A new methodology for constructing convex optimization models called disciplined convex programming is introduced. The methodology enforces a set of conventions upon the models constructed, in turn allowing much of the work required to analyze and solve the models to be automated.

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Disciplined Convex Programming

The algorithm presented aims to segment the foreground objects in video (e.g., people) given time-varying, textured backgrounds. Examples of time-varying backgrounds include waves on water, clouds moving, trees waving in the wind, automobile traffic, moving crowds, escalators, etc. We have developed a novel foreground-background segmentation algorithm that explicitly accounts for the nonstationary nature and clutter-like appearance of many dynamic textures. The dynamic texture is modeled by an autoregressive moving average model (ARMA). A robust Kalman filter algorithm iteratively estimates the intrinsic appearance of the dynamic texture, as well as the regions of the foreground objects. Preliminary experiments with this method have demonstrated promising results.

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Segmenting foreground objects from a dynamic textured background via a robust Kalman filter

In this paper, a robust finite-horizon Kalman filter is designed for discrete time-varying uncertain systems with both additive and multiplicative noises. The system under consideration is subject to both deterministic and stochastic uncertainties. Sufficient conditions for the filter to guarantee an optimized upper bound on the state estimation error variance for admissible uncertainties are established in terms of two discrete Riccati difference equations. A numerical example is given to show the applicability of the presented method.

/pdf/robust-kalman-filtering-for-discrete-time-varying-uncertain-1yaktz0bn0.pdf

Robust Kalman filtering for discrete time-varying uncertain systems with multiplicative noises

A convex optimization based beampattern synthesis method with antenna selection is proposed for linear and planar arrays. Conjugate symmetric beamforming weights are used so that the upper and non-convex lower bound constraints on the beampattern can be convex. Thus, a mainlobe of an arbitrary beamwidth and response ripple can be obtained. This method can achieve completely arbitrary sidelobe levels. By minimizing a re-weighted objective function based on the magnitudes of the elements in the beamforming weight vector iteratively, the proposed method selects certain antennas in an array to satisfy the prescribed beampattern specifications precisely. Interestingly, a sparse array with fewer antennas (compared to other methods) is produced. This method can design non-uniformly spaced arrays with inter-element spacings larger than one half-wavelength, without the appearance of grating lobes in the resulting beampattern. Simulations are shown using arrays of up to a few hundred antennas to illustrate the practicality of the proposed method.

Beampattern Synthesis for Linear and Planar Arrays With Antenna Selection by Convex Optimization

Preface- Optimization under Composite Monotonic Constraints and Constrained Optimization over the Efficient Set (H Tuy, NT Hoai-Phuong)- On a Local Search for Reverse Convex Problems (A Strekalovsky)- Some Transformation Techniques in Global Optimization (T Westerlund)- Solving Nonlinear Mixed Integer Stochastic Problems: A Global Perspective (ME Bruni)- Application of Quasi Monte Carlo Methods in Global Optimization (S Kucherenko)- GLOB-A New VNS-Based Software for Global Optimization (M Drazic, V Kovacevic-Vujcic, M Cangalovic, N Mladenovic)- Disciplined Convex Programming (M Grant, S Boyd, Y Ye)- Writing Global Optimization Software (L Liberti)- MathOptimizer Professional: Key Features and Illustrative Applications (JD Pinter, FJ Kampas)- Variable Neighborhood Search for Extremal Graphs 14: The AutoGraphiX 2 System (M Aouchiche, JM Bonnefoy, A Fidahoussen, G Caporossi, P Hansen, L Hiesse, J Lachere, A Monhait)- From Theory to Implementation: Applying Metaheuristics (IJ Garcia del Amo, F Garcia Lopez, M Garcia Torres, B Melian Batista, JA Moreno Perez, JM Moreno Vega)- ooMILP-A C++ Callable Object-Oriented Library and the Implementation of Its Parallel Version Using CORBA (P Tsiakis, B Keeping)- Global Order-Value Optimization by Means of a Multistart Harmonic Oscillator Tunneling Strategy (R Andreani, JM Martinez, M Salvatierra, F Yano)- On Generating Instances for the Molecular Distance Geometry Problem (C Lavor)- Index

Global optimization : from theory to implementation

We present a robust recursive Kalman filtering algorithm that addresses estimation problems that arise in linear time-varying systems with stochastic parametric uncertainties. The filter has a one-step predictor-corrector structure and minimizes an upper bound of the mean square estimation error at each step, with the minimization reduced to a convex optimization problem based on linear matrix inequalities. The algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant. A numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional Kalman filtering techniques.

Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties

We present new algorithms for robust stability analysis and gain-scheduled controller synthesis for linear systems affected by time-varying parametric uncertainties. These new techniques can also be applied to parameter-dependent nonlinear systems with real-rational nonlinearities. Sufficient conditions for robust stability, as well as conditions for the existence of a robustly stabilizing gain-scheduled controller, are given in terms of a finite number of linear matrix inequalities (LMIs); explicit formulas for constructing robustly stabilizing gain-scheduled controllers are given in terms of the feasible set of these LMIs. The improvement offered by our approach over existing methods for stability analysis and gain-scheduled controller synthesis for parameter-dependent linear systems are analyzed in theory. Numerical examples demonstrate that our approach can offer significant improvement in practice.

Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems

We present algorithms for the solution of two problems in array pattern synthesis. The first is the design of nonuniform arrays with a desired magnitude response. The second is that of robust design, i.e., design in the presence of uncertainties. Constraints such as power limitation can be addressed with both problems. The algorithms that we present are based on semidefinite programming, for which efficient software is readily available. We present examples that illustrate the effectiveness of our approach.

Optimal array pattern synthesis using semidefinite programming

We present a new technique to solve array pattern synthesis problems by using semidefinite programming. We first formulate (or reformulate) the array design problems into semidefinite programming problems, and then use the recently developed efficient numerical algorithms and software to compute the numerical solution of antenna array weights. Using this approach, we can directly solve not only the standard synthesis problems for nonuniform arrays, but also the synthesis problems for arrays having power restrictions and uncertainties. Numerical examples are presented to illustrate our approach.

For uncertain systems containing both deterministic and stochastic uncertainties, we consider two problems of optimal filtering. The first is the design of a linear time-invariant filter that minimizes an upper bound on the mean energy gain between the noise affecting the system and the estimation error. The second is the design of a linear time-invariant filter that minimizes an upper bound on the asymptotic mean square estimation error when the plant is driven by a white noise. We present filtering algorithms that solve each of these problems, with the filter parameters determined via convex optimization based on linear matrix inequalities. We demonstrate the performance of these robust algorithms on a numerical example consisting of the design of equalizers for a communication channel.

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Fan Wang

Papers

Robust Kalman filters for linear time-varying systems with stochastic parametric uncertainties

Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems

Optimal array pattern synthesis using semidefinite programming

Optimal array pattern synthesis using semidefinite programming

Robust steady-state filtering for systems with deterministic and stochastic uncertainties