The paper studies a distributed constrained optimization problem, where multiple agents connected in a network collectively minimize the sum of individual objective functions subject to a global constraint being an intersection of the local constraint sets assigned to the agents. Based on the augmented Lagrange method, a distributed primal-dual algorithm with a projection operation included is proposed to solve the problem. It is shown that with appropriately chosen constant step size, the local estimates derived at all agents asymptotically reach a consensus at an optimal solution. In addition, the value of the cost function at the time-averaged estimate converges with rate O( 1 k ) to the optimal value for the unconstrained problem. By these properties the proposed primal-dual algorithm is distinguished from the existing algorithms for distributed constrained optimization. The theoretical analysis is justified by numerical simulations.

The electrochemical cyclic voltammetric (CV) oxidation behavior of an arc-derived singlewall carbon nanotube (SWNT) sample in potassium hydroxide solution was investigated. Amorphous carbon in the as-grown SWNT sample was effectively removed by the CV oxidation, as confirmed by analyzing the sp 3 /sp 2 carbon ratio from C1s XPS spectra and HRTEM observations. The removal of the amorphous carbon led to the exposure of metal nanoparticles, hence facilitating the elimination of the metal impurities by subsequent HCl washing. The CV oxidation can be applied as an alternative oxidative treatment for the purification of SWNT samples. Redox peaks were observed during the CV oxidation. The reduction peaks in the range of -0.96 to -1.0 V and the oxidation peaks in the range of -0.61 to -0.49 V were attributed to the electrochemical redox transformations between metallic Fe and Fe(II) oxide, Fe(II) oxide and Fe(III) oxide, as well as Ni and Ni(II) oxide, and the observed reduction peaks in the potential range of 0.29-0.13 V were believed to be caused by the electrochemical reduction of NiOOH into Ni(OH) 2 . The intensity of all of the redox peaks was dependent on the cycle number because more and more metal nanoparticles could be exposed as a result of the incremental removal or damage of the amorphous carbon coating during the CV oxidation, while the intensity remained almost unchanged after 80 cycles because of the completion of the amorphous carbon removal. Therefore, the redox peaks from the electrochemical redox reactions of Fe and Ni impurities can be considered as a benchmark for the removal extent of the amorphous carbon, and optimal electrochemical oxidation time for the purification of the as-grown SWNT sample can be determined in real time during the CV oxidation treatment. This is a predominant advantage of the CV oxidation over common oxidation methods using air or other oxidizing reagents for SWNT purification.

We consider the expected residual minimization formulation of the stochastic R 0 matrix linear complementarity problem. We show that the involved matrix being a stochastic R 0 matrix is a necessary and sufficient condition for the solution set of the expected residual minimization problem to be nonempty and bounded. Moreover, local and global error bounds are given for the stochastic R 0 matrix linear complementarity problem. A stochastic approximation method with acceleration by averaging is applied to solve the expected residual minimization problem. Numerical examples and applications of traffic equilibrium and system control are given.