Optimization and Control (math.OC)

Abstract: A multiobjective optimization method is proposed for obtaining the optimal plane trusses simultaneously for various aspect ratios of the initial ground structure as a set of Pareto optimal solutions generated through a single optimization process. The shape and topology are optimized simultaneously to minimize the compliance under constraint on the total structural volume. The strain energy of each member is divided into components of two coordinate directions on the plane. The force density method is used for alleviating difficulties due to existence of coalescent or melting nodes. It is shown in the numerical example that sufficiently accurate optimal solutions are obtained by comparison with those obtained by the linear weighted sum approach that requires solving a single-objective optimization problem many times.

Abstract: A persistent challenge with the development of electric vertical take-off and landing vehicles (eVTOL) to meet flight power and energy demands is the mass of the load and thermal management systems for batteries. One possible strategy to overcome this problem is to employ optimization techniques to obtain a lightweight battery pack while satisfying structural and thermal requirements. In this work, a structural battery pack with high-energy-density cylindrical cells is optimized using the level-set topology optimization method. The heat generated by the batteries is predicted using a high-fidelity electrochemical model for a given eVTOL flight profile. The worst-case scenario for the battery's heat generation is then considered as a source term in the weakly coupled steady-state thermomechanical finite element model used for optimization. The objective of the optimization problem is to minimize the weighted sum of thermal compliance and structural compliance subjected to a volume constraint. The methodology is demonstrated with numerical examples for different sets of weights. The optimized results due to different weights are compared, discussed, and evaluated with thermal and structural performance indicators. The optimized pack topologies are subjected to a transient thermal finite element analysis to assess the battery pack's thermal response.

Abstract: We study the problem of decentralized constrained POMDPs in a team-setting where the multiple non-strategic agents have asymmetric information. Strong duality is established for the setting of infinite-horizon expected total discounted costs when the observations lie in a countable space, the actions are chosen from a finite space, and the immediate cost functions are bounded. Following this, connections with the common-information and approximate information-state approaches are established. The approximate information-states are characterized independent of the Lagrange-multipliers vector so that adaptations of the multiplier (during learning) will not necessitate new representations. Finally, a primal-dual multi-agent reinforcement learning (MARL) framework based on centralized training distributed execution (CTDE) and three time-scale stochastic approximation is developed with the aid of recurrent and feedforward neural-networks as function-approximators.

Abstract: Golden-section search and bisection search are the two main principled algorithms for 1d minimization of quasiconvex (unimodal) functions. The first one only uses function queries, while the second one also uses gradient queries. Other algorithms exist under much stronger assumptions, such as Newton's method. However, to the best of our knowledge, there is no principled exact line search algorithm for general convex functions -- including piecewise-linear and max-compositions of convex functions -- that takes advantage of convexity. We propose two such algorithms: $\Delta$-Bisection is a variant of bisection search that uses (sub)gradient information and convexity to speed up convergence, while $\Delta$-Secant is a variant of golden-section search and uses only function queries. While bisection search reduces the $x$ interval by a factor 2 at every iteration, $\Delta$-Bisection reduces the (sometimes much) smaller $x^*$-gap $\Delta^x$ (the $x$ coordinates of $\Delta$) by at least a factor 2 at every iteration. Similarly, $\Delta$-Secant also reduces the $x^*$-gap by at least a factor 2 every second function query. Moreover, the $y^*$-gap $\Delta^y$ (the $y$ coordinates of $\Delta$) also provides a refined stopping criterion, which can also be used with other algorithms. Experiments on a few convex functions confirm that our algorithms are always faster than their quasiconvex counterparts, often by more than a factor 2. We further design a quasi-exact line search algorithm based on $\Delta$-Secant. It can be used with gradient descent as a replacement for backtracking line search, for which some parameters can be finicky to tune -- and we provide examples to this effect, on strongly-convex and smooth functions. We provide convergence guarantees, and confirm the efficiency of quasi-exact line search on a few single- and multivariate convex functions.

Abstract: This paper studies the privacy-preserving distributed optimization problem under limited communication, where each agent aims to keep its cost function private while minimizing the sum of all agents' cost functions. To this end, we propose two differentially private distributed algorithms under compressed communication. We show that the proposed algorithms achieve sublinear convergence for smooth (possibly nonconvex) cost functions and linear convergence when the global cost function additionally satisfies the Polyak--Lojasiewicz condition, even for a general class of compressors with bounded relative compression error. Furthermore, we rigorously prove that the proposed algorithms ensure $\epsilon$-differential privacy. Noting that the definition of $\epsilon$-differential privacy is stricter than the definition of ($\epsilon$, $\delta$)-differential privacy used in the literature. Simulations are presented to demonstrate the effectiveness of our proposed approach.

Abstract: The task of state estimation faces a major challenge due to the inherent lack of real-time observability, as certain measurements can only be acquired with a delay. As a result, power systems are essentially unobservable in real time, indicating the existence of multiple states that result in identical values for the available measurements. Certain existing approaches utilize historical data to infer the relationship between real-time available measurements and the state. Other learning-based methods aim at generating the pseudo-measurements required to make the system observable. Our paper presents a methodology that utilizes the outcome of an unobservable state estimator to exploit information on the joint probability distribution between real-time available measurements and delayed ones. Through numerical simulations conducted on a realistic electricity network with insufficient real-time measurements, the proposed procedure showcases superior performance compared to existing state forecasting approaches and those relying on inferred pseudo-measurements.

Abstract: This paper introduces a novel computational framework for solving alternating current optimal power flow (ACOPF) problems using graphics processing units (GPUs). While GPUs have demonstrated remarkable performance in various computing domains, their application in AC OPF has been limited due to challenges associated with porting sparse automatic differentiation (AD) and sparse linear solver routines to GPUs. We aim to address these issues with two key strategies. First, we utilize a single-instruction, multiple-data (SIMD) abstraction of nonlinear programs (NLP). This approach enables the specification of model equations while preserving their parallelizable structure, and in turn, facilitates the implementation of AD routines that can exploit such structure. Second, we employ a condensed-space interior-point method (IPM) with an inequality relaxation strategy. This technique involves relaxing equality constraints to inequalities and condensing the Karush-Kuhn-Tucker system into a much smaller positive definite system. This strategy offers the key advantage of being able to factorize the KKT matrix without numerical pivoting, which in the past has hampered the parallelization of the IPM algorithm. By combining these two strategies, we can perform the majority of operations on GPUs while keeping the data residing in the device memory only. Comprehensive numerical benchmark results showcase the substantial computational advantage of our approach. Remarkably, for solving large-scale AC OPF problems to a moderate accuracy, our implementations -- MadNLP.jl and ExaModels.jl -- running on NVIDIA GPUs achieve an order of magnitude speedup compared to state-of-the-art tools running on contemporary CPUs.

Abstract: This paper summarises the main multi-year investment modelling approaches in energy planning models. Therefore, here we will go from a simple (basic) formulation to a more complex (general) one to understand different levels of detail, including examples to make more accessible the understanding of the concepts.