This article deals with the scenario approach to robust optimization. This relies on a random sampling of the possibly infinite number of constraints induced by uncertainties in the parameters of an optimization probl...
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This article deals with the scenario approach to robust optimization. This relies on a random sampling of the possibly infinite number of constraints induced by uncertainties in the parameters of an optimization problem. Solving the resulting random program yields a solution for which the quality is measured in terms of the probability of violating the constraints for a random value of the uncertainties, typically unseen before. Another central issue is the determination of the sample complexity, i.e., the number of random constraints (or scenarios) that one must consider in order to guarantee a certain reliability. In this article, we introduce an additional margin in the constraints and analyze the probability of violation of solutions to the modified random programs. In particular, using tools from statistical learning theory, we show that the sample complexity of a class of problems does not explicitly depend on the number of variables. In addition, within this class, that includes polynomial constraints among others, the same guarantees hold for both convex and nonconvex instances.
We propose a randomized second-order method for optimization known as the Newton sketch: it is based on performing an approximate Newton step using a randomly projected Hessian. For self-concordant functions, we prove...
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We propose a randomized second-order method for optimization known as the Newton sketch: it is based on performing an approximate Newton step using a randomly projected Hessian. For self-concordant functions, we prove that the algorithm has superlinear convergence with exponentially high probability, with convergence and complexity guarantees that are independent of condition numbers and related problem-dependent quantities. Given a suitable initialization, similar guarantees also hold for strongly convex and smooth objectives without self-concordance. When implemented using randomized projections based on a subsampled Hadamard basis, the algorithm typically has substantially lower complexity than Newton's method. We also describe extensions of our methods to programs involving convex constraints that are equipped with self-concordant barriers. We discuss and illustrate applications to linear programs, quadratic programs with convex constraints, logistic regression, and other generalized linear models, as well as semidefinite programs.
In this paper, we study "complete instability" of interval polynomials, which is the counterpart of classical robust stability. That is, the objective is to check if all polynomials in the family are unstabl...
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In this paper, we study "complete instability" of interval polynomials, which is the counterpart of classical robust stability. That is, the objective is to check if all polynomials in the family are unstable. If not, a subsequent goal is to find a stable polynomial. To this end, we first propose a randomized algorithm which is based on a (recursive) necessary condition for Hurwitz stability. The second contribution of this paper is to provide a probability-one estimate of the volume of stable polynomials. These results are based on a combination of deterministic and randomized methods. Finally, we present two numerical examples and simulations showing the efficiency of the proposed methodology for small and medium-size problems. (c) 2006 Elsevier B.V. All rights reserved.
Computation of the sign of the determinant of a matrix and the determinantitself is a challenge for both numerical and exact methods. We survey the complexity of existingmethods to solve these problems when the input ...
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Computation of the sign of the determinant of a matrix and the determinantitself is a challenge for both numerical and exact methods. We survey the complexity of existingmethods to solve these problems when the input is an n * n matrix A with integer entries. We studythe bit-complexities of the algorithms asymptotically in n and the norm of A. Existing approachesrely on numerical approximate computations, on exact computations, or on both types of arithmetic incombination.
The objective of this paper is twofold. First, the problem of generation of real random matrix samples with uniform distribution in structured (spectral) norm bounded sets is studied. This includes an analysis of the ...
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The objective of this paper is twofold. First, the problem of generation of real random matrix samples with uniform distribution in structured (spectral) norm bounded sets is studied. This includes an analysis of the distribution of the singular values of uniformly distributed real matrices, and an efficient (i.e. polynomial-time) algorithm for their generation. Second, it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty. (C) 2002 Elsevier Science Ltd. All rights reserved.
Backoff algorithms are used in many distributed systems where multiple devices contend for a shared resource. For the classic balls-into-bins problem, the number of singletons- those bins with a single ball-is importa...
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Backoff algorithms are used in many distributed systems where multiple devices contend for a shared resource. For the classic balls-into-bins problem, the number of singletons- those bins with a single ball-is important to the analysis of several backoff algorithms;however, existing analyses employ advanced probabilistic tools. Here, we show that standard Chernoff bounds can be used instead, and the simplicity of this approach is illustrated by re-analyzing some well-known backoff algorithms. (C) 2022 Elsevier B.V. All rights reserved.
We study the problem of efficiently correcting an erroneous product of two n x n matrices over a ring. Among other things, we provide a randomized algorithm for correcting a matrix product with at most k erroneous ent...
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We study the problem of efficiently correcting an erroneous product of two n x n matrices over a ring. Among other things, we provide a randomized algorithm for correcting a matrix product with at most k erroneous entries running in (O) over tilde (n(2) + kn) time and a deterministic (O) over tilde (kn(2))-time algorithm for this problem (where the notation (O) over tilde suppresses polylogarithmic terms in n and k).
We present two randomized algorithms. One solves linear programs involving m constraints in d variables in expected time O(m). The other constructs convex hulls of n points in R(d), d > 3, in expected time O(n[d/2]...
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We present two randomized algorithms. One solves linear programs involving m constraints in d variables in expected time O(m). The other constructs convex hulls of n points in R(d), d > 3, in expected time O(n[d/2]). In both bounds d is considered to be a constant. In the linear programming algorithm the dependence of the time bound on d is of the form d!. The main virtue of our results lies in the utter simplicity of the algorithms as well as their analyses.
In this paper, a new iterative approach to probabilistic robust controller design is presented, which is applicable to any robust controller/filter design problem that can be represented as an LMI feasibility problem....
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In this paper, a new iterative approach to probabilistic robust controller design is presented, which is applicable to any robust controller/filter design problem that can be represented as an LMI feasibility problem. Recently, a probabilistic Subgradient Iteration algorithm was proposed for solving LMIs. It transforms the initial feasibility problem to an equivalent convex optimization problem, which is subsequently solved by means of an iterative algorithm. While this algorithm always converges to a feasible solution in a finite number of iterations, it requires that the radius of a non-empty ball contained into the solution set is known a priori. This rather restrictive assumption is released in this paper, while retaining the convergence property. Given an initial ellipsoid that contains the solution set, the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the solution set. At each iteration a random uncertainty sample is generated with a specified probability density, which parameterizes an LMI. For this LMI the next minimum-volume ellipsoid that contains the solution set is computed. An upper bound on the maximum number of possible correction steps, that can be performed by the algorithm before finding a feasible solution, is derived. A method for finding an initial ellipsoid containing the solution set, which is necessary for initialization of the optimization, is also given. The proposed approach is illustrated on a real-life diesel actuator benchmark model with real parametric uncertainty, for which a H-2 robust state-feedback controller is designed. (C) 2003 Elsevier B.V. All rights reserved.
We consider the problem of determining whether a given function f : {0, 1}(n) --> {0, 1} belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to ...
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We consider the problem of determining whether a given function f : {0, 1}(n) --> {0, 1} belongs to a certain class of Boolean functions F or whether it is far from the class. More precisely, given query access to the function f and given a distance parameter epsilon, we would like to decide whether f is an element of F or whether it differs from every g is an element of F on more than an c-fraction of the domain elements. The classes of functions we consider are singleton ("dictatorship") functions, monomials, and monotone disjunctive normal form functions with a bounded number of terms. In all cases we provide algorithms whose query complexity is independent of n (the number of function variables), and linear in 1/epsilon.
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