Sums of squares based approximation algorithms for MAX-SAT

We investigate the Semidefinite Programming based sums of squares (SOS) decomposition method, designed for global optimization of polynomials, in the context of the (Maximum) Satisfiability problem. To be specific, we examine the potential of this theory for providing tests for unsatisfiability and providing MAX-SAT upper bounds. We compare the SOS approach with existing upper bound and rounding techniques for the MAX-2-SAT case of Goemans and Williamson [Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming, J. Assoc. Comput. Mach. 42(6) (1995) 1115-1145] and Feige and Goemans [Approximating the value of two prover proof systems, with applications to MAX2SAT and MAXDICUT, in: Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182-189] and the MAX-3-SAT case of Karloff and Zwick [A 7/8-approximation algorithm for MAX 3SAT? in: Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, Miami Beach, FL, USA, IEEE Press, New York, 1997], which are based on Semidefinite Programming as well. We prove that for each of these algorithms there is an SOS-based counterpart which provides upper bounds at least as tight, but observably tighter in particular cases. Also, we propose a new randomized rounding technique based on the optimal solution of the SOS Semidefinite Program (SDP) which we experimentally compare with the appropriate existing rounding techniques. Further we investigate the implications to the decision variant SAT and compare experimental results with those yielded from the higher lifting approach of Anjos [On semidefinite programming relaxations for the satisfiability problem, Math. Methods Oper. Res. 60(3) (2004) 349-367; An improved semidefinite programming relaxation for the satisfiability problem, Math. Programming 102(3) (2005) 589-608; Semidefinite optimization approaches for satisfiability and maximum-satisfiability problems, J. Satisfiability Boolean Modeling Comput. 1 (2005) 1-47].We give some impression of the fraction of the so-called unit constraints in the various SDP relaxations. From a mathematical viewpoint these constraints should be easily dealt within an algorithmic setting, but seem hard to be avoided as extra constraints in an SDP setting. Finally, we briefly indicate whether this work could have implications in finding counterexamples to uncovered cases in Hilbert's Positivstellensatz.     

Dealing with the multiplicity of solutions of the ℓ1 and ℓ∞ regression models

It is well known that the least absolute value (?_∞) and the least sum of absolute deviations (?₁) algorithms produce estimators that are not necessarily unique. In this paper it is shown how the set of all solutions of the ?₁ and ?_∞ regression problems for moderately large sample sizes can be obtained. In addition, if the multiplicity of solutions wants to be avoided, two new methods giving the same optimal ?₁ and ?_∞ values, but supplying unique solutions, are proposed. The idea consists of using two steps: in the first step the optimal values of the ?₁ and ?_∞ errors are calculated, and in the second step, in case of non-uniqueness of solutions, one of the multiple solutions is selected according to a different criterion. For the ?_∞ the procedure is used sequentially but removing, in each iteration, the data points with maximum absolute residual and adding the corresponding constraints for keeping these residuals, and this process is repeated until no change in the solution is obtained. In this way not only the maximum absolute residual values are minimized in the modified method, but also the maximum absolute residual values of the remaining points sequentially, until no further improvement is possible. In the ?₁ case a least squares criterion is used but restricted to the ?₁ residual condition. Thus, in the modified ?₁ method not only the ?₁ residual is minimized, but also the sum of squared residuals subject to the ?₁ residual. The methods are illustrated by their application to some well known examples and their performances are tested by some simulations, which show that the lack of uniqueness problem cannot be corrected for some experimental designs by increasing the sample size.     

Noise and nonlinear estimation with optimal schemes in DTI

In general, the estimation of the diffusion properties for diffusion tensor experiments (DTI) is accomplished via least squares estimation (LSE). The technique requires applying the logarithm to the measurements, which causes bad propagation of errors. Moreover, the way noise is injected to the equations invalidates the least squares estimate as the best linear unbiased estimate. Nonlinear estimation (NE), despite its longer computation time, does not possess any of these problems. However, all of the conditions and optimization methods developed in the past are based on the coefficient matrix obtained in a LSE setup. In this article, NE for DTI is analyzed to demonstrate that any result obtained relatively easily in a linear algebra setup about the coefficient matrix can be applied to the more complicated NE framework. The data, obtained using non-optimal and optimized diffusion gradient schemes, are processed with NE. In comparison with LSE, the results show significant improvements, especially for the optimization criterion. However, NE does not resolve the existing conflicts and ambiguities displayed with LSE methods.     

Near-infrared spectra combined with partial least squares for pH determination of toothpaste of different brands

Near-infrared spectroscopy(NIR),which is generally used for online monitoring of the food analysis and production process, was applied to determine the internal quality of toothpaste samples.It is acknowledged that the spectra can be significantly influenced by non-linearities introduced by light scatter,therefore,four data preprocessing methods,including off-set correction, 1st-derivative,standard normal variate(SNV) and multiplicative scatter correction(MSC),were employed before the date analysis. The multivariate calibration model of partial least squares(PLS) was established and then was used to predict the pH values of the toothpaste samples of different brand.The results showed that the spectral date processed by MSC was the best one for predicting the pH value of the toothpaste samples.     

Efficient robust estimation of time-series regression models

The paper studies a new class of robust regression estimators based on the two-step least weighted squares (2S-LWS) estimator which employs data-adaptive weights determined from the empirical distribution or quantile functions of regression residuals obtained from an initial robust fit. Just like many existing two-step robust methods, the proposed 2S-LWS estimator preserves robust properties of the initial robust estimate. However, contrary to the existing methods, the first-order asymptotic behavior of 2S-LWS is fully independent of the initial estimate under mild conditions. We propose data-adaptive weighting schemes that perform well both in the cross-section and time-series data and prove the asymptotic normality and efficiency of the resulting procedure. A simulation study documents these theoretical properties in finite samples.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

Weighted Banzhaf power and interaction indexes through weighted approximations of games

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.     

Evaluation of important fatty acid ratios in poultry feed lipids by ATR FTIR spectroscopy

A rapid, simple and reproducible single bounce attenuated total reflectance (SB-ATR) Fourier transform infrared (FTIR) spectroscopic method was developed for determination of the n − 6, n − 3 and ratio of n − 6:n − 3 polyunsaturated fatty acids (PUFAs) in poultry feed lipids using partial least squares (PLS) regression. Data for n − 6, n − 3 and ratio of n − 6:n − 3 was acquired by gas chromatography (GC) and used as a standard values for FTIR calibration. The best regression results were achieved using first derivatives of the 1475–650 cm⁻¹ spectral region for n − 6, n − 3 and ratio of n − 6:n − 3 with high regression coefficients (R²) of 0.999, 0.994 and 0.998, respectively and low RMSEP values of 1, 0.06 and 0.83, respectively. The results of the present study revealed that FTIR could be used for rapid and accurate determination of n − 6, n − 3 and ratio of n − 6:n − 3 PUFAs present in poultry feed lipids.     

Noise analysis and suppression method in attitude determination using the global positioning system (GPS)

A noise suppression method is developed for attitude determination using the global positioning system. The influence of noise on attitude determination application is analyzed to determine the relationship of errors. In order to suppress the noise, the total least squares method is utilized for double difference carrier phase measurements and unit vectors between satellites and antennas. Experimental results indicate that compared to the traditional least squares method without noise suppression, the accuracy of the measurement of attitude angles is increased by about 30-50%. The increased computation time of this method does not significantly influence the real time performance for land vehicle application.