scholarly journals Minimax Estimation in Regression under Sample Conformity Constraints

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1080
Author(s):  
Andrey Borisov
Keyword(s):  
Optimization Problem ◽  
Probability Distributions ◽  
Essential Feature ◽  
Estimation Error ◽  
Saddle Points ◽  
Minimax Problem ◽  
Estimation Of Parameters ◽  
Minimax Optimization ◽  
Optimal Estimator ◽  
Regression Parameters

The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.

scholarly journals Geometric methods for optimal sensor design

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150312 ◽  
Author(s):  
M.-A. Belabbas
Keyword(s):  
Structural Equation ◽  
Optimization Problem ◽  
Dual Problem ◽  
Morse Function ◽  
Ad Hoc ◽  
Signal To Noise Ratio ◽  
Estimation Error ◽  
Gradient Algorithm ◽  
Optimal Estimator ◽  
Actuator Design

The Kalman–Bucy filter is the optimal estimator of the state of a linear dynamical system from sensor measurements. Because its performance is limited by the sensors to which it is paired, it is natural to seek optimal sensors. The resulting optimization problem is however non-convex. Therefore, many ad hoc methods have been used over the years to design sensors in fields ranging from engineering to biology to economics. We show in this paper how to obtain optimal sensors for the Kalman filter. Precisely, we provide a structural equation that characterizes optimal sensors. We furthermore provide a gradient algorithm and prove its convergence to the optimal sensor. This optimal sensor yields the lowest possible estimation error for measurements with a fixed signal-to-noise ratio. The results of the paper are proved by reducing the optimal sensor problem to an optimization problem on a Grassmannian manifold and proving that the function to be minimized is a Morse function with a unique minimum. The results presented here also apply to the dual problem of optimal actuator design.

scholarly journals Duality Theorem for a Minimax Vector Optimization Problem

2021 ◽  
Author(s):  
Jacob Atticus Armstrong Goodall
Keyword(s):  
Vector Optimization ◽  
Duality Theorem ◽  
Optimization Problem ◽  
Optimal Transport ◽  
Probability Distributions ◽  
Vector Optimization Problem ◽  
Complexity Science ◽  
Polish Spaces ◽  
Leibler Divergence ◽  
Mathematical Optimisation

Abstract A duality theorem is stated and proved for a minimax vector optimization problem where the vectors are elements of the set of products of compact Polish spaces. A special case of this theorem is derived to show that two metrics on the space of probability distributions on countable products of Polish spaces are identical. The appendix includes a proof that, under the appropriate conditions, the function studied in the optimisation problem is indeed a metric. The optimisation problem is comparable to multi-commodity optimal transport where there is dependence between commodities. This paper builds on the work of R.S. MacKay who introduced the metrics in the context of complexity science in [4] and [5]. The metrics have the advantage of measuring distance uniformly over the whole network while other metrics on probability distributions fail to do so (e.g total variation, Kullback–Leibler divergence, see [5]). This opens up the potential of mathematical optimisation in the setting of complexity science.

The multiple scattering of waves in irregular media

1968 ◽  
Vol 262 (1133) ◽  
pp. 609-640 ◽  
Keyword(s):  
Wave Field ◽  
Probability Distributions ◽  
Essential Feature ◽  
Mean Square ◽  
Angular Power Spectrum ◽  
Absorbing Medium ◽  
Phase Fluctuations ◽  
Field Fluctuations ◽  
The Mean ◽  
Wave Normal

When a wave passes through a large thickness of a non-absorbing medium containing weak random irregularities of refractive index, large amplitude and phase fluctuations of the wave field can develop. The probability distributions of these fluctuations are important, since they may be readily observed and from them can be found the mean square amplitudes of the fluctuations. This paper shows how to calculate these distributions and also the ‘ angular power spectrum ’ for an assembly of media which are statistically stationary with respect to variations in time, and in space for directions perpendicular to the wave normal of the incident wave. The scattered field at a given point is resolved into two components in phase and in quadrature with the residual unscattered wave at that point. The assembly averages of the powers in these two components, and of their correlation coefficient are found, and a set of three integro-differential equations is constructed which show how these three quantities vary as the medium is traversed. The probability distributions of amplitude and phase of the wave field at any point in the medium are functions of these three quantities which are found by integrating the equations through the medium. An essential feature of these equations is that they include waves which have been scattered several or m any times (multiple scatter). The equations are solved analytically for some particular cases. Solutions for the general case have been obtained numerically and are presented, together with the corresponding probability distributions of the field fluctuations and their average values.

scholarly journals ϵ-Henig Saddle Points and Duality of Set-Valued Optimization Problems in Real Linear Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhi-Ang Zhou
Keyword(s):  
Saddle Point ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Saddle Points ◽  
Linear Spaces ◽  
Equivalent Characterization ◽  
Generalized Cone Subconvexlikeness ◽  
Real Linear ◽  
The Relationship

We studyϵ-Henig saddle points and duality of set-valued optimization problems in the setting of real linear spaces. Firstly, an equivalent characterization ofϵ-Henig saddle point of the Lagrangian set-valued map is obtained. Secondly, under the assumption of the generalized cone subconvexlikeness of set-valued maps, the relationship between theϵ-Henig saddle point of the Lagrangian set-valued map and theϵ-Henig properly efficient element of the set-valued optimization problem is presented. Finally, some duality theorems are given.

scholarly journals Criteria of saddle points for the general form of vector optimization problem in complex space

Filomat ◽  
2020 ◽  
Vol 34 (1) ◽  
pp. 221-230
Author(s):  
Mamdouh Elbrolosy
Keyword(s):  
Vector Optimization ◽  
Optimization Problem ◽  
Complex Space ◽  
Saddle Points ◽  
Vector Optimization Problem

scholarly journals A new three-term conjugate gradient method for solving the finite minimax problems

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 737-758
Author(s):  
Yue Hao ◽  
Shouqiang Du ◽  
Yuanyuan Chen
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Optimization Problem ◽  
Minimax Problems ◽  
Minimax Problem ◽  
Constrained Optimization Problem ◽  
Nonlinear Conjugate Gradient Method ◽  
Nonlinear Conjugate Gradient ◽  
Finite Minimax

In this paper, we consider the method for solving the finite minimax problems. By using the exponential penalty function to smooth the finite minimax problems, a new three-term nonlinear conjugate gradient method is proposed for solving the finite minimax problems, which generates sufficient descent direction at each iteration. Under standard assumptions, the global convergence of the proposed new three-term nonlinear conjugate gradient method with Armijo-type line search is established. Numerical results are given to illustrate that the proposed method can efficiently solve several kinds of optimization problems, including the finite minimax problem, the finite minimax problem with tensor structure, the constrained optimization problem and the constrained optimization problem with tensor structure.

Restricted Complexity Controller Design via Multi-Objective Optimization

2003 ◽  
Author(s):  
Wenjun Yan ◽  
Yi Cao
Keyword(s):  
Optimization Problem ◽  
Controller Design ◽  
Least Square ◽  
Controller Synthesis ◽  
Multiple Objective Optimization ◽  
Multi Objective Optimization ◽  
Minimax Optimization ◽  
Optimal Controllers ◽  
Simulation And Experiment ◽  
Synthesis Approach

The problem of restricted complexity controller synthesis is recast into a multiple objective optimization problem. The general H∞ synthesis approach is introduced first to reveal the limitation of this method. Using nonlinear least square and minimax optimization algorithms, sub-optimal controllers have been developed to satisfy specifications of the frequency performance and transient performance. The simulation and experiment results illustrate that these two methods are very effective in finding low order controllers.

Robustness of Design Through Minimum Sensitivity

1992 ◽  
Vol 114 (2) ◽  
pp. 213-217 ◽  
Author(s):  
A. D. Belegundu ◽  
Shenghua Zhang
Keyword(s):  
Optimization Problem ◽  
Probability Distributions ◽  
Uncertain Variable ◽  
Design Stage ◽  
Cross Sectional ◽  
Uncertain Variables ◽  
Design Variables ◽  
Variable Reduction ◽  
Minimum Sensitivity ◽  
Selection Of

The problem of designing mechanical systems or components under uncertainty is considered. The basic idea is to ensure quality control at the design stage by minimizing sensitivity of the response to uncertain variables by proper selection of design variables. The formulation does not involve probability distributions. It is proved, however, that when the response is linear in the uncertain variable, reduction in sensitivity implies lesser probability of failure. The proof is generalized to the non-linear case under certain restrictions. In one example, the design of a three-bar truss is considered. The length of one of the bars is considered to be the uncertain variable while cross-sectional areas are the design variables. The sensitivity of the x-displacement is minimized. The constrained optimization problem is solved using a nonlinear programming code. A criterion which can help identify some of the problems where robustness in design is critical is discussed.

scholarly journals Genetic algorithms applied to a faster distance protection of transmission lines

2011 ◽  
Vol 22 (4) ◽  
pp. 334-344
Author(s):  
Denis V. Coury ◽  
Mário Oleskovicz ◽  
Silvio A. Souza
Keyword(s):  
Genetic Algorithms ◽  
Power Systems ◽  
Transmission Lines ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Estimation Error ◽  
Distance Protection ◽  
Selection Principles ◽  
Program Software ◽  
Speed And Accuracy

The main purpose of this paper is to implement a new methodology based on Genetic Algorithms (GAs) to extract the fundamental voltage and current phasors from noisy waves in power systems to be applied to a faster distance protection. GAs solve optimization problems based on natural selection principles. This application was then formulated as an optimization problem, and the aim was to minimize the estimation error. A 440 kV, 150 km transmission line was simulated using the ATP (Alternative Transients Program) software in order to show the efficiency of the new method. The results from this application show that the global performance of GAs was highly satisfactory concerning speed and accuracy of response, if compared to the traditional Discrete Fourier Transform (DFT).

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