Showing papers by "Aharon Ben-Tal published in 2005"

Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach

Abstract: We propose the use of robust optimization (RO) as a powerful methodology for multiperiod stochastic operations management problems. In particular, we study a two-echelon multiperiod supply chain problem, known as the retailer-supplier flexible commitment (RSFC) problem with uncertain demand that is only known to reside in some uncertainty set. We adopt a min-max criterion, whereby the cost function is minimized against the worst case demand occurrence. To solve the min-max RSFC problem we employ a recent extension of the RO method adapted to dynamic decision problems and known as the affinely adjustable robust counterpart (AARC) methodology. The AARC solution is tested by a large simulation study and found to provide excellent results.

Robust mean-squared error estimation in the presence of model uncertainties

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.

Non-euclidean restricted memory level method for large-scale convex optimization

Abstract: We propose a new subgradient-type method for minimizing extremely large-scale nonsmooth convex functions over “simple” domains. The characteristic features of the method are (a) the possibility to adjust the scheme to the geometry of the feasible set, thus allowing to get (nearly) dimension-independent (and nearly optimal in the large-scale case) rate-of-convergence results for minimization of a convex Lipschitz continuous function over a Euclidean ball, a standard simplex, and a spectahedron (the set of positive semidefinite symmetric matrices, of given size, with unit trace); (b) flexible handling of accumulated information, allowing for tradeoff between the level of utilizing this information and iteration’s complexity. We present extensions of the scheme for the cases of minimizing non-Lipschitzian convex objectives, finding saddle points of convex-concave functions and solving variational inequalities with monotone operators. Finally, we report on encouraging numerical results of experiments with test problems of dimensions up to 66,000.

A Global Solution for the Structured Total Least Squares Problem with Block Circulant Matrices

Abstract: We study the structured total least squares (STLS) problem of system of linear equations Ax = b, where A has a block circulant structure with N blocks. We show that by applying the discrete Fourier transform (DFT), the STLS problem decomposes into N unstructured total least squares (TLS) problems. The N solutions of these problems are then assembled to generate the optimal global solution of the STLS problem. Similar results are obtained for elementary block circulant matrices. Here the optimal solution is obtained by assembling two solutions: one of an unstructured TLS problem and the second of a multidimensional TLS problem.

MSE estimation of multichannel signals with model uncertainties

Abstract: We consider the problem of multichannel estimation, in which we seek to estimate multiple input vectors that are observed through a set of linear transformations and corrupted by additive noise. The input vectors x/sub k/ are known to satisfy a weighted norm constraint. We discuss both the case where the linear transformations are fixed (certain) and the case where they are only known to reside in some deterministic uncertainty set. We seek the linear estimator that minimizes the worst-case mean-squared error (MSE) across all possible values of the linear transformations and possible values of x/sub k/. We show that for an arbitrary choice of weighting matrix, the minimax MSE estimator can be formulated as a solution to a semidefinite programming problem (SDP). In the case in which the linear transformations are fixed and the norms are unweighed, the minimax MSE multichannel estimator has an explicit closed from solution. Finally, we demonstrate through examples, that the minimax MSE estimator can significantly increase the performance over conventional least-squares based methods.