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2012

Volume 22, Issue 1 (partial)


A Distance For Multistage Stochastic Optimization Models

Georg Ch. Pflug and Alois Pichler

SIAM J. Optim. 22, pp. 1-23 (23 pages)

Online Publication Date: January 05, 2012

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We describe multistage stochastic programs in a purely in-distribution setting, i.e., without any reference to a concrete probability space. The concept is based on the notion of nested distributions, which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made. The nested distance between these distributions is introduced and turns out to be a generalization of the Wasserstein distance for stochastic two-stage problems. We give characterizations of this distance and show its usefulness in examples. The main result states that the difference of the optimal values of two multistage stochastic programs, which are Lipschitz and differ only in the nested distribution of the stochastic parameters, can be bounded by the nested distance of these distributions. This theorem generalizes the well-known Kantorovich–Rubinstein theorem, which is applicable only in two-stage situations, to multistage. Moreover, a dual characterization for the nested distance is established. The setup is applicable both for general stochastic processes and for finite scenario trees. In particular, the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation. Minimizing—at least heuristically—this distance is what good scenario tree generation is all about.

Fixed Points of Averages of Resolvents: Geometry and Algorithms

Heinz H. Bauschke, Xianfu Wang, and Calvin J. S. Wylie

SIAM J. Optim. 22, pp. 24-40 (17 pages)

Online Publication Date: January 13, 2012

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To provide generalized solutions if a given problem admits no actual solution is an important task in mathematics and the natural sciences. It has a rich history dating back to the early 19th century, when Carl Friedrich Gauss developed the method of least squares of a system of linear equations—its solutions can be viewed as fixed points of averaged projections onto hyperplanes. A powerful generalization of this problem is to find fixed points of averaged resolvents (i.e., firmly nonexpansive mappings). This paper concerns the relationship between the set of fixed points of averaged resolvents and certain fixed point sets of compositions of resolvents. It partially extends recent work for two mappings on a question of C. Byrne. The analysis suggests a reformulation in a product space. Algorithmic consequences are also presented.

Perturbation Analysis of Error Bounds for Quasi-subsmooth Inequalities and Semi-infinite Constraint Systems

Xi Yin Zheng and Zhou Wei

SIAM J. Optim. 22, pp. 41-65 (25 pages)

Online Publication Date: January 13, 2012

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The stability of error bounds is significant in optimization theory and applications. Based on either the linearity assumption or the convexity and finite dimension assumption, several authors have focused on perturbation analysis of error bounds and obtained valuable results. Mainly motivated by Ngai, Kruger, and Théra [SIAM J. Optim., 20 (2010), pp. 2080–2096], in a general Banach space, we study the stability of error bounds for inequalities determined by proper lower semicontinuous quasi-subsmooth functions which are a very large class of nonconvex functions (in particular, approximate convex functions, primal-lower-nice functions, and convexly composite functions satisfying the Robinson qualification). We also consider the stability of error bounds for infinite constraint systems determined by infinitely many uniformly quasi-subsmooth functions. In particular, we extend the main results of Ngai, Kruger, and Théra to the infinite dimensional and nonconvex setting.

On the Oracle Complexity of First-Order and Derivative-Free Algorithms for Smooth Nonconvex Minimization

Coralia Cartis, Nicholas I. M. Gould, and Philippe L. Toint

SIAM J. Optim. 22, pp. 66-86 (21 pages)

Online Publication Date: January 17, 2012

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The (optimal) function/gradient evaluations worst-case complexity analysis available for the adaptive regularization algorithms with cubics (ARC) for nonconvex smooth unconstrained optimization is extended to finite-difference versions of this algorithm, yielding complexity bounds for first-order and derivative-free methods applied on the same problem class. A comparison with the results obtained for derivative-free methods by Vicente [Worst Case Complexity of Direct Search, Technical report, Preprint 10-17, Department of Mathematics, University of Coimbra, Coimbra, Portugal, 2010] is also discussed, giving some theoretical insight into the relative merits of various methods in this popular class of algorithms.

Maximum Block Improvement and Polynomial Optimization

Bilian Chen, Simai He, Zhening Li, and Shuzhong Zhang

SIAM J. Optim. 22, pp. 87-107 (21 pages)

Online Publication Date: January 24, 2012

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In this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem. The new approach has the following three main ingredients. First, we establish a block coordinate descent type search method for nonlinear optimization, with the novelty being that we accept only a block update that achieves the maximum improvement, hence the name of our new search method: maximum block improvement (MBI). Convergence of the sequence produced by the MBI method to a stationary point is proved. Second, we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem; thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the MBI approach. Third, we propose a scheme to reach a KKT point of the polynomial optimization, provided that a stationary solution for the relaxed tensor problem is available. Numerical experiments have shown that our new method works very efficiently: for a majority of the test instances that we have experimented with, the method finds the global optimal solution at a low computational cost.

On the Solution of the GPS Localization and Circle Fitting Problems

Amir Beck and Dror Pan

SIAM J. Optim. 22, pp. 108-134 (27 pages)

Online Publication Date: January 26, 2012

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We consider the problem of locating a user's position from a set of noisy pseudoranges to a group of satellites. We consider both the nonlinear least squares formulation of the problem, which is nonconvex and nonsmooth, and the nonlinear squared least squares variant, in which the objective function is smooth, but still nonconvex. We show that the squared least squares problem can be reformulated as a generalized trust region subproblem and as such can be solved efficiently. Conditions for attainment of the optimal solutions of both problems are derived. The nonlinear least squares problem is shown to have tight connections to the well-known geometric circle fitting and orthogonal regression problems. Finally, a fixed point method for the nonlinear least squares formulation is derived and analyzed.

Projection-like Retractions on Matrix Manifolds

P.-A. Absil and Jérôme Malick

SIAM J. Optim. 22, pp. 135-158 (24 pages)

Online Publication Date: January 26, 2012

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This paper deals with constructing retractions, a key step when applying optimization algorithms on matrix manifolds. For submanifolds of Euclidean spaces, we show that the operation consisting of taking a tangent step in the embedding Euclidean space followed by a projection onto the submanifold is a retraction. We also show that the operation remains a retraction if the projection is generalized to a projection-like procedure that consists of coming back to the submanifold along “admissible” directions, and we give a sufficient condition on the admissible directions for the generated retraction to be second order. This theory offers a framework in which previously proposed retractions can be analyzed, as well as a toolbox for constructing new ones. Illustrations are given for projection-like procedures on some specific manifolds for which we have an explicit, easy-to-compute expression.
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