An efficient nonlinear interval uncertain optimization method using Legendre polynomial chaos expansion

Chapter 25 - Uncertain optimization (with a special focus on data envelopment analysis)

Group elevator scheduling (GES) problem is an optimization problem. In this work, a robust optimization scheduling method is proposed to solve elevator group control system problem with passenger traffic flow uncertainty. An uncertain optimization model of elevator group scheduling is set up based on considering both energy consumption and waiting time. When considering the uncertainty of passenger traffic flow, the model is an uncertain optimization problem. It is tuned into its robust counterpart (RC) based on robust optimization theory. The simulation experiment presents the proposed scheduling method's scheduling performance.

Generally speaking, the navigation of underwater targets is solved by the principle of spherical intersection. The number of array elements required for this method is fixed, but in the case of redundant array elements, they can’t be fully utilized. This paper presents two optimization methods based on redundancy measurement. One approach is the Gauss-Newton method and the other is Non linear Optimization method. The former, a non-linear least-squares optimization method that constantly revise regression coefficients through multiple iterations to minimize the sum of squared residuals, and the later, getting least squares solution of non linear redundant equations based on penalty constants. Simulation show that both methods are greatly affected by the initial position, the closer the distance between initial position to the target is, the smaller navigation error is. Under the same simulation conditions, the navigation accuracy of these two methods is higher than that of the ordinary spherical intersection method. The experiment proves these two methods can satisfy the navigation accuracy of the experiment in actual project.

In this paper, the minimization of an unknown but measurable cost function of the state variables of nonlinear systems governed by uncertain dynamics is considered. An extremum-seeking algorithm is proposed to solve this uncertain dynamic optimization problem without the need for a time-scale separation. The Lie derivatives of the convex cost function with respect to nonlinear dynamics of the system are regarded as time-varying parameters. A new technique based on the concept of invariant manifolds is proposed for the adaptive estimation of the time-varying parameters. A nonlinear proportional-integral approach is then used to formulate the extremum-seeking controller. This approach is shown to avoid the need for a time-scale separation in real-time optimization problem. The effectiveness of the proposed method is illustrated with a simulation example.

In this paper we consider general form of a optimization problem in face of data uncertainty. We determine a necessary and sufficient condition ensuring the equality between the worst value and the robust value of this uncertain problem with attainment of the worst value. And we deduce robust strong duality and robust stable strong duality property. Keywords: Uncertain optimization problem; Robust counterpart; Robust value; Worst value; Robust strong duality. Mathematics Subject Classification: 90C17, 90C31, 90C46

A calibration method for stereo vision sensor with large FOV based on 1D targets

An experimental study of neutron elastic scattering at forward angles

The coordinate-transformation-based differential method developed by Chandezon et al. is recognized as one of the simplest and most versatile approaches for modeling surface-relief gratings. In this study, we present a novel numerical solution using Legendre polynomial expansion, enabling us to deal efficiently with faceted gratings. Additionally, we introduce an oblique coordinate transformation to analyze overhanging faceted gratings. Notably, the C-method with polynomial expansion (CPE) demonstrates a dramatic improvement in convergence speed compared to the Fourier Modal Method (FMM).

The radiative transfer equation (RTE) is fundamental in modelling photon transport across various scientific and engineering domains, including biomedical optics, atmospheric physics, and nuclear engineering. This study investigates photon propagation in a homogeneous medium using the Henyey-Greenstein (HG) scattering function, which is widely applied in biophysics to describe light scattering in biological tissues. The anisotropy parameter g characterizes scattering properties, playing a crucial role in photon movement. To solve the RTE for a steady-state homogeneous system, we employed two polynomial approximation methods: Legendre polynomial expansion (PN method) and second-kind Chebyshev polynomial expansion (UN method). The eigenvalues play a central role in classical problems such as the albedo problem, the Milne problem, and criticality analysis in slab geometry. We iteratively solve the RTE up to the fifth order and compare the convergence and accuracy of both methods. The results indicated that increasing the number of iterations improves accuracy, with both methods exhibiting stable convergence behaviour. A detailed comparison was conducted and the numerical results were validated against existing literature. Additionally, the tabulated results highlighted the forward and backward scattering components of the HG function, illustrating the influence of the anisotropy parameter on photon transport. The consistency between the proposed methods and the literature was confirmed by comparing discrete eigenvalues in tables. The maximum deviation between the P₉ and U₉ methods remained below 0.08, while both approaches closely matched the benchmark results in literature, supporting the reliability and numerical robustness of the polynomial expansions used. This study offers a valuable reference for future research, enabling comparisons and validation of computational approaches in radiative transfer modelling, particularly in biophysics and optical applications.

Synthetic Aperture Sonar (SAS) achieves high-resolution imaging of underwater targets by moving a small-aperture array and coherently processing the signals, thereby generating a virtual large-aperture imaging effect. The conventional Range-Doppler (RD) algorithm offers advantages such as high computational efficiency, simple processing flow, and ease of engineering implementation. However, due to the limitations of its underlying assumptions and its heavy reliance on interpolation for range walk correction, the RD algorithm cannot achieve both imaging efficiency and accuracy. To address this issue, this paper proposes an improved RD algorithm based on a sparse Bayesian learning two-dimensional deconvolution (SBL-dCv2) framework. The mathematical model of the algorithm is derived, and comparative simulation experiments with the traditional RD algorithm are conducted for multi-point targets. The simulation results show that the SBL-dCv2 improved RD algorithm delivers both high resolution and computational efficiency, offers stronger resolution performance for multi-point targets, and is better suited for practical engineering applications of SAS.

This survey reviews the recent development of gradient domain mesh deformation method. Different to other deformation methods, the gradient domain deformation method is a surface-based, variational optimization method. It directly encodes the geometric details in differential coordinates, which are also called Laplacian coordinates in literature. By preserving the Laplacian coordinates, the mesh details can be well preserved during deformation. Due to the locality of the Laplacian coordinates, the variational optimization problem can be casted into a sparse linear system. Fast sparse linear solver can be adopted to generate deformation result interactively, or even in real-time. The nonlinear nature of gradient domain mesh deformation leads to the development of two categories of deformation methods: linearization methods and nonlinear optimization methods. Basically, the linearization methods only need to solve the linear least-squares system once. They are fast, easy to understand and control, while the deformation result might be suboptimal. Nonlinear optimization methods can reach optimal solution of deformation energy function by iterative updating. Since the computation of nonlinear methods is expensive, reduced deformable models should be adopted to achieve interactive performance. The nonlinear optimization methods avoid the user burden to input transformation at deformation handles, and they can be extended to incorporate various nonlinear constraints, like volume constraint, skeleton constraint, and so on. We review representative methods and related approaches of each category comparatively and hope to help the user understand the motivation behind the algorithms. Finally, we discuss the relation between physical simulation and gradient domain mesh deformation to reveal why it can achieve physically plausible deformation result.

A new nonlinear optimal control method is proposed for solving the problem of control and stabilization of a two-link robotic manipulator where the first link is rigid and the second link is flexible. The dynamic model of this robot describes both the rotational motion of the two links and the vibration modes which are excited at the flexible link. The control and stabilization problem of this 2-DOF rigid-flexible robot is nontrivial due to complex nonlinear dynamics and underactuation. To apply the proposed nonlinear optimal control method, the dynamic model of the two-link rigid-flexible robotic manipulator undergoes first approximate linearization around a temporary operating point that is updated at each iteration of the control algorithm. The linearization takes place through first-order Taylor series expansion and through the computation of the Jacobian matrices of the flexible manipulator’s state-space description. For the approximately linearized model of the two-link rigid-flexible robot, an H-infinity feedback controller is designed. Actually, the H-infinity controller stands for the solution of the optimal control problem for the rigid-flexible robot under uncertainty and external perturbations. For the computation of the feedback gains of the H-infinity controller, an algebraic Riccati equation is solved at each time-step of the control method. The stability properties of the control algorithm are proven through Lyapunov analysis. First, it is shown that the control scheme achieves H-infinity tracking performance which signifies robustness for the control loop of the two-link rigid-flexible robot under uncertainties and external perturbations. Next, it is also shown that the control loop of the 2-DOF rigid-flexible robot is globally asymptotically stable. The proposed control method achieves fast and accurate tracking of setpoints under moderate variations of the control inputs. The concept of the proposed nonlinear optimal control method has been to approximate locally and at each sampling interval the nonlinear dynamics of the rigid-flexible robotic manipulator by an equivalent linear state-space description and to solve the H-infinity control problem for the linearized model. Finally, the control inputs are applied to the initial nonlinear dynamics and stability proof is given for the initial nonlinear state-space model.

Data in many real life engineering and economic problems suffer from inexactness. In the real world there are many forms of uncertainty that affect the utility function. Uncertainty always exists in practical engineering problems. To deal with uncertain optimization problems, fuzzy, stochastic and interval numbers approaches are commonly used to describe the imprecise characteristics. Herein, we consider the estimation of the scaling constants of the additive utility function under a fuzzy environment. The investigated procedure uses the Kuhn-Tucker necessary optimality conditions. An illustrative numerical example is given to clarify the developed procedure.

This paper presents a novel real-time algorithm to find unknown maximum power of photovoltaic (PV) systems. The maximum power of a typical PV system is continuously changing with variations in solar irradiance and PV module temperature. There is no precise mathematical equation to show relation between PV output power and environmental variables. Since power of a PV cell/module is unknown nonlinear and time-varying function of irradiance and temperature, real-time approaches are required to find maximum power values through time. Based on strong concavity feature of PV output power, one can consider this measurable signal as an unknown objective function that needs to be maximized. Hence, an estimation-based extremum-seeking control algorithm is proposed to solve this nonlinear and uncertain optimization problem. First, gradient of the objective function with respect to the PV voltage is treated as an unknown time-varying parameter. An adaptive estimation technique is utilized to identify this parameter. Then, a direct gradient-based adaptive controller is designed to find optimum voltage value and the corresponding maximum power point at uniform and uncertain weather condition in real time. Unlike most model-free approaches, the proposed algorithm has proof of convergence with precise steady-state and fast-transient performance.

Plant-wide byproduct gas distribution under uncertainty in iron and steel industry via quantile forecasting and robust optimization