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Approximation related to quotient functionals

JOURNAL OF APPROXIMATION THEORY 2010年第3期162卷 545-558页

作者： Setzer, S. Steidl, G. Teuber, T. Moerkotte, G. Univ Mannheim Dept Math & Comp Sci D-6800 Mannheim 1 Germany

We examine the best approximation of componentwise positive vectors or positive continuous functions f by linear combinations (f) over cap = Sigma(j)alpha(j)phi(j) of given vectors or functions phi(j) with respect to atnctionals Q(p), 1 <= p <= infinity, involving quotients max {f/(f) over cap.(f) over cap /f) rather than differences vertical bar f - (f) over cap vertical bar. We verify the existence of a best approximating function under mild conditions on {phi(j)}(j=1)(n). For discrete data, we compute a best approximating function with respect to Q(p), p = 1, 2, infinity by second order cone programming. Special attention is paid to the Q(infinity), functional in both the discrete and the continuous setting. Based on the computation of the subdifferential of our convex functional Q(infinity) we give an equivalent characterization of the hest approximation by using its extremal set. Then we apply this characterization to prove the uniqueness of the best Q(infinity) approximation for Chebyshev sets {phi(j)}(j=1)(n). (C) 2009 Elsevier Inc. All rights reserved.

关键词： Best approximation Polynomial approximation Chebyshev sets Convex optimization second order cone programming

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L₂-norm multiple kernel learning and its application to biomedical data fusion

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BMC BIOINFORMATICS 2010年第1期11卷 309-309页

作者： Yu, Shi Falck, Tillmann Daemen, Anneleen Tranchevent, Leon-Charles Suykens, Johan A. K. De Moor, Bart Moreau, Yves Katholieke Univ Leuven Bioinformat Grp Dept Elect Engn B-3001 Heverlee Belgium Katholieke Univ Leuven Syst Models & Control Grp Dept Elect Engn B-3001 Heverlee Belgium

Background: This paper introduces the notion of optimizing different norms in the dual problem of support vector machines with multiple kernels. The selection of norms yields different extensions of multiple kernel learning (MKL) such as L-infinity, L-1, and L-2 MKL. In particular, L-2 MKL is a novel method that leads to non-sparse optimal kernel coefficients, which is different from the sparse kernel coefficients optimized by the existing L-infinity MKL method. In real biomedical applications, L-2 MKL may have more advantages over sparse integration method for thoroughly combining complementary information in heterogeneous data sources. Results: We provide a theoretical analysis of the relationship between the L-2 optimization of kernels in the dual problem with the L-2 coefficient regularization in the primal problem. Understanding the dual L-2 problem grants a unified view on MKL and enables us to extend the L-2 method to a wide range of machine learning problems. We implement L-2 MKL for ranking and classification problems and compare its performance with the sparse L-infinity and the averaging L-1 MKL methods. The experiments are carried out on six real biomedical data sets and two large scale UCI data sets. L-2 MKL yields better performance on most of the benchmark data sets. In particular, we propose a novel L-2 MKL least squares support vector machine (LSSVM) algorithm, which is shown to be an efficient and promising classifier for large scale data sets processing. Conclusions: This paper extends the statistical framework of genomic data fusion based on MKL. Allowing non-sparse weights on the data sources is an attractive option in settings where we believe most data sources to be relevant to the problem at hand and want to avoid a "winner-takes-all" effect seen in L-infinity MKL, which can be detrimental to the performance in prospective studies. The notion of optimizing L-2 kernels can be straightforwardly extended to ranking, classification, regression, a

关键词： Little Square Support Vector Machine Multiple Kernel Multiple Kernel Learning second order cone programming Gene Prioritization

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Adjustable robust counterpart of conic quadratic problems

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2008年第2期68卷 211-233页

作者： Boni, Odellia Ben-Tal, Aharon Technion Israel Inst Technol Fac Ind Engn & Management MINERVA Optimizat Ctr IL-32000 Technion Haifa Israel

This paper presents an approximate affinely adjustable robust counterpart for conic quadratic constraints. The theory is applied to obtain robust solutions to the problems of subway route design with implementation errors and a supply chain management with uncertain demands. Comparison of the adjustable solutions with the nominal and non-adjustable robust solutions shows that the adjustable (dynamic) robust solution maintains feasibility for all possible realizations, while being less conservative than the usual (static) robust counterpart solution.

关键词： robust optimization adjustable variables conic quadratic programming second order cone programming flexible commitments contracts

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Optimal oblivious routing under linear and ellipsoidal uncertainty

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OPTIMIZATION AND ENGINEERING 2008年第3期9卷 257-271页

作者： Belotti, Pietro Pinar, Mustafa C. Carnegie Mellon Univ Tepper Sch Business Pittsburgh PA 15213 USA Bilkent Univ Dept Ind Engn Ankara Turkey

In telecommunication networks, a common measure is the maximum congestion (i.e., utilization) on edge capacity. As traffic demands are often known with a degree of uncertainty, network management techniques must take into account traffic variability. The oblivious performance of a routing is a measure of how congested the network may get, in the worst case, for one of a set of possible traffic demands. We present two models to compute, in polynomial time, the optimal oblivious routing: a linear model to deal with demands bounded by box constraints, and a second-order conic program to deal with ellipsoidal uncertainty, i.e., when a mean-variance description of the traffic demand is given. A comparison between the optimal oblivious routing and the well-known OSPF routing technique on a set of real-world networks shows that, for different levels of uncertainty, optimal oblivious routing has a substantially better performance than OSPF routing.

关键词： traffic engineering oblivious routing linear programming second order cone programming

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Stochastic 0-1 linear programming under limited distributional information

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OPERATIONS RESEARCH LETTERS 2008年第2期36卷 150-156页

作者： Wagner, Michael R. Calif State Univ Hayward Dept Management Hayward CA 94542 USA

We consider the problem min(X epsilon(0, 1)n) {c'x : a'(j)x <= b(j), j = 1,..., m), where the a(j) are random vectors with unknown distributions. The only information we are given regarding the random vectors aj are their moments, up to order k. We give a robust formulation, as a function of k, for the 0-1 integer linear program under this limited distributional information. (C) 2007 Elsevier B.V. All rights reserved.

关键词： stochastic integer programming second order cone programming semidefinite programming

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Robust classification of noisy data using second order cone programming approach

Robust classification of noisy data using second order cone ...

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International Conference on Intelligent Sensing and Information Processing

作者： Bhattacharyya, C Indian Inst Sci Dept Comp Sci & Automat Bangalore 560012 Karnataka India

ISBN: (纸本)0780382439

Assuming an ellipsoidal model of uncertainty a robust formulation for classifying noisy data is presented. The formulation is a convex optimization problem, in particular it is a instance of second order cone Programmimg problem. The formulation is derived from a worst case consideration and the robustness properties hold for a large class of distributions. The equivalence of ellipsoidal uncertainty and Gaussian noise models is also discussed. The Generalized Optimal hyperplane is recovered as a special case of the robust formulation. Experiments on real world datasets illustrates the efficacy of the formulation.

关键词： convex programming pattern classification Gaussian noise support vector machines robust classification second order cone programming ellipsoidal uncertainty model convex optimization problem Gaussian noise models generalized optimal hyperplane robust formulation robustness Support Vector Network Gaussian noise convex programming noisy data Pattern recognition Noise models Robustness Convex function

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Transmit/receive beamformer design and power control in MIMO MC-CDMA systems

Transmit/receive beamformer design and power control in MIMO...

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IEEE International Symposium on Circuits and Systems

作者： Chan, S. C. Zhao, S. H. Univ Hong Kong Dept Elect & Elect Engn Pokfulam Rd Hong Kong Peoples R China

ISBN: (纸本)9780780393899

In this paper, a joint transmitter and receiver beamformers design algorithm for downlink multiple input multiple output multicarrier code-division multiple access (MIMO MC-CDMA) system is proposed. The algorithm is iterative in nature where the transmitter beamformers and the receiver beamformers are determined alternately. The transmitter beamforming problem with a given receiver beamformer is formulated as a convex programming problem, which can be solved optimally using second order cone programming (SOCP), while the receiver beamforming problem is formulated as a constrained optimization problem with an analytical solution. The convergence of the algorithm is analyzed and the performance of the proposed algorithm is evaluated by computing simulation.

关键词： MIMO systems code division multiple access power control transceivers MIMO convex programming problem iterative algorithm multicarrier code-division multiple access receiver beamformers second order cone programming transmit/receive beamfor

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Efficient algorithms for the smallest enclosing ball problem

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2005年第2期30卷 147-160页

作者： Zhou, G Toh, KC Sun, B Curtin Univ Technol Dept Math & Stat Bentley WA 6102 Australia Natl Univ Singapore Dept Math Singapore 117543 Singapore Natl Univ Singapore Dept Decis Sci Singapore 117548 Singapore

Consider the problem of computing the smallest enclosing ball of a set of m balls in R-n. Existing algorithms are known to be inefficient when n > 30. In this paper we develop two algorithms that are particularly suitable for problems where n is large. The first algorithm is based on log-exponential aggregation of the maximum function and reduces the problem into an unconstrained convex program. The second algorithm is based on a second-order cone programming formulation, with special structures taken into consideration. Our computational experiments show that both methods are efficient for large problems, with the product mn on the order of 10(7). Using the first algorithm, we are able to solve problems with n = 100 and m = 512, 000 in about 1 hour.

关键词： computational geometry smoothing approximation second order cone programming

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Avoiding numerical cancellation in the interior point method for solving semidefinite programs

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MATHEMATICAL programming 2003年第2期95卷 219-247页

作者： Sturm, JF Tilburg Univ Dept Econometr NL-5000 LE Tilburg Netherlands

The matrix variables in a primal-dual pair of semidefinite programs are getting increasingly ill-conditioned as they approach a complementary solution. Multiplying the primal matrix variable with a vector from the eigenspace of the non-basic part will therefore. result in heavy numerical cancellation. This effect is amplified by the scaling operation in interior point methods. A complete example illustrates these numerical issues. In order to avoid numerical problems in interior point methods, we propose to maintain the matrix variables in a Cholesky form. We discuss how the factors of the v-space Cholesky form can be updated after a main iteration of the interior point method with Nesterov-Todd scaling. An analogue for second order cone programming is also developed. Numerical results demonstrate the success of this approach.

关键词： semidefinite programming second order cone programming

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An independent benchmarking of SDP and SOCP solvers

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MATHEMATICAL programming 2003年第2期95卷 407-430页

作者： Mittelmann, HD Arizona State Univ Dept Math & Stat Tempe AZ 85287 USA

This work reports the results of evaluating all computer codes submitted to the Seventh DIMACS Implementation Challenge on Semidefinite and Related Optimization Problems. The codes were run on a standard platform and on all the benchmark problems provided by the organizers of the challenge. A total of ten codes were tested on fifty problems in twelve categories. For each code the most important information is summarized. Together with the tabulated and commented benchmarking results this provides an overview of the state of the art in this field.

关键词： semidefinite programming second order cone programming optimization software performance evaluation

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