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Matrix filter design using semi-infinite programming with application to DOA estimation

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2000年第1期48卷 267-271页

作者： Zhu, ZW Wang, S Leung, H Ding, Z McMaster Univ Commun Res Lab Hamilton ON L8S 4K1 Canada Nanjing Univ Sci & Technol Dept Elect Engn Nanjing Peoples R China Univ Calgary Dept Elect & Comp Engn Calgary AB T2N 1N4 Canada

In this correspondence, we propose using a semi-infinite programming technique to design a matrix filter. The idea is to formulate the design problem into a semi-infinite optimization model where the mean square error between the desired response and the designed filter in the passband and stopband is minimized subject to a set of nonlinear functional inequalities, These inequality constraints are used to ensure that the stop-band attenuation and the passband deviation satisfy the prescribed specifications. Simulations showed that the proposed method was better than the conventional matrix filter design techniques. The matrix filters based on the proposed design method was also applied to the direction-of-arrival (DOA) estimation problem. It was shown that the filter greatly improved the estimation accuracy at low signal-to-noise ratios (SNR's).

关键词： direction-of-arrival estimation matrix filter semi-infinite programming

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A dual parametrization method for convex semi-infinite programming

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ANNALS OF OPERATIONS RESEARCH 2000年第1期98卷 189-213页

作者： Ito, S Liu, Y Teo, KL Inst Stat Math Ctr Dev Stat Comp Minato Ku 4-6-7 Minami Azabu Tokyo 1068569 Japan Hong Kong Polytech Univ Dept Appl Math Kowloon Peoples R China

We formulate convex semi-infinite programming problems in a functional analytic setting and derive optimality conditions and several duality results, based on which we develop a computational framework for solving con... 详细信息

关键词： semi-infinite programming convex programming optimality condition duality converse duality quadratic semi-infinite programming linear semi-infinite programming

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Sparse regression ensembles in infinite and finite hypothesis spaces

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MACHINE LEARNING 2002年第1-3期48卷 189-218页

作者： Rätsch, G Demiriz, A Bennett, KP GMD FIRST D-12489 Berlin Germany Rensselaer Polytech Inst Dept Decis Sci & Engn Syst Troy NY 12180 USA Rensselaer Polytech Inst Dept Math Sci Troy NY 12180 USA

We examine methods for constructing regression ensembles based on a linear program (LP). The ensemble regression function consists of linear combinations of base hypotheses generated by some boosting-type base learning algorithm. Unlike the classification case, for regression the set of possible hypotheses producible by the base learning algorithm may be infinite. We explicitly tackle the issue of how to define and solve ensemble regression when the hypothesis space is infinite. Our approach is based on a semi-infinite linear program that has an infinite number of constraints and a finite number of variables. We show that the regression problem is well posed for infinite hypothesis spaces in both the primal and dual spaces. Most importantly, we prove there exists an optimal solution to the infinite hypothesis space problem consisting of a finite number of hypothesis. We propose two algorithms for solving the infinite and finite hypothesis problems. One uses a column generation simplex-type algorithm and the other adopts an exponential barrier approach. Furthermore, we give sufficient conditions for the base learning algorithm and the hypothesis set to be used for infinite regression ensembles. Computational results show that these methods are extremely promising.

关键词： ensemble learning boosting regression sparseness semi-infinite programming

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On generalized semi-infinite optimization and bilevel optimization

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2002年第3期142卷 444-462页

作者： Stein, O Still, G Univ Twente Fac Math Sci NL-7500 AE Enschede Netherlands Rhein Westfal TH Aachen Dept Math D-52056 Aachen Germany

The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL) We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions For the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： semi-infinite programming bilevel programming optimality conditions genericity behavior numerical methods

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Gap functions and existence of solutions to set-valued vector variational inequalities

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2002年第2期115卷 407-417页

作者： Yang, XQ Yao, JC Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Natl Sun Yat Sen Univ Dept Appl Math Kaohsiung 80424 Taiwan

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.

关键词： vector variational inequalities set-valued mappings gap functions existence of a solution semi-infinite programming

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Generalized semi-infinite programming: Theory and methods

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 1999年第2期119卷 301-313页

作者： Still, G Univ Twente NL-7500 AE Enschede Netherlands

Generalized semi-infinite optimization problems (GSIP) are considered. The difference between GSIP and standard semi-infinite problems (SIP) is illustrated by examples. By applying the 'Reduction Ansatz', optimality conditions for GSIP are derived. Numerical methods for solving GSIP are considered in comparison with methods for SIP. From a theoretical and a practical point of view it is investigated, under which assumptions a GSIP can be transformed into an SIP. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： semi-infinite programming optimality conditions numerical methods

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A dual parameterization approach to linear-quadratic semi-infinite programming problems

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OPTIMIZATION METHODS & SOFTWARE 1999年第3期10卷 471-495页

作者： Liu, Y Teo, KL Ito, S Curtin Univ Technol Sch Math & Stat Perth WA 6001 Australia Inst Stat Math Ctr Dev Stat Comp Minato Ku Tokyo 106 Japan

semi-infinite programming problems are special optimization problems in which a cost is to be minimized subject to infinitely many constraints. This class of problems has many real-world applications. In this paper, we consider a class of linear-quadratic semiinfinite programming problems. Using the duality theory, the dual problem is obtained, where the decision variables are measures. A new parameterization scheme is developed for approximating these measures. On this bases, an efficient algorithm for computing the solution of the dual problem is obtained. Rigorous convergence results are given to supper? the algorithm. The solution of the primal problem is easily obtained from that of the dual problem. For illustration, three numerical examples are included.

关键词： semi-infinite programming duality theory parameterization of measures optimization problem convergence result

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Stability and well-posedness in linear semi-infinite programming

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SIAM JOURNAL ON OPTIMIZATION 1999年第1期10卷 82-98页

作者： Cánovas, MJ López, MA Parra, J Todorov, MI Miguel Hernandez Univ Dept Appl Math & Stat Alicante 03202 Spain Univ Alicante Dept Stat & Operat Res Alicante 03071 Spain Bulgarian Acad Sci Inst Math Plovdiv 4002 Bulgaria

This paper presents an approach to the stability and the Hadamard well-posedness of the linear semi-infinite programming problem (LSIP). No standard hypothesis is required in relation to the set indexing of the constraints and, consequently, the functional dependence between the linear constraints and their associated indices has no special property. We consider, as parameter space, the set of all LSIP problems whose constraint systems have the same index set, and we define in it an extended metric to measure the size of the perturbations. Throughout the paper the behavior of the optimal value function and of the optimal set mapping are analyzed. Moreover, a certain type of Hadamard well-posedness, which does not require the boundedness of the optimal set, is characterized. The main results provided in the paper allow us to point out that the lower semicontinuity of the feasible set mapping entails high stability of the whole problem, mainly when this property occurs simultaneously with the boundedness of the optimal set. In this case all the stability properties hold, with the only exception being the lower semicontinuity of the optimal set mapping.

关键词： stability Hadamard well-posedness semi-infinite programming feasible set mapping optimal set mapping optimal value function

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Convex semi-infinite parametric programming: Uniform convergence of the optimal value functions of discretized problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1999年第1期101卷 191-201页

作者： Gugat, M Univ Trier Dept Math Trier Germany

The continuity of the optimal value function of a parametric convex semi-infinite program is secured by a weak regularity condition that also implies the convergence of certain discretization methods for semi-infinite problems. Since each discretization level yields a parametric program, a sequence of optimal value functions occurs. The regularity condition implies that, with increasing refinement of the discretization, this sequence converges uniformly with respect to the parameter to the optimal value function corresponding to the original semi-infinite problem. Our result is applicable to the convergence analysis of numerical algorithms based on parametric programming, for example, rational approximation and computation of the eigenvalues of the Laplacian.

关键词： semi-infinite programming parametric optimization discretization optimal value function continuity uniform convergence rational approximation defect minimization methods

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semi-infinite linear programming: A unified approach to digital filter design with time- and frequency-domain specifications

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-ANALOG AND DIGITAL SIGNAL PROCESSING 1999年第6期46卷 765-775页

作者： Nordebo, S Zang, ZQ Univ Karlskrona Ronneby Dept Signal Proc S-37225 Ronneby Sweden Curtin Univ Technol Australian Telecommun Res Inst Perth WA 6845 Australia

Using the recently developed semi-infinite linear programming techniques and Caratheodory's dimensionality theory, we present a unified approach to digital filter design with time and/or frequency-domain specifications. Through systematic analysis and detailed numerical design examples, we demonstrate that the proposed approach exhibits several salient features compared to traditional methods: 1) using the unified approach, complex responses can be handled conveniently without resorting to discretization;2) time-domain constraints can be included easily;and 3) any filter structure, recursive or nonrecursive, can be employed, provided that the frequency response can be represented by a finite-complex basis. More importantly, the solution procedure is based on the numerically efficient simplex extension algorithms. As numerical examples, a discrete-time Laguerre network is used in a frequency-domain design with additional group-delay specifications, and in a H-infinity-optimal envelope constrained filter design problem. Finally, a finite impulse response phase equalizer is designed with additional frequency domain H-infinity, robustness constraints.

关键词： digital filter design Laguerre filters linear programming semi-infinite programming

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