Lagrangian Duality Research Articles

Certifiably optimal rotation and pose estimation based on the Cayley map

We present novel, convex relaxations for rotation and pose estimation problems that can a posteriori guarantee global optimality for practical measurement noise levels. Some such relaxations exist in the literature for specific problem setups that assume the matrix von Mises-Fisher distribution (a.k.a., matrix Langevin distribution or chordal distance) for isotropic rotational uncertainty. However, another common way to represent uncertainty for rotations and poses is to define anisotropic noise in the associated Lie algebra. Starting from a noise model based on the Cayley map, we define our estimation problems, convert them to Quadratically Constrained Quadratic Programs (QCQPs), then relax them to Semidefinite Programs (SDPs), which can be solved using standard interior-point optimization methods; global optimality follows from Lagrangian strong duality. We first show how to carry out basic rotation and pose averaging. We then turn to the more complex problem of trajectory estimation, which involves many pose variables with both individual and inter-pose measurements (or motion priors). Our contribution is to formulate SDP relaxations for all these problems based on the Cayley map (including the identification of redundant constraints) and to show them working in practical settings. We hope our results can add to the catalogue of useful estimation problems whose solutions can be a posteriori guaranteed to be globally optimal.

Discretization and quantification for distributionally robust optimization with decision-dependent ambiguity sets

ABSTRACT In this paper, we investigate the discrete approximation and the quantitative analysis for a class of the distributionally robust optimization problems with decision-dependent ambiguity sets. We establish the local Lipschitz continuity of the decision-dependent ambiguity set, measured by the Hausdorff distance, under a broad class of metrics known as ζ-structure and the Slater condition. Furthermore, we employ Lagrange duality and first-order growth conditions to derive quantitative analysis for the optimal value and optimal solution. We also examine the application of a classical and widely-used ambiguity set within the theoretical framework of this paper. Finally, we conduct experiments to demonstrate the computational times and variations in the optimal value.

Optimal economic configuration by sharing hydrogen storage while considering distributed demand response in hydrogen-based renewable microgrid

With the issues of the greenhouse effect and environmental pollution globally, a high proportion of renewable energy has been integrated into microgrids. Its interconnection brings many challenges to the electric power industry. This paper proposes a new distributed response strategy through sharing hydrogen storage resources, aiming to solve the supply-demand imbalance in microgrids. First, the uneven power distribution from shared energy storage stations necessitates ensuring a fair power dispatch among users. Next, a microgrid demand response model with multi-constraint conditions and high penetration of wind and solar power is established, incorporating constraints like user power consumption, energy storage capacity, and charging/discharging power while considering differences in user electricity behaviors. Finally, the optimal economic configuration uses Lagrange duality theory to process the integrated models, including the demand response, hydrogen sharing, and power generation models, to make the optimization goals fairer. Results demonstrate that our strategy can adjust user electricity usage according to real-time supply-demand conditions, reducing users' total costs by 39.4 %. Compared with the existing configurations of sharing hydrogen storage configurations, our model uses more accurate utility functions, demonstrating better economic advantages, with users' total costs reduced by 4.89 % and revenue of the shared storage station increased by 4.02 %.

Computing the alpha complex using dual active set quadratic programming

The alpha complex is a fundamental data structure from computational geometry, which encodes the topological type of a union of balls B(x;r)⊂Rm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B(x;r) \\subset {\\mathbb {R}}^m$$\\end{document} for x∈S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in S$$\\end{document}, including a weighted version that allows for varying radii. It consists of the collection of “simplices” σ={x0,...,xk}⊂S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma =\\{x_0,...,x_k\\} \\subset S$$\\end{document}, which correspond to nonempty (k+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k+1)$$\\end{document}-fold intersections of cells in a radius-restricted version of the Voronoi diagram Vor(S,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Vor}\\,}}(S,r)$$\\end{document}. Existing algorithms for computing the alpha complex require that the points reside in low dimension because they begin by computing the entire Delaunay complex, which rapidly becomes intractable, even when the alpha complex is of a reasonable size. This paper presents a method for computing the alpha complex without computing the full Delaunay triangulation by applying Lagrangian duality, specifically an algorithm based on dual quadratic programming that seeks to rule simplices out rather than ruling them in.

Lagrange Duality and Saddle-Point Optimality Conditions for Nonsmooth Interval-Valued Multiobjective Semi-Infinite Programming Problems with Vanishing Constraints

This article deals with a class of nonsmooth interval-valued multiobjective semi-infinite programming problems with vanishing constraints (NIMSIPVC). We introduce the VC-Abadie constraint qualification (VC-ACQ) for NIMSIPVC and employ it to establish Karush–Kuhn–Tucker (KKT)-type necessary optimality conditions for the considered problem. Regarding NIMSIPVC, we formulate interval-valued weak vector, as well as interval-valued vector Lagrange-type dual and scalarized Lagrange-type dual problems. Subsequently, we establish the weak, strong, and converse duality results relating to the primal problem, NIMSIPVC, and the corresponding dual problems. Moreover, we introduce the notion of saddle points for the interval-valued vector Lagrangian and scalarized Lagrangian of NIMSIPVC. Furthermore, we derive the saddle-point optimality criteria for NIMSIPVC by establishing the relationships between the solutions of NIMSIPVC and the saddle points of the corresponding Lagrangians of NIMSIPVC, under convexity assumptions. Non-trivial illustrative examples are provided to demonstrate the validity of the established results. The results presented in this paper extend the corresponding results derived in the existing literature from smooth to nonsmooth optimization problems, and we further generalize them for interval-valued multiobjective semi-infinite programming problems with vanishing constraints.

A verified training support vector machine in bearing fault diagnosis

Abstract The combination of support vector machine (SVM) and deep learning is widely used in bearing fault diagnosis. The SVM-based bearing fault diagnosis method usually uses features extracted from bearing vibration signals as input. However, environmental noise in industrial applications can affect the feature representation, potentially leading to failures in SVM-based fault diagnosis methods. Adversarial robustness verification methods can evaluate the robustness of models to small perturbations in feature representations. Certified defense methods achieve robustness enhancement by embedding adversarial robustness verification into the learning process of the model. Thus, we propose a certified defense method named verified training SVM (VT-SVM) based on Lagrange duality. The proposed method incorporates adversarial robustness verification into the SVM learning framework, thereby enhancing the robustness of SVM in noisy environments. First, the certified defense optimization problem of SVM is constructed by combining the original optimization problem of SVM with the adversarial robustness verification problem. Then, the certified defense optimization problem is converted into an equivalent convex quadratic programming problem by introducing slack variables. Next, the Lagrange duality is used to transform the convex quadratic programming problem into a dual problem. Finally, the solution to the original optimization problem can be obtained by solving the dual problem. We validate the effectiveness of VT-SVM on the artificial dataset, the rolling bearing dataset from Case Western Reserve University, and the VBL-VA001 dataset from Sepuluh Nopember Institute of Technology. Experimental results demonstrate that our method both ensures high predictive performance and improves the resistance of the model to small noise.

On the strength of Lagrangian duality in multiobjective integer programming

AbstractThis paper investigates the potential of Lagrangian relaxations to generate quality bounds on non-dominated images of multiobjective integer programs (MOIPs). Under some conditions on the relaxed constraints, we show that a set of Lagrangian relaxations can provide bounds that coincide with every bound generated by the convex hull relaxation. We also provide a guarantee of the relative quality of the Lagrangian bound at unsupported solutions. These results imply that, if the relaxed feasible region is bounded, some Lagrangian bounds will be strictly better than some convex hull bounds. We demonstrate that there exist Lagrangian multipliers which are sparse, satisfy a complementary slackness property, and generate tight relaxations at supported solutions. However, if all constraints are dualized, a relaxation can never be tight at an unsupported solution. These results characterize the strength of the Lagrangian dual at efficient solutions of an MOIP.

An Innovative Double-Frontier Approach to Measure Sustainability Efficiency Based on an Energy Use and Operations Management Perspective

China’s economic development has achieved great success in recent years, but the problems of energy scarcity and environmental pollution have become increasingly serious. To enhance the reliability and efficiency between energy, the environment and the economy, sustainable development is an inevitable choice. In the context of measuring sustainability efficiency, a network data envelopment analysis model is proposed to formulate the two-stage process of energy use and operations management. A double frontier is derived to optimize the available energy for sustainable development. Due to nonlinearity, previous linear methods are not directly applicable to identify the double frontier and calculate stage efficiencies for inefficient decision-making units. To address this problem, this study develops the primal-dual relationship between multiplicative and envelopment network models based on the Lagrange duality principle of parametric linear programming. The newly developed approach is used to evaluate the sustainability efficiency of 30 administrative regions in China. The results show that insufficient sustainability efficiency is a systemic problem. Different regions should take different measures to conserve energy and reduce pollutant emissions for sustainable development. To increase sustainability efficiency, regions should support energy-saving and emission-reducing technologies in production processes and strengthen their capacity for technological innovation. Compared with energy use efficiency, operations management efficiency in China has a wider range of changes. During the operations management stage, there is not much difference between the capacity and quantity of each region. Based on benchmark regions at the efficiency frontier, there is an opportunity to improve operations management in the near future. Blockchain technology can effectively improve energy allocation efficiency.

Stochastic linear–quadratic control problems with affine constraints

This paper investigates the stochastic linear–quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality theory, both the dual problem and the state feedback form of the solution are obtained for the primal problem. Under the Slater condition, the strong duality is proved between the dual problem and the primal problem, and the KKT condition is also provided for solving the primal problem. Moreover, a new sufficient condition is given for the invertibility assumption, which ensures the uniqueness of the solutions to the dual problem. Finally, two numerical examples are provided to illustrate our main results.

Research on Risk-Averse Procurement Optimization of Emergency Supplies for Mine Thermodynamic Disasters

Reducing procurement risks to ensure the supply of emergency supplies is crucial for mitigating the losses caused by mine thermodynamic disasters. The risk preference of decision-makers and supply chain collaboration are the important aspects for this reductiom. In this study, a novel P-CVaR (Piecewise conditional risk value) distributionally robust optimization model is proposed to accurately assist the decision-makers’ decision of risk preference for reducing procurement risks. Meanwhile, the role of cooperation between procurement and reserves are considered for the weakening procurement risks. A risk-averse bi-level optimization model is proposed to obtain the optimal procurement strategy. Furthermore, by applying the Lagrange duality theorem, the complexity of the bi-level optimization model is simplified then solved using a PSO algorithm. Using empirical analysis, it has been verified that the model presented in this paper serves as a valuable guideline for mine thermodynamic pre-disaster emergency material procurement strategies for the prevention of thermodynamic disasters.

UAV-Assisted Mobile Edge Computing: Dynamic Trajectory Design and Resource Allocation.

The recent advancements of mobile edge computing (MEC) technologies and unmanned aerial vehicles (UAVs) have provided resilient and flexible computation services for ground users beyond the coverage of terrestrial service. In this paper, we focus on a UAV-assisted MEC system in which the UAV equipped with MEC servers is used to assist user devices in computing their tasks. To minimize the weighted average energy consumption and delay in the UAV-assisted MEC system, a LQR-Lagrange-based DDPG (LLDDPG) algorithm, which jointly optimizes the user task offloading and the UAV trajectory design, is proposed. To be specific, the LLDDPG algorithm consists of three subproblems. The DDPG algorithm is used to address the issue of UAV desired trajectory planning, and subsequently, the LQR-based algorithm is employed to achieve the real-time tracking control of UAV desired trajectory. Finally, the Lagrange duality method is proposed to solve the optimization problem of computational resource allocation. Simulation results indicate that the proposed LLDDPG algorithm can effectively improve the system resource management and realize the real-time UAV trajectory design.

Active Distribution Network Expansion Planning Based on Wasserstein Distance and Dual Relaxation

In the future, a high proportion of distributed generations (DG) will be integrated into the distribution network. The existing active distribution network (ADN) planning methods have not fully considered multiple uncertainties, differentiated regulation modes or the cost of multiple types of interconnection switches. Meanwhile, it is difficult to solve large-scale problems at small granularity. Therefore, an expansion planning method of ADN considering the selection of multiple types of interconnection switches is proposed. Firstly, a probability distribution ambiguity set of DG output and electrical-load consumption based on the Wasserstein distance is established for dealing with the issue of source-load uncertainty. Secondly, a distributionally robust optimization model for collaborative planning of distribution network lines and multiple types of switches based on the previously mentioned ambiguity set is established. Then, the original model is transformed into a mixed integer second-order cone programming (SOCP) model by using the convex relaxation method, the Lagrangian duality method and the McCormick relaxation method. Finally, the effectiveness of the proposed method is systematically verified using the example of Portugal 54. The results indicate that the proposed method raises the annual net profit by nearly 5% compared with the traditional planning scheme and improves the reliability and low-carbon nature of the planning scheme.

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis

Polar duality and the reconstruction of quantum covariance matrices from partial data

We address the problem of the reconstruction of quantum covariance matrices using the notion of Lagrangian and symplectic polar duality introduced in previous work. We apply our constructions to Gaussian quantum states which leads to a non-trivial generalization of Pauli’s reconstruction problem and we state a simple tomographic characterization of such states.

Energy-efficient mobile edge computing assisted by layered UAVs based on convex optimization

The utilization of unmanned aerial vehicles (UAVs) as aerial mobile edge computing (MEC) servers presents an effective approach for mitigating the computational limitations of intelligent mobile devices. However, the limited battery capacity of UAVs poses a significant challenge in ensuring sustained service provision. To counter this obstacle and enhance energy efficiency, we introduce a partial offloading MEC scheme assisted by layered UAVs (POAL). Our objective is to minimize the total energy consumption of UAVs by jointly optimizing several factors, including wireless channels, transmit power, task partition, computing power, and UAV trajectories. Due to the non-convexity of the problem, we decompose it into three sub-problems and address them iteratively. Firstly, we employ the CVX optimization tool to allocate wireless channels and utilize Lagrangian duality to determine the optimal transmit power. Secondly, leveraging the monotonicity of the optimization objective, we derive the optimal computing power. Lastly, we employ the successive convex approximation (SCA) technique to tackle the trajectory planning sub-problem. Numerical results demonstrate that the proposed algorithm can effectively position UAVs within 15 iterations. Furthermore, our layered scheme significantly reduces overall energy consumption.

Robust optimal power flow considering uncertainty in wind power probability distribution

This paper proposes an optimal power flow model that takes into account the uncertainty in the probability distribution of wind power. The model can schedule controllable generators under any possible distribution of wind power to ensure the safe and economic operation of the system. Firstly, considering the incompleteness of historical wind power data, the paper models the uncertainty of wind power using second-order moments of probability distribution and their fluctuation intervals. Subsequently, a robust optimal power flow model based on probability distribution model and joint chance constraints is established. The Lagrangian duality theorem is then employed to eliminate random variables from the optimization model, transforming the uncertainty model into a deterministic linear matrix inequality problem. Finally, a convex optimization algorithm is used to solve the deterministic problem, and the results are compared with traditional chance-constrained optimal power flow model. The feasibility and effectiveness of the proposed method are validated through case study simulations.

ALM for insurers with multiple underwriting lines and portfolio constraints: a Lagrangian duality approach

We investigate how an insurance firm can optimally allocate its assets to back up liabilities from multiple non-life business lines. The insurance risks are modeled by a multidimensional jump-diffusion process that accounts for simultaneous claims in different insurance lines with policy limits. We use Lagrangian convex duality techniques to derive optimal investment-underwriting strategies that maximize the expected utility from dividends and final wealth over a finite horizon. We examine how risk aversion, prudence, portfolio constraints, and multivariate insurance risk affect the firm’s earnings retention. We obtain explicit solutions for optimal strategies under constant relative risk aversion preferences. Finally, we illustrate our results with numerical examples and show the impact of co-integration for asset-liability management with multiple sources of insurance risk.

Classically Computing Performance Bounds on Depolarized Quantum Circuits

Quantum computers and simulators can potentially outperform classical computers in finding ground states of classical and quantum Hamiltonians. However, if this advantage can persist in the presence of noise without error correction remains unclear. In this paper, by exploiting the principle of Lagrangian duality, we develop a numerical method to classically compute a certifiable lower bound on the minimum energy attainable by the output state of a quantum circuit in the presence of depolarizing noise. We provide theoretical and numerical evidence that this approach can provide circuit-architecture-dependent bounds on the performance of noisy quantum circuits. Published by the American Physical Society 2024

Intelligent Reflecting Surface Assisted Secure Computation of Wireless Powered MEC System

The integration of mobile edge computing (MEC) and wireless power transfer (WPT) can effectively improve the computing ability and energy sustainability of energy-constrained wireless devices in the Internet of Things (IoT) networks. Intelligent reflecting surface (IRS) has recently emerged as an effective technique to improve the performance of wireless systems by intelligently reconfiguring wireless environments. This paper studies the exploitation of IRS to improve the secure computation performance of WPT-MEC systems with a passive eavesdropper. A wireless access point (AP) first charge multiple users with the emitted energy signals, and then the users perform local computing and partial offloading to complete their computation tasks with the harvested energy in the presence of an eavesdropper, where the local computing can be executed during the whole process of WPT and offloading. Meanwhile, deploying IRS can improve the energy capture and secure offloading performance of the users. We maximize the secure computation task bits of users by jointly optimizing the AP energy transmit beamforming, the IRS phase shifts, the transmit power, users' offloading time, and the local computation frequency of users, which are tangled with each other. An iterative optimal algorithm is developed to solve this non-convex problem by combining Taylor expansion method, semidefinite relaxation (SDR) algorithm, the Lagrange duality theory and Karush-Kuhn-Tucker (KKT) conditions. The numerical results show that the proposed scheme can effectively increase the secure computation task bits compared with other benchmark schemes, especially for the maximum transmit power of AP, the improvement is above 45 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\%$</tex-math></inline-formula> .

Offloading Decision and Resource Allocation in Mobile Edge Computing for Cost and Latency Efficiencies in Real-Time IoT

Internet of Things (IoT) devices can integrate with applications requiring intensive contextual data processing, intelligent vehicle control, healthcare remote sensing, VR, data mining, traffic management, and interactive applications. However, there are computationally intensive tasks that need to be completed quickly within the time constraints of IoT devices. To address this challenge, researchers have proposed computation offloading, where computing tasks are sent to edge servers instead of being executed locally on user devices. This approach involves using edge servers located near users in cellular network base stations, and also known as Mobile Edge Computing (MEC). The goal is to offload tasks to edge servers, optimizing both latency and energy consumption. The main objective of this paper mentioned in the summary is to design an algorithm for time- and energy-optimized task offloading decision-making in MEC environments. Therefore, we developed a Lagrange Duality Resource Optimization Algorithm (LDROA) to optimize for both decision offloading and resource allocation for tasks, whether to locally execute or offload to an edge server. The LDROA technique produces superior simulation outcomes in terms of task offloading, with improved performance in computation latency and cost usage compared to conventional methods like Random Offloading, Load Balancing, and the Greedy Latency Offloading scheme.