRE uncertainty, and the third stage reacts to N-k contingencies additionally by line switching and load shedding. The VRE and N-k contingencies, considering different availability of their probability distributions, are tackled by stochastic and distributionally robust optimization, respectively. By adopting stage-wise realization of uncertainties in VRE and contingencies, the associated corrective controls with different mechanisms can be handled separately and properly, which makes the proposed OTS model more realistic than existing two-stage ones. For solving the proposed OTS model, its tractable reformulation is derived, and a solution approach that combines the nested column-and-constraint generation algorithm and Dantzig-Wolfe procedure is developed. Finally, case studies include a simple IEEE network for illustrative purposes and then real system networks to demonstrate the efficacy of the proposed approach.

3.Convergence of Adam Under Relaxed Assumptions

Authors:Haochuan Li, Ali Jadbabaie, Alexander Rakhlin

Abstract: In this paper, we provide a rigorous proof of convergence of the Adaptive Moment Estimate (Adam) algorithm for a wide class of optimization objectives. Despite the popularity and efficiency of the Adam algorithm in training deep neural networks, its theoretical properties are not yet fully understood, and existing convergence proofs require unrealistically strong assumptions, such as globally bounded gradients, to show the convergence to stationary points. In this paper, we show that Adam provably converges to $\epsilon$-stationary points with $\mathcal{O}(\epsilon^{-4})$ gradient complexity under far more realistic conditions. The key to our analysis is a new proof of boundedness of gradients along the optimization trajectory, under a generalized smoothness assumption according to which the local smoothness (i.e., Hessian norm when it exists) is bounded by a sub-quadratic function of the gradient norm. Moreover, we propose a variance-reduced version of Adam with an accelerated gradient complexity of $\mathcal{O}(\epsilon^{-3})$.

4.Modeling the Complexity of City Logistics Systems for Sustainability

Authors:Taiwo Adetiloye, Anjali Awasthi

Abstract: The logistics of urban areas are becoming more sophisticated due to the fast city population growth. The stakeholders are faced with the challenges of the dynamic complexity of city logistics(CL) systems characterized by the uncertainty effect together with the freight vehicle emissions causing pollution. In this conceptual paper, we present a research methodology for the environmental sustainability of CL systems that can be attained by effective stakeholders' collaboration under non-chaotic situations and the presumption of the human levity tendency. We propose the mathematical axioms of the uncertainty effect while putting forward the notion of condition effectors, and how to assign hypothetical values to them. Finally, we employ a spider network and causal loop diagram to investigate the system's elements and their behavior over time.

5.Propagating Kernel Ambiguity Sets in Nonlinear Data-driven Dynamics Models

Authors:Jia-Jie Zhu

Abstract: This paper provides answers to an open problem: given a nonlinear data-driven dynamical system model, e.g., kernel conditional mean embedding (CME) and Koopman operator, how can one propagate the ambiguity sets forward for multiple steps? This problem is the key to solving distributionally robust control and learning-based control of such learned system models under a data-distribution shift. Different from previous works that use either static ambiguity sets, e.g., fixed Wasserstein balls, or dynamic ambiguity sets under known piece-wise linear (or affine) dynamics, we propose an algorithm that exactly propagates ambiguity sets through nonlinear data-driven models using the Koopman operator and CME, via the kernel maximum mean discrepancy geometry. Through both theoretical and numerical analysis, we show that our kernel ambiguity sets are the natural geometric structure for the learned data-driven dynamical system models.

6.How to avoid ordinal violations in incomplete pairwise comparisons

Authors:László Csató

Abstract: Assume that some ordinal preferences can be represented by a weakly connected directed acyclic graph. The data are collected into an incomplete pairwise comparison matrix, the missing entries are estimated, and the priorities are derived from the optimally filled pairwise comparison matrix. Our paper studies whether these weights are consistent with the partial order given by the underlying graph. According to previous results from the literature, two popular procedures, the incomplete eigenvector and the incomplete logarithmic least squares methods fail to satisfy the required property. Here, it is shown that the recently introduced lexicographically optimal completion combined with any of these weighting methods avoids ordinal violation in the above setting. This finding provides a powerful argument for using the lexicographically optimal completion to determine the missing elements in an incomplete pairwise comparison matrix.

7.Detection of a very serious error in the paper: "On identifiability of nonlinear ODE models and applications in viral dynamics"

Authors:Agostino Martinelli

Abstract: This erratum highlights a very serious error in a paper published by SIAM Review in 2011. The error is in Section 6.2 of [1]. It is very important to notify this error because of the following two reasons: (i) [1] is one of the most cited contributions in the field of identifiability of viral dynamics models, and (ii)the error is relevant because, as a result of it, a very popular viral model (perhaps the most popular in the field of HIV dynamics) has been classified as identifiable. In contrast, three of its parameters are not identifiable, even locally. This erratum first proves the non uniqueness of the three unidentifiable parameters by exhibiting infinitely many distinct but indistinguishable values of them. The non uniqueness is even local. Then, this erratum details the error made by the authors of [1] which produced the claimed (but false) local identifiability of all the model parameters.