Nonnegative ill-conditioned linear systems and GMRES method

In this paper we consider solving ill-conditioned linear systems under nonnegativity constraints with noisy right hand sides. The classical approaches to solve such systems are constrained least square (quadratic programming) and barrier methods. First we present these classical methods. Then a modified version of the GMRES method (NGMRES) is presented. Since we assume that the coefficient matrices are ill-conditioned, then the Tikhonov regularization of the problem is considered. Our computational experiments show that the NGMRES provides us high quality solutions much faster than the other two approaches.     

Convex optimization approach to a single quadratically constrained quadratic minimization problem

In this paper, first we show that for rank deficient matrices, the optimal solution of a single equality constrained quadratic minimization problem can be found by relaxing the equality constraint to the inequality one, which makes the problem a convex problem. Then we show that for full rank matrices, an optimal solution can be obtained using semidefinite optimization framework.     

Robust least squares solution of linear inequalities

Least squares solution of linear inequalities appears in many disciplines such as linear separability problems and inconsistency correction. In this paper we consider this problem with uncertainty in its data. Then we prove that its robust counterpart is equivalent to a second order conic linear optimization problem, which can be efficiently solved using interior point methods.     

Comparative of the removal of Pb2+, Cd2+ and Ni2+ by nano crystallite hydroxyapatite from aqueous solutions: Adsorption isotherm study

Release of heavy metal onto the water and soil as a result of agricultural and industrial activities may pose a serious threat to the environment. In this study, the adsorption behavior of nano hydroxyapatite with respect to Pb²⁺, Cd²⁺ and Ni²⁺ has been studied in order to consider its application to purity metal finishing wastewater. The batch method has been employed, using metal concentrations in solution ranging from 100 to 400 mg/L. The uptake capacity and distribution coefficients (K_d) were determined for the adsorption system as a function of sorbate concentration. The Langmuir, Freundlich, and Dubinin–Kaganer–Radushkevich (DKR) isotherms applied for sorption studies showed that the amount of metal sorbed on nano hydroxyapatite. It was found that the adsorption phenomena depend on charge density and hydrated ion diameter. According to the equilibrium studies, the selectivity sequence can be given as Pb²⁺ > Cd²⁺ > Ni²⁺. These results show that nano hydroxyapatite holds great potential to remove cationic heavy metal species from industrial wastewater.     

Removal of divalent nickel cations from aqueous solution by multi-walled carbon nano tubes: equilibrium and kinetic processes

Release of heavy metals into water as a result of industrial activity may pose a serious threat to the environment. In this study, the potential of multi-walled carbon nano tubes (MWCNT) to remove Ni²⁺ cations from aqueous solutions was investigated in a batch reactor under different experimental conditions. The effects on the removal process of conditions such as initial concentration of Ni²⁺ ions, temperature, and adsorbent mass were investigated. Nickel uptake was quantitatively evaluated by use of the Langmuir, Freundlich, and Dubinin?CKaganer?CRadushkevich isotherm models. For 20?mg/L initial Ni²⁺ cation concentration, adsorption capacity increased from 8.12 to 11.75?mg/g when the temperature was increased from 25 to 65?°C, an indication of the endothermic nature of adsorption process. In addition, the adsorption equilibrium was well described by the Langmuir isotherm model; maximum adsorption capacity was 17.86?mg/g Ni²⁺ cations on HNO₃-treated MWCNT (t-MWCNT). The results obtained in this study show that adsorption of Ni²⁺ on t-MWCNT is a spontaneous and endothermic process. By use of second-order kinetic constants and the Arrhenius equation, the activation energy of adsorption (E _a) was determined as 5.56?kJ?mol^?1.     

A finite termination Mehrotra-type predictor–corrector algorithm

Mehrotra-type predictor–corrector algorithms are the backbone of most Interior Point Methods based software packages. Salahi et al. [M. Salahi, J. Peng, T. Terlaky, On Mehrotra-type predictor–corrector algorithms, Technical Report 2005/4, Advanced Optimization Lab., McMaster University, Hamilton, ON, Canada. http://www.cas.mcmaster.ca/~oplab/publication] in their recent works have shown some ill behaviors of Mehrotra’s original algorithm which motivated them to modify it in order to achieve the polynomial iteration complexity while preserving its practical efficiency. In this paper we analyze the same algorithm from a different perspective and give a condition on the maximum feasible step size in the predictor step, violation of which might lead to a very small or even zero step size in the corrector step. If the maximum step size in the predictor step is above a certain threshold, then we cut it to satisfy the derived condition. This enables us to prove that the algorithm terminates finitely.     

Postponing the choice of the barrier parameter in Mehrotra-type predictor–corrector algorithms

Recently Salahi et al. have considered a variant of Mehrotra’s celebrated predictor–corrector algorithm. By a numerical example they showed that this variant might make very small steps in order to keep the iterate in a certain neighborhood of the central path, that itself implies the inefficiency of the algorithm. This observation motivated them to incorporate a safeguard in their algorithmic scheme that gives a lower bound for the step size at each iteration and thus imply polynomial iteration complexity. In this paper we propose a different approach that enables us to have control on the iterates.