We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, it is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.

Abstract—We consider the problem of estimating an unknown parameter vector x in a linear model that may be subject to uncertainties, where the vector x is known to satisfy a weighted norm constraint. We first assume that the model is known exactly and seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of x. We show that for an arbitrary choice of weighting, the optimal minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP), which can be solved very efficiently. We then develop a closed form expression for the minimax MSE estimator for a broad class of weighting matrices and show that it coincides with the shrunken estimator of Mayer and Willke, with a specific choice of shrinkage factor that explicitly takes the prior information into account. Next, we consider the case in which the model matrix is subject to uncertainties and seek the robust linear estimator that minimizes the worst-case MSE across all possible values of x and all possible values of the model matrix. As we show, the robust minimax MSE estimator can also be formulated as a solution to an SDP. Finally, we demonstrate through several examples that the minimax MSE estimator can significantly increase the performance over the conventional least-squares estimator, and when the model matrix is subject to uncertainties, the robust minimax MSE estimator can lead to a considerable improvement in performance over the minimax MSE estimator. Index Terms—Data uncertainty, linear estimation, mean squared error estimation, minimax estimation, robust estimation. I.

This paper develops a framework for state-space estimation when the parameters of the underlying linear model are subject to uncertainties. Compared with existing robust filters, the proposed filters perform regularization rather than de-regularization. It is shown that, under certain stabilizability and detectability conditions, the steady-state filters are stable and that, for quadratically-stable models, the filters guarantee a bounded error variance. Moreover, the resulting filter structures are similar to various (time- and measurement-update, prediction, and information) forms of the Kalman filter, albeit ones that operate on corrected parameters rather than on the given nominal parameters. Simulation results and comparisons with H1 , guaranteed-cost, and set-valued state estimation filters are provided.

Abstract. We consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints. When the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions. We then develop a connection between the image of the real and complex spaces under a quadratic mapping, which together with the results in the complex case lead to a condition that ensures strong duality in the real setting. Preliminary numerical simulations suggest that for random instances of the extended trust region subproblem, the sufficient condition is satisfied with a high probability. Furthermore, we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems. Finally, we discuss an application of our results to robust least squares problems.

We develop an algorithm for the solution of indefinite least-squares problems. Such problems arise in robust estimation, filtering, and control, and numerically stable solutions have been lacking. The algorithm developed herein involves the QR factorization of the coefficient matrix and is provably numerically stable. keywords Indefinite least-squares problems, error analysis, backward stability. 1 Introduction Many optimization criteria have been used for parameter estimation, starting with the standard least-squares formulation of Gauss (ca. 1795) and moving to more recent works on total leastsquares (TLS) and robust (or H 1 ) estimation (see, e.g., [3, 4, 6, 7, 8, 9]). The latter formulations have been motivated by an increasing interest in estimators that are less sensitive to data uncertainties and measurement errors. They can both be shown to require the minimization of indefinite quadratic forms, where the standard inner product of two vectors, say a T b, is replaced by an...

An analysis of a class of data tting problems, where the data uncertainties are subject to known bounds, is given in a very general setting. It is shown how such problems can be posed in a computationally convenient form, and the connection with other more conventional data fitting problems is examined. The problems have attracted interest so far in the special case when the underlying norm is the least squares norm. Here the special structure can be exploited to computational advantage, and we include some observations which contribute to algorithmic development for this particular case. We also consider some variants of the main problems and show how these too can be posed in a form which facilitates their numerical solution.

We formulate and solve new least squares type problems for parameter estimation in the presence of bounded data uncertainties. The new methods are suitable when a priori bounds on the uncertain data are available, and their solutions lead to more meaningful results especially when compared with other methods such as total least squares and robust estimation. Their superior performance is due to the fact that the new methods guarantee that the effect of the uncertainties will never be unnecessarily over-estimated, beyond what is reasonably assumed by the a priori bounds. Geometric interpretations of the solutions are provided, along with closed-form expressions for them.

Abstract—We consider the problem of estimating an unknown deterministic parameter vector in a linear model with a Gaussian model matrix. We derive the maximum likelihood (ML) estimator for this problem and show that it can be found using a simple line-search over a unimodal function that can be efficiently evaluated. We then discuss the similarity between the ML, the total least squares (TLS), the regularized TLS, and the expected least squares estimators. Index Terms—Errors in variables (EIV), linear models, maximum likelihood (ML) estimation, random model matrix, total least squares (TLS). I.

We consider the following problem min x2R n min kEk 2 j k(A E)x \Gamma bk2 where A is an m \Theta n real matrix and b is an n-dimensional real column vector, when it has multiple global minima. An efficient algorithm is presented to find the global solution of minimum Euclidean norm.

. We pose and solve a parameter estimation problem in the presence of bounded data uncertainties. The problem involves a minimization step and admits a closed form solution in terms of the positive root of a secular equation. Key words. Least-squares estimation, total least-squares, modeling errors, secular equation. AMS subject classifications. 15A06, 65F05, 65F10, 65F35, 65K10, 93C41, 93E10, 93E24 1. Introduction. Parameter estimation in the presence of data uncertainties is a problem of considerable practical importance, and many estimators have been proposed in the literature with the intent of handling modeling errors and measurement noise. Among the most notable is the total least-squares method [1, 2, 3, 4], also known as orthogonal regression or errors-in-variables method in statistics and system identification [5]. In contrast to the standard least-squares problem, the TLS formulation allows for errors in the data matrix. Its performance may degrade in some situations where ...