Nonlinear Perturbation of Linear Programs

Mangasarian, O. L.; Meyer, Robert R.

doi:10.1137/0317052

Cited by 150 publications

(98 citation statements)

References 2 publications

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“…To obtain a unique set of commodities with minimum lifetime in this step and the subsequent steps, we use the regularization method. Details of the regularization technique can be found in [16]. The regularization term, δ k∈C q k 2 , is the Euclidean norm of the commodity lifetime vector, where δ is the regularization coefficient.…”

Section: B First Stepmentioning

confidence: 99%

Multicommodity Lifetime Routing for Wireless Sensor Networks with Multiple Sinks

Shah-Mansouri

Rad

Wong

2008

2008 IEEE International Conference on Communications

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Abstract-Wireless sensor networks (WSNs) have recently received increasing attention from research and development communities. In a WSN, the field information (e.g., temperature, humidity, airflow) is acquired via several battery-equipped wireless devices and is relayed towards a sink node. As the size of the WSNs increases, it becomes inefficient to gather all information in one sink. To tackle this problem, the number of sinks can be increased. The data information flow towards each of the sinks is called a commodity. In this paper, we formulate a lexicographically optimal commodity lifetime (LOCL) routing problem. A stepwise algorithm is proposed to obtain the optimal routing solution which can lead to lexicographical fairness among commodity lifetimes. Simulation results show that our proposed algorithm increases the normalized commodity lifetime compared to MLMS [1] and LMM [2] routing algorithms.

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Section: B First Stepmentioning

confidence: 99%

Multicommodity Lifetime Routing for Wireless Sensor Networks with Multiple Sinks

Shah-Mansouri

Rad

Wong

2008

2008 IEEE International Conference on Communications

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“…For this problem ten databases were used from the Irvine repository and the Star/Galaxy database. Table 1 gives the percent of correctly separated points as well as CPU times using an average of ten SLA runs on the smooth misclassi cation minimization problem (20). These quantities are compared with those of a parametric minimization method (PMM) applied to an LPEC associated with the misclassi cation minimization 18, 1].…”

Section: Numerical Testsmentioning

confidence: 99%

“…By replacing the variables (w; ) by the nonnegative variables (w 1 ; 1 ; 1 ) using the standard transformation w = w 1 e 1 ; = 1 1 ; the smooth problems (20) and (21) can be transformed to the following concave minimization problem: min x ff(x) j Ax < = b; x > = 0g; (22) where f: R`! R, is a di erentiable, concave function bounded below on the nonempty polyhedral feasible region of (22), A 2 R p `a nd b 2 R p : By 27, Corollary 32.3.4] it follows that f attains its minimum at a vertex of the feasible region of (22).…”

Section: Successive Linearization Of Polyhedral Concave Programsmentioning

confidence: 99%

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Machine Learning via Polyhedral Concave Minimization

Mangasarian¹

1996

Applied Mathematics and Parallel Computing

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Two fundamental problems of machine learning, misclassi cation minimization 10, 24, 18] and feature selection, 25,29,14] are formulated as the minimization of a concave function on a polyhedral set. Other formulations of these problems utilize linear programs with equilibrium constraints 18, 1, 4, 3] which are generally intractable. In contrast, for the proposed concave minimization formulation, a successive linearization algorithm without stepsize terminates after a maximum average of 7 linear programs on problems with as many as 4192 points in 14-dimensional space. The algorithm terminates at a stationary point or a global solution to the problem. Preliminary numerical results indicate that the proposed approach is quite e ective and more e cient than other approaches.

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“…We now allow ε to be a a nonnegative variable in the optimization problem above that will be driven to some positive tolerance error determined by the size of the parameter µ. By making use of linear programming perturbation theory (Mangasarian & Meyer, 1979) we parametrically maximize ε in the objective function with a positive parameter µ to obtain our basic linear programming formulation of the nonlinear support vector regression (SVR) problem:…”

Section: The Support Vector Regression Problemmentioning

confidence: 99%

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Mangasarian

Musicant

2002

Machine Learning

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Abstract. The problem of tolerant data fitting by a nonlinear surface, induced by a kernel-based support vector machine is formulated as a linear program with fewer number of variables than that of other linear programming formulations. A generalization of the linear programming chunking algorithm for arbitrary kernels is implemented for solving problems with very large datasets wherein chunking is performed on both data points and problem variables. The proposed approach tolerates a small error, which is adjusted parametrically, while fitting the given data. This leads to improved fitting of noisy data (over ordinary least error solutions) as demonstrated computationally. Comparative numerical results indicate an average time reduction as high as 26.0% over other formulations, with a maximal time reduction of 79.7%. Additionally, linear programs with as many as 16,000 data points and more than a billion nonzero matrix elements are solved.

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Nonlinear Perturbation of Linear Programs

Cited by 150 publications

References 2 publications

Multicommodity Lifetime Routing for Wireless Sensor Networks with Multiple Sinks

Multicommodity Lifetime Routing for Wireless Sensor Networks with Multiple Sinks

Machine Learning via Polyhedral Concave Minimization

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